Bending Strain Measures Research Articles

Mid-Surface Scaling Invariance of Some Bending Strain Measures

The mid-surface scaling invariance of bending strain measures proposed in (Int. J. Solids Struct. 37(39):5517–5528, 2000) is discussed in light of the work of (J. Elast. 146(1):83–141, 2021).

High-Precision large deformation measurement of array SAR based on FBG strain monitoring and initial state reconstruction

Array Synthetic Aperture Radar (SAR) as the novel generation airborne remote sensing system is installed below the wing, has received more attention because it can realize all-weather three-dimensional (3-D) imaging. Its high-precision imaging needs to measure large deformation of wing accurately. The exiting methods based on Fiber Bragg grating (FBG) sensors ignore the effects of wing initial state, which will generate non-negligible non-linear measurement error. To solve this problem, a high-precision large deformation measurement method of array SAR based on FBG strain monitoring and initial state reconstruction is proposed. Firstly, initial angle reconstruction approach is presented to obtain initial state. Then, bending strain measurement method is used to obtain continuous strain. Finally, large deformation model based on initial angle and bending strain is established to achieve large deformation. Experimental results indicate proposed method can effectively measure large deformation, and can acquire the higher displacement accuracy compared to traditional method.

Analysis of Flexible Screen Bending Strain Measurement Based on Digital Image Correlation (Invited)

Analysis of Flexible Screen Bending Strain Measurement Based on Digital Image Correlation (Invited)

Effect of Suspension Clamp Types on the Dynamic Bending Stress of All Aluminium Alloy 1120 Overhead Conductors

Determination of the bending stress is one of the important factors for the design or fatigue assessment of transmission power lines. Researchers and designers calculate the bending stress of overhead conductors by using the Poffenberger–Swart (P-S) formula, which was established for the conductor/metal suspension clamp system. This publication presents experimental work on the effect of different configurations of suspension clamps on the dynamic bending stress that causes fatigue in the conductor. For each type of clamp configuration, specific points of the conductors were instrumented with strain gauges for the measurement of bending strain. The prediction of bending stress by the P-S formula presented errors of up to 140% for clamp different from the one established during the formulation. A better correlation with experimental data was obtained by adjusting considering an effective length of the lever arm for elastomeric clamps.

Parametric reflection of the quasi-distributed optical fiber sensors with flexible packaging layer for bending strain measurement

The health condition assessment of long-span structures (i.e., pipes) has always been important for tracking the service safety state. Effectively monitoring the large-scale deformation of the pipe structures can be adopted to diagnose the structural performance and identify the possible damage. Considering the local interfacial debonding between the distributed optical fiber and the pipe induced testing failure, improved packaging design is expected to enhance the feasibility and durability of the sensor. Therefore, a flexible silicone elastomer packaged quasi-distributed optical fiber sensor with cross section in semi-circle shape is proposed and tested on the pipe structures. Comparison analysis on the bending strain measurement with bare fiber Bragg grating (FBG) and packaged FBGs in series has been conducted. Parametric reflection method is theoretically proposed for the packaged sensor subjected to bending action. Simulation analysis based on software ANSYS has also been performed to validate the effectiveness of the proposed sensor and the corrected calibration coefficient deduced from the theoretical analysis. Results show that the flexible packaged optical fiber sensor has stable performance for measuring the bending induced compressive or tensile deformation of the pipe. The proposed parametric reflection method can accurately calibrate the quasi-distributed and distributed optical fiber sensors. The study can promote the precise and feasible application of distributed optical fiber sensors in bending strain measurement.

