Large Deflection Beam Research Articles

Large Structural Shear Deformation and Failure Monitoring Using Bend Losses in Polymer Optical Fibre

Rapid identification of structural damage positions is essential to the post-disaster rehabilitation of structures and infrastructures. Large shear deformation, e.g., shear failure of bridge piers, shear-slip of slopes, and shear cracking of structural walls, is often the cause of structural instability. Distributed optical fibre sensing (DOFS) techniques have an advantage over point-based sensors in terms of spatial continuous structural condition monitoring. This paper presents the development of new measurement theory and algorithm to evaluate the structural shear deflection based on the large beam deflection and optical bend loss theories. The proposed technique adopted a photon-counting Optical Time Domain Reflectometer (ν-OTDR) with polymer optical fibres (POFs) which has a large deformation measurement range and high spatial resolution. In the experiment, shear deformation events can be successfully detected and evaluated from the proposed technique. When the normalised shear deformation is larger than 0.2, both the event locations and the magnitudes can be accurately determined. When normalised shear deformation is lesser than 0.2, the error in the magnitude evaluation increased, but the event location can be found with an absolute error <0.5 m. Multiple shear events can be treated as independent events when they are separated by more than 5 m. Various configurations of POFs attached to concrete beam specimens for rupture failure monitoring were also studied. The configuration that could maximise the POF curvature at the beam failure produced the largest ν-OTDR signals. In other configurations in which the POFs were only stretched at failure, the signals were less strong and were influenced by the POF-structure bonding strength.

Geometrically nonlinear mechanical properties of auxetic double-V microstructures with negative Poisson's ratio

Double-V microstructure (DVM) is a type of auxetic cellular material with negative Poisson's ratio (NPR). Accurate predictions of the mechanical properties of these microstructures at large strains are critical for many engineering applications. In this paper, nonlinear theoretical models of two-dimensional (2D) and three-dimensional (3D) DVM based on a large beam deflection model are established to predict the normalized Young's modulus and Poisson's ratio in the principal directions. The theoretical solutions are compared to solutions obtained from numerical finite element analyses and quasi-static compression experiments of a 2D prototype manufactured using additive printing technique. It is found that there is good agreement between these results, validating the accuracy of the proposed theoretical model. Effects of geometry parameters on the mechanical properties of the DVM are also investigated to understand the mechanical behavior at large strains. This study provides validated models for predicting the behavior of these microstructures at large strains, useful for engineering designs.

Simulation of shape memory cycle of a polymeric beam undergoing large deflection using a simple incremental approach

Shape memory polymers are a group of polymers which can store a temporary shape at a relatively cold temperature and recover when heated above its glass transition temperature. Various constitutive models exist in literature to simulate the uniaxial shape memory cycle. One such popular model is based on the concept of multiple natural configurations. In the present work, the stress-strain-based model is adapted for bending application by converting it into a scalar moment-curvature-based relationship. The adapted model is used in a numerical framework for large deflection of beams to simulate the shape memory cycle. The employed numerical framework is based on linearising the non-linear governing differential equation and subsequently solving it in steps by numerical integration. The results are presented in suitable non-dimensional form for generality. Since coefficient of thermal expansion plays a minimal role in bending, it is found that the bending results are considerably different from their uniaxial counterpart. The approach may be claimed to be computationally economic as compared to finite element method because here large matrix inversion is avoided.

3D small strain large deflection beam shape sensing including poisson effect

This article presents a new method to monitor beam shapes from strain measures. The beam is instrumented with strain sensors placed on its surface with an angle of orientation. The method developed is a 3D large deflection beam shape reconstruction based on a beam model incorporating Poisson effect and is thus able to take into account bending, torsion, shearing and tensile-compression deformations. Exemple of application on circular beams is conducted with beam shape reconstructed from FEM simulations strain measures. Results show that, compared to beam shape sensing methods using axial strain, the beam shape method proposed provides a significant increase in the reconstruction accuracy when the beam is subject to deformations containing torsion.