A new approach to curvature measures in linear shell theories

The paper presents a coordinate-free analysis of deformation measures for shells modeled as 2D surfaces. These measures are represented by second-order tensors. As is well-known, two types are needed in general: the surface strain measure (deformations in tangential directions), and the bending strain measure (warping). Our approach first determines the 3D strain tensor E of a shear deformation of a 3D shell-like body and then linearizes E in two smallness parameters: the displacement and the distance of a point from the middle surface. The linearized expression is an affine function of the signed distance from the middle surface: the absolute term is the surface strain measure and the coefficient of the linear term is the bending strain measure. The main result of the paper determines these two tensors explicitly for general shear deformations and for the subcase of Kirchhoff-Love deformations. The derived surface strain measures are the classical ones: Naghdi’s surface strain measure generally and its well-known particular case for the Kirchhoff-Love deformations. With the bending strain measures comes a surprise: they are different from the traditional ones. For shear deformations our analysis provides a new tensor , which is different from the widely used Naghdi’s bending strain tensor . In the particular case of Kirchhoff–Love deformations, the tensor reduces to a tensor introduced earlier by Anicic and Léger (Formulation bidimensionnelle exacte du modéle de coque 3D de Kirchhoff–Love. C R Acad Sci Paris I 1999; 329: 741–746). Again, is different from Koiter’s bending strain tensor (frequently used in this context). AMS 2010 classification: 74B99

Development the method of pipeline bending strain measurement based on microelectromechanical systems inertial measurement unit.

With the development of pipeline construction, the additional stress and strain becomes the key factor to induce the damage for oil and gas pipeline. The in-line inspection of pipeline bending strain which is based on high-end tactical-grade inertial measurement unit has become routine practice for the oil and gas pipelines over recent years. However, these accurate inertial measurement units are large size and high cost limit to use in small diameter pipelines of bending strain inspection. Microelectromechanical systems-based inertial navigation has been applied to mapping the centerline of the small size pipeline, and the accurate trajectory and attitude information become key factors to calculate the bending strain of pipelines. This article proposed a method not only to calculate the pipeline bending strain but also to improve the accuracy for the bending strain based on the wavelet analysis. Tests show that this method can be effectively used in the calculation and optimization of the bending strain, and it will increase the accuracy to within 19.1% of the actual bending strain.

A novel algorithm for pipeline displacement and bending strain of in-line inspection based on inertia measurement technology

The in-line inspection tool with Inertial Measurement Unit tool is becoming a routine and important practice for many pipeline companies and is effective for whole-line bending strain measurement. However, the measurements of Inertial Measurement Unit tool are always affected by noises and errors, which are caused by inherent inaccuracies and deficiencies of the experimental techniques and measuring devices. For the calculations of the bending strain, the results are very sensitive to the noises and errors. A filtering algorithm based on cubic spline interpolation was proposed for Inertial Measurement Unit data processing to eliminate noises and errors for bending strain, and the effectiveness is validated through the pipeline field test. The results showed that the average pipeline displacement deviation declined 13.82% in the three tests, and the bending strain error reduced from 0.037% to 0.014%. The proposed method effectively improves the inspection accuracy and provides an effective method for pipeline displacement and strain inspection, which ensures the safe operation of the pipeline.

Accuracy of Quasi-Static Bending Strain Measurement using Scanning Laser Doppler Vibrometry (SLDV)

This paper presents an investigation into the use of 1D scanning laser Doppler vibrometry for the measurement of quasi-static bending strain, on a beam, from measured out-of-plane deflections. To calculate the bending strain from measured deflections, a Savitzky-Golay differentiation filter is used. An experimental investigation was conducted to determine how the strain estimate would be affect by the spatial interval between measurement points, amplitude of loading, and the numerical parameters used within the Savitzky-Golay differentiation filter. A number of findings, from the investigations undertaken are presented. The bending strain measurement technique is best approximated by using a third-order polynomial in the Savitzky-Golay differentiator filter. An accurate bending strain measurement can be produced at any spatial interval, as long as the filter width spans a fixed distance. In addition, changes in applied load have no effect on the accuracy of the technique. This technique has significant application for the measurement of localised bending strain on the surface of structures or materials that are cyclically loaded and subjected to harsh environmental conditions, such as elevated temperatures.

Elimination of self-straining in isogeometric formulations of curved Timoshenko beams in curvilinear coordinates

Isogeometric formulations of curved Timoshenko beams in curvilinear coordinates often result in self-straining of membrane and shear strains, due to the discretization of the local displacement field. Self-straining means a failure in exact representation of rigid body motions, which consequently deteriorates response quality. To overcome the difficulty of self-straining, we propose an invariant formulation that discretizes the global displacement field. It turns out that the approximated membrane, shear, and bending strain measures are invariant regardless of initial geometry in the proposed formulation. For effective applications to any arbitrarily curved structures and locking-free formulations to alleviate membrane and shear locking, the proposed invariant formulation is combined with selective reduced integration (SRI) and B̄ projection method. Numerical examples demonstrate the effectiveness and applicability of the proposed invariant formulation, which gives much more accurate results together with both SRI and B̄ projection method, in comparison to the existing isogeometric formulations.

Pipeline Bending Strain Measurement and Compensation Technology Based on Wavelet Neural Network

The bending strain of long distance oil and gas pipelines may lead to instability of the pipeline and failure of materials, which seriously deteriorates the transportation security of oil and gas. To locate the position of the bending strain for maintenance, an Inertial Measurement Unit (IMU) is usually adopted in a Pipeline Inspection Gauge (PIG). The attitude data of the IMU is usually acquired to calculate the bending strain in the pipe. However, because of the vibrations in the pipeline and other system noises, the resulting bending strain calculations may be incorrect. To improve the measurement precision, a method, based on wavelet neural network, was proposed. To test the proposed method experimentally, a PIG with the proposed method is used to detect a straight pipeline. It can be obtained that the proposed method has a better repeatability and convergence than the original method. Furthermore, the new method is more accurate than the original method and the accuracy of bending strain is raised by about 23% compared to original method. This paper provides a novel method for precisely inspecting bending strain of long distance oil and gas pipelines and lays a foundation for improving the precision of inspection of bending strain of long distance oil and gas pipelines.

Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part II. The effective boundary value problem and exemplary solutions

The compliance minimization of transversely homogeneous plates with predefined Kelvin moduli leads to the equilibrium problem of an effective hyperelastic plate with the hyperelastic potential expressed explicitly in terms of both the membrane and bending strain measures, as derived in Part I of the present paper. The aim of this second part of the paper is to show convexity of this potential and, consequently, uniqueness of solutions of the minimum compliance problem considered. Theoretical results are illustrated by numerically calculated optimal trajectories of the eigenstate corresponding to the largest Kelvin modulus.

A Robust, Multiplexable Fiber Sensor for Simultaneous Measurement of Bend and Strain

We report on the design and implementation of a novel optical fiber sensor for simultaneous measurement of bending and longitudinal strain. An all-fiber transducer is comprised of a fiber Bragg grating preceded by a short region of conventional fiber. Peak Bragg reflections experienced by orthogonal linearly polarized modes traveling within the sensor region shift identically with strain and temperature but oppositely with fiber bending. A novel switching interrogator allows isolation of the common mode strain signal to facilitate a robust measurement of both bend and strain, while retaining the grating-based transducer's capability for serial multiplexing.

Analysis of active closed cross-section slender beams based on asymptotically correctthin-wall beam theory

An asymptotically correct theory for multi-cell thin-wall anisotropic slender beams thatincludes the shell bending strain measures is extended to include embedded active fibrecomposites (AFCs). A closed-form solution of the asymptotically correct cross-sectionalactuation force and moments is obtained. Active thin-wall beam theories found in theliterature neglect the shell bending strains, which lead to incorrect predictions for certaincross-sections, while the theory presented is shown to overcome this shortcoming. Thetheory is implemented and verified against single-cell examples that were solved using theUniversity of Michigan/Variational Beam Sectional Analysis (UM/VABS) software.The stiffness constants and the actuation vector are obtained for two-cell andthree-cell active cross-sections. The theory is argued to be reliable for efficientinitial design analysis and interdisciplinary parametric or optimization studies ofthin-wall closed cross-section slender beams with no initial twist or obliqueness.

Structural Behavior of Thin- and Thick-Walled Composite Blades with Multicell Sections

A mixed beam approach that combines both the stiffness and the flexibility formulation in a unified manner has been performed to model and analyze coupled composite blades with closed, multiple-celled cross sections. The analysis model includes the effects of elastic couplings, shell wall thickness, transverse shear deformation, torsion warping, and constrained warping. Reissner's semicomplementary energy functional is used to derive the beam force-displacement relations. The influence of the shell bending strain measures as well as the membrane strain measures are incorporated in the formulation. For completeness required in a rigorous beam theory, four separate continuity conditions are imposed on each cell of the closed, multicelled sections. The theory is validated against experimental test data, detailed finite element analysis results, and other analytical results found in the literature for coupled composite beams and blades with various cross sections. These include two-cell box beams with bending-torsion and/or extension-torsion couplings and bending-torsion coupled composite blades with two-cell airfoils. The correlation between the present theory and other methods is found to be good, dependent on the geometries and material distributions adopted in the blades. Numerical results showing the effects of including the shell bending strain measures are examined. The effects of inappropriate treatment of the direction of integration for two-celled composite beams are also investigated in the current framework of the analysis.