The effect of graphene on the deflections of multiscale composites under thermo-mechanical loading

The present investigation aims to shed light on an important aspect of graphene reinforced multiscale composites that may appear in the form of reduced dimensional stability of the structure when exposed to thermal gradients. The study consists of two main parts: an extensive review of the related literature and a thermo-mechanical analysis of a laminated composite beam made of three-phases: arbitrary oriented Graphene Nanoplatelets (GNPs) embedded in an epoxy resin (ER) polymer matrix, reinforced with E-Glass fibers.The review of related literature reveals that the problem of finding the equivalent coefficient of thermal expansion (CTE) for multiscale composites is not as easy as the conventional composites, the trend is different, and more complicating parameters are involved. The analysis includes the predictions of the effective thermo-elastic properties of the multiscale composite laminate, based on Halpin-Tsai theory and hierarchy modeling, and the formulation of the beam deflection governing equation, based on Timoshenko nonlinear (large deflection) beam theory. Different GNPs weight fractions, different E-glass fiber volume fractions, and three lamination layups are investigated. It is found that the addition of GNPs increases the beam deflections when exposed to temperature gradients and that the laminate longitudinal CTE is the main factor responsible for this effect.

Large deflection of functionally graded porous beams based on a geometrically exact theory with a fully intrinsic formulation

• 3D large deflections of FG porous beams are studied using a geometrically exact beam theory. • Constitutive relation is obtained based on the concepts of continuum mechanics. • For solution, the Chebyshev collocation method is adopted for the numerical discretisation . • Two types of porosity distributions, namely cross-sectional and span-wise are considered. • To simulate different loading conditions, conservative and non-conservative (follower) loading scenarios are considered. Porous graded materials found in nature can be regarded as variable stiffness optimised load carrier elements that exhibit beneficial properties for real-life engineering designs. In order to investigate the nonlinear behaviour of variable stiffness bioinspired materials, the large deflection of functionally graded beams made from porous materials is considered in this work. Our purpose is to present an efficient and accurate methodology capable of capturing spatially large deflections of these structures with different types of loading conditions and porosity distributions. A geometrically exact beam model with fully intrinsic formulation is employed for the first time to study the large deflection behaviour of functionally graded beams under conservative and non-conservative (follower) loading scenarios. An orthogonal Chebyshev collocation method is used for the discretisation of the fully intrinsic formulation. Two types of porosity distributions, namely cross-sectional and span-wise, are considered and the effect of porosity distribution has been investigated for various benchmark classical test cases. For a given level of accuracy, it is shown that the span-wise functionally graded beam is computationally more demanding compared to the cross-sectional functionally graded beam. In addition to classical problems, two examples demonstrating 3D deflections of highly flexible structures made from porous material subject to combined loads are investigated. It is shown that the current paradigm, while being computationally efficient, can effectively capture the large deflections of functionally graded beams with excellent accuracy.

Saturated analysis of pulse-loaded beams based on Membrane Factor Method

Under intense transverse loading, beams or thin plates will experience large elastic-plastic deflection. The membrane force induced by large deflection will enhance the structure's load-carrying capacity. Since 1950s, various analytical tools have been developed to account for the effect of membrane force and predict the final deflection of beams and plates under quasi-static or dynamic loading. Among those tools, Membrane Factor Method (MFM) has been proved to be a powerful one. On the other hand, it has been revealed that beams and plates under pulse loading exhibit a kind of saturation phenomena, i.e. there exist saturated impulse and saturated deflection due to the membrane enhancement of the beams and plates. The present paper will, for the first time, show that a combination of MFM and SA (Saturation Analysis) can provide an advanced theoretical tool for the study of large plastic deflection of beams under pulse loading. Then the complete solutions obtained by this MFM+SA approach are compared with modal solutions and FE simulations. Finally, a set of simple formulae are proposed for predicting the saturated duration and saturated deflection of beams to facilitate engineering applications.

Large Deflections of Beams, Arches and Panels in an Elastic Medium with Regard to Deformation Shifts

The supercritical bending of arches in an elastic medium loaded with a lateral distributed load or concentrated force, including arches obtained by pre-bending by means of compressing a beam, is investigated. A resolving nonlinear equation is obtained, its analytical solution and numerical solutions are given.

High-Speed Time-Domain En Face Optical Coherence Tomography System Using KTN Optical Beam Deflector

We developed high-speed time-domain (TD) en face optical coherence tomography (OCT) system using KTN optical beam deflector. The KTN optical beam deflector operates at a high repetition rate of 200 kHz with a fairly large beam deflection angle. We proposed a high-speed en face OCT system that used a KTN optical deflector as the sample beam scanning. In the experiment, we obtained en face OCT images of human fingerprint with a frame rate of 800 fps, which is the fastest speed obtained by a TD-OCT imaging. Furthermore, a 3D-OCT image was also obtained at 0.2 s (=5 volumes/s) by our imaging system.

An inverse problem methodology for multiple parameter estimation in bend stiffeners

The flexible riser top connection is a critical region for lifetime assessment due to large tension/curvature variations and modeling uncertainties. The bend stiffener polyurethane mechanical response not only presents a nonlinear loading rate and temperature dependency but is also subjected to weather ageing during operation, which may affect its mechanical behavior over time. The top tension, employed for riser local cross-section stress calculation, is usually obtained from global dynamic analyses performed under selected environmental conditions, if direct measurement is not available. As a consequence, both the bend stiffener effect on the curvature distribution and the top tension time series present inherent uncertainties for riser lifetime (re)assessment. In the present work, a proposed monitoring approach composed by gyrometers installed along flexible riser/bend stiffener top connection system length combined with an inverse problem methodology is numerically investigated to estimate the following parameters: (i) polyurethane hyperelastic response and (ii) effective top tension. The top connection system is modeled using a large deflection beam bending model and the parameters are estimated using a damped least-square minimization approach with the Levenberg–Marquardt algorithm. For the preliminary feasibility investigation, the gyrometer experimental data is numerically estimated through Monte Carlo simulations. A case study is carried out to investigate the influence that the number of sensors, sensors arrangement, loading conditions and top connection model have on the inverse parameters estimation. The results indicate that the proposed monitoring approach and inverse parameter estimation methodology may effectively reduce flexible riser lifetime calculation uncertainties.

A new physical model for empirical fiber snubbing effect in cementitious composites based on large deflection beam theory

A single fiber crosses a crack in Strain-Hardening Cementitious Composites (SHCC) can be either perpendicular or inclined to the crack surface. When the single fiber pullout test is performed on inclined polymeric fibers, the peak pull-out load is commonly observed to increase with the inclination angle. Usually the fiber is modeled as a flexible string passing over a frictional pulley, and the observed phenomenon is explained by the high local friction (or snubbing effect) near the fiber exiting point. However, such an explanation is not perfectly satisfactory because the bending stiffness of the fiber is not necessarily negligible and the pulley does not physically exist. In this study, a new model which treats the fiber as a beam and emphasize on the large deflection effect of the fiber is developed. The entire fiber bridging stress vs crack opening relationship can be derived and the empirical snubbing effect is also revealed, even without assuming the presence of a frictional pulley. Moreover, at small crack opening, the predicted trend of bridging force with inclination angle from the new model is in better agreement with test results than the snubbing friction model.

Analytical model of the dynamic response of clamped metallic sandwich beam subjected to underwater impulsive loading

An analytical model is developed for the dynamic response of clamped metallic sandwich beam subjected to underwater impulsive loading. Considering the coupling interaction of different response stages, the theoretical dynamic response is achieved for the large deflection of metallic sandwich beams with a new yield criterion during the structural deformation stage. The computational and theoretical dynamic response of the sandwich beam are compared with the previous experimental results in terms of the response time, response rate, velocity of the faces, and transverse deflection. The results indicate that the tighter solution of transverse deflection from the new circumscribing and inscribing yield surfaces exhibits a good agreement with the experimental and numerical results, demonstrating that the analytical prediction is more accurate than the previous analytical model. The optimization shows that sandwich beams not always outperform the monolithic beams. The optimal sandwich beam designs are sensitive to the span of the beam and the longitudinal strength of the core.