Bending and axial strain dependence of the critical current in superconducting BSCCO tapes

A comparison is made between the critical current (Ic) versus bending strain and axial strain of superconducting multi-filamentary Bi2Sr2Ca2Cu3Ox (Bi-2223) AgMg sheathed tapes. For the bending strain measurement the tape is sandwiched between a curved base and cover plate. Six sets of bending plates introduce bending strains ranging from 0% to 1.0%. The measurements show a slight decrease in Ic after the first bending step after which the degradation becomes more pronounced. The Ic in a bent conductor is calculated assuming a linear axial strain profile inside the conductor. For this calculation the Ic degradation determined in an axial compression and elongation experiment is used. The model predicts an immediate decrease of Ic, caused by the compressive strain dependence. There is a good agreement (within 5%) between the measured data and the calculated values. Based on this good agreement it can be concluded that a possible shift in the neutral line or the formation of additional cracks due to bending has no significant influence on the Ic degradation. It is concluded that the influence of thermal contraction is crucial for a good calculation.

Single- and Multicelled Composite Thin-Walled Beams

structed from a general variational-asymptotic framework. The results of the theory are contained in closed-form expressions for the stiffness matrices of single- and double-celled composite thin-walled beams. The accuracy of the approach is demonstrated by several examples. The most important feature of these solutions,which distinguishes them from those previously published in the literature,is that shell bending strain measures are consistently taken into consideration. This is shown to be important for correct treatment of certain closed-cell configurations,the torsional stiffness of which can be off by a factor of two when bending strain measures or hoop moments are neglected. Correlation of the present formulae with finite element analysis is demonstrated to be excellent.

Theory of Anisotropic Thin-Walled Beams

Asymptotically correct, linear theory is presented for thin-walled prismatic beams made of generally anisotropic materials. Consistent use of small parameters that are intrinsic to the problem permits a natural description of all thin-walled beams within a common framework, regardless of whether cross-sectional geometry is open, closed, or strip-like. Four “classical” one-dimensional variables associated with extension, twist, and bending in two orthogonal directions are employed. Analytical formulas are obtained for the resulting 4×4 cross-sectional stiffness matrix (which, in general, is fully populated and includes all elastic couplings) as well as for the strain field. Prior to this work no analytical theories for beams with closed cross sections were able to consistently include shell bending strain measures. Corrections stemming from those measures are shown to be important for certain cases. Contrary to widespread belief, it is demonstrated that for such “classical” theories, a cross section is not rigid in its own plane. Vlasov’s correction is shown to be unimportant for closed sections, while for open cross sections asymptotically correct formulas for this effect are provided. The latter result is an extension to a general contour of a result for I-beams previously published by the authors. [S0021-8936(00)03003-8]

FULL-FIELD RANDOM BENDING STRAIN MEASUREMENT OF A PLATE FROM VIBRATION MEASUREMENT

Experimental results of a non-contacting, full-field measurement of random bending strain over the surface of a two-dimensional structure are presented. The method is based on a combination of the measurment of random vibration at discrete points on the structure using a scanning laser vibrometer along with an analysis of the measured data using a surface smoothing technique. The power spectral densities of the random bending strains at all points on the structure are estimated by the numerical differentiation of cross-spectral densities of the vibration measured by two transducers, a fixed reference transducer and a roving (or scanning) transducer. The experimental results are compared with those measured directly using strain gages, which shows very good agreement.

Estimation of Random Bending Strain in a Beam from Discrete Vibration Measurements

A method is presented for using vibration measurements obtained at discrete locations on a randomly excited beam to estimate the power spectral density of bending strain. The technique is intended to be applied using a scanning laser vibrometer to allow a non-contacting measurement of random bending strain over the surface of a structure. The theory behind the measurement method is presented, along with results of numerical simulations. The results indicate that the method is practical and can lead to reliable estimates.