Low-velocity impact of sandwich beams with fibre-metal laminate face-sheets

Low-velocity impact of fully clamped sandwich beams with fibre-metal laminate face-sheets and metal foam core struck by a heavy mass is investigated. Analytical solutions are developed for the dynamic response of sandwich beams with fibre-metal laminate face-sheets considering the interaction of bending and stretching induced by large deflections. According to the rigid-plastic material approximation with modifications, simple formulae are obtained for the large deflection of sandwich beams with fibre-metal laminate face-sheets. Numerical calculations are carried out, and analytical solutions capture numerical results reasonably. The effects of the composite volume fraction, the ratio of metal layer strength to composite layer strength, and core strength on the structural response are discussed. Using the analytical formulae, optimal design charts are constructed to minimize the mass of sandwich beams. It is demonstrated that the present analytical model can predict the post-yield behaviour of sandwich beams with fibre-metal laminate face-sheets reasonably.

Analytical Solution for Large Deflection of Multilayered Composite Cantilever Beams with Interlayer Slip

The solution methodologies for nonlinear differential equations play a significant role in the area of theoretical mechanics, in particular large deflection of beams. In this paper, Euler–Bernoulli’s equation is adapted along with R. Da Silva method for solving the problem of large deflection on multilayered cantilever beams with interlayer slip analytically. Interlayer slip, horizontal displacement and large deflection nonlinear differential equations of multilayered Euler–Bernoulli cantilever beams were derived. The equations are applied to the typical partially interacted multilayered cantilever beam problem with point and uniformly distributed load. Solutions of the geometrically linear theory and geometrically nonlinear theory are compared.

Frequency and mode change in the large deflection and post-buckling of compact and thin-walled beams

This paper deals with the investigation of normal modes change of metallic structures, when subjected to geometrical nonlinearities in the large displacement/rotations field. Namely, a unified framework based on the Carrera Unified Formulation (CUF) and a total Lagrangian approach are employed to formulate higher order beam theories including geometric nonlinearities. Thus, a finite element approximation is used along with a path-following method to perform nonlinear analyses. Linearized vibration modes around equilibrium states and along the whole equilibrium path of structures subjected to bending and compression loadings are evaluated by solving a classical eigenvalue problem. In order to show the capabilities of the proposed methodology, both solid and thin-walled cross-section beams are considered. The analyses demonstrate that, with some differences depending on the geometry and both boundary and loading conditions, natural frequencies and modal shapes may change significantly as the structure is subjected to large displacements and rotations.

Large deflections of multilayer sandwich beams with metal foam cores under transverse loading

In this paper, large deflections of fully clamped multilayer sandwich beams with metal foam cores are investigated. Based on the yield criteria, analytical solutions are derived for the plastic behavior of fully clamped multilayer sandwich beams loaded by a punch, in which the effects of local denting and core strength on the overall bending are considered. The finite element method is employed to predict the large deflection behavior of multilayer sandwich beams, and the analytical model captures numerical results reasonably well. The effects of core strength, multilayer factor, the size of the loading punch and loading location on the plastic analysis of multilayer sandwich beams are discussed. It is demonstrated that the load-carrying capacity of multilayer sandwich beams increases with a decrease in the multilayer factor and an increase in the core strength and that loading location and the size of the loading punch play important roles in plastic analysis.

Modeling and mechanical analysis of multiscale fiber-reinforced graphene composites: Nonlinear bending, thermal post-buckling and large amplitude vibration

In this paper, a mathematical model was developed to predict the effective material properties of graphene nanoplatelets/fiber/polymer multiscale composites (GFPMC). The large deflection, post-buckling and free nonlinear vibration of graphene nanoplatelets-reinforced multiscale composite beams were studied through a theoretical study. The governing equations of laminated nanocomposite beams were derived from the Euler–Bernoulli beam theory with von Kármán geometric nonlinearity. Halpin–Tsai equations and fiber micromechanics were used in hierarchy to predict the bulk material properties of the multiscale composite. Graphene nanoplatelets (GNPs) were assumed to be uniformly distributed and randomly oriented through the epoxy resin matrix. A semi-analytical approach was used to calculate the large static deflection and critical buckling temperature of multiscale multifunctional nanocomposite beams. A perturbation scheme was also employed to determine the nonlinear dynamic response and the nonlinear natural frequencies of the beams with clamped–clamped, and hinged–hinged boundary conditions. The effects of weight percentage of graphene nanoplatelets, volume fraction of fibers, and boundary conditions on the static deflection, thermal buckling and post-buckling and linear and nonlinear natural frequencies of the GFPMC beams were investigated in detail. The numerical results showed that the central deflection and natural frequency were significantly improved by a small percentage of GNPs. However, addition of GNPs led to a lower critical buckling temperature.

Hybrid modeling and verification of disk-stacked shock absorber valve

The mathematical model of valve deformation of hydraulic shock absorber is established using the hybrid method of large deflection and beam deflection. The model can be used for design purposes and helps in developing damping valve based on single disk and shim stack. The model of the hydraulic shock absorber with valve system consists of shim stack and a spring preloaded disk is developed using AMESim. The correctness of the mathematical model of the valve deformation based on the hybrid method and the simulation model using AMESim is verified by the shock absorber bench test, which proves the method proposed in this article is reasonable and reliable. It is significant for reference in the design and development of hydraulic shock absorber.

Design, fabrication, and analysis of lattice exhibiting energy absorption via snap-through behavior

In this study, an energy absorption lattice, comprised of multiple tetra-beam-plate unit cells with negative stiffness, was designed, fabricated by selective laser sintering method, and analyzed both numerically and experimentally. Snap-through behavior of the unit cell developed due to negative stiffness caused by geometric nonlinearity from large deflection of the constituent elastic beams, resulting in energy absorption. A criterion for the unit cell to achieve the snap-through behavior was investigated numerically in terms of the beam slenderness ratio and the inclined angle. This approach was chosen to facilitate control of energy dissipation performance and further design space such as tuning force threshold. The unit cell with the selected geometric parameters was then created and used to construct the energy absorption lattice. Load-displacement relationships of the lattices obtained from cyclic loading tests disclosed an area enclosed by two distinct loading and unloading curves, which indicates energy dissipation. This was shown both numerically and experimentally. Drop tests were also performed to investigate energy loss of the lattices due to an impact. An energy absorption phenomenon was revealed by observing a reduced rebound height when the lattice exhibited the snap-through behavior.

Large Deflection of Prismatic Cantilever Beam Exposed to Combination of End Inclined Force and Tip Moment

The large deflection of a prismatic Euler-Bernoulli cantilever beam under a combination of end-concentrated coplanar inclined force and tip-concentrated moment is investigated. The angle of inclination of the applied force with respect to the horizontal axis remains unchanged during deformation. The cantilever beam is assumed to be naturally straight, slender, inextensible and elastic. The large deflection of the cantilever beam induces geometrical nonlinearity; hence, the nonlinear theory of bending and the exact expression of curvature are used. Based on an elliptic integral formulation, an accurate numerical solution is obtained in terms of an integration constant that should satisfy the boundary conditions associated with the cantilever beam. For some special cases this integration constant is exactly found, which leads to closed form solution. The numerical solution obtained is quite simple, accurate and involves less computational time compared with other techniques available in literature. The details of elastica and its corresponding orientation curves are presented and analyzed for extremely large load combinations. A comparative study with pre-obtained results has been made to verify the accuracy of the presented solution; an excellent agreement has been obtained.