Curved Beam Theory Research Articles

Analysis of mode II strain energy release rates in end-notched flexure tests of carbon fiber-reinforced plastic curved beams

The present study aims to formulate strain energy release rates (SERR) for curved beams, which are supposed to be cut from tanks or cylinders, in End-Notched Flexure (ENF) tests based on Euler-Bernoulli’s and Timoshenko’s curved beam theories. The derived SERRs based on the curved beam theories were compared with those based on the flat beam theory for specimens with various geometries to prove the validity of using the curved beam theory in the ENF tests of curved beams. The validity of the derived compliance and SERRs was verified for the specimens with various geometries using a two-dimensional finite element analysis, in which virtual crack closure techniques were used to calculate the SERRs. The results revealed that Euler–Bernoulli’s curved beam theory was effective in cases with large radius-of-curvature-to-thickness (R/h) values but not in those with small R/h values, whereas Timoshenko’s curved beam theory was effective even for small R/h values.

A new higher order shear and normal deformation theory for the free vibration analysis of sandwich curved beams

The free vibration analysis of functionally graded sandwich beams curved in elevation is presented in this paper using a fifth-order curved beam theory accounting for the effects of transverse shear and normal strains i.e. thickness stretching effects. The theory assumes higher-order variations of axial (tangential) as well as transverse (radial) displacements. Equations of motion are derived from the dynamic version of the principle of virtual work and solved analytically by using Navier’s technique. The present theory is applied for the free vibration analysis of sandwich curved beams with functionally graded face sheets and the homogenous core. The material properties of face sheets are graded in the thickness direction according to the power law. The non-dimensional fundamental frequencies are obtained for different parameters such as radius of curvature, the power-law index, and lamination scheme and compared with previously published results wherever possible and found in good agreement with those. It is observed that as the radius of curvature is decreased keeping length and thickness constant the value of non-dimensional fundamental frequency increases. It is also observed that as the power-law index increased the non-dimensional frequency decreases for all length to thickness ratios.

Development of an Adjustable Frequency Device for a Test Vehicle Based on Curved Beam Theory

In this article, an adjustable frequency device based on curved beam theory is designed to control vertical stiffness of an instrumented vehicle that it can detect dynamic data when moving on a test beam for frequency measurement. The adjustable frequency device consists of a set of two-layer cantilever semi-circular thin-beams to support a lumped mass for vibrations, in which a rotatable U-frame is used to change its subtended angle for adjustment of the supporting stiffness and corresponding vertical frequencies of the vehicle. Based on curved beam theory, an analytical frequency equation of the single-degree-of-freedom test vehicle was derived and applied to mobile frequency measurement of a simple beam. To determine the sectional rigidity of the semi-circular thin-beams, both theoretical and experimental studies were be carried out in the ITAM laboratory of the Academy of Science in Czech. The analytical and experimental results indicated that the present semi-circular beam model with guided ends is applicable to prediction of natural frequencies of the test vehicle considering different supporting stiffness

Deformation and stress analysis of a U-shaped pipe compensator using a 3D scanner

The paper shows a comparison of the three different methods to determine stress and strain in a U-shaped pipe compensator which is used to decrease stress in long pipes due to the constrained temperature dilatations. The stress and strain are analyzed analytically first with some parts of the analytical solution obtained numerically, such as integrals with no analytical solution in a closed form, i.e., functional series can be involved as a tool to solve those integrals. The pipe is analyzed as a beam or a planar frame using the Castigliano's method to determine displacements. Since there are curved parts of the U compensator, the curved beam theory is applied. The alternative method to determine the strains and stresses along the pipe is shown using the numerical simulations in SolidWorks. The results are compared with the analytical solution. Finally, the experimental method using a 3D scanner is involved for a comparison to check the applied conditions in the analytical and the simulation model.

On the Determination of Material Mechanical Properties of St 37 Steel Pipes Using O-Ring Specimens

This paper presents experimental, analytical and numerical studies of testing O-rings specimens cut from an St 37 steel pipe. These studies are meant to determine the mechanical properties of the St 37 material and to predict the behavior of its pipes when subjected to internal pressure. This was done by applying transversely concentrated force on relatively small rings (outer diameter = 60.3 mm and the wall thickness = 3.9 mm) until the rupture of the ring. A universal tensile testing machine along with specially designed holders were used to measure the load-line displacement of the rings. In addition, formulas to determine the displacement of the loading pin were derived using the straight beam theory and the curved beam theory. Both elastic and plastic deformations were calculated and compared in the results. Furthermore, finite element analyses (FEA) were carried out using strain hardening material model, to obtain the full deformation behavior of the O-ring during testing until rupture. From the experimental results, it was possible to determine the material yield strength and modulus of elasticity. The analytically obtained results predicted the material modulus of elasticity which was in very good agreement with the published values. The FE analyses were able to predict the full behavior of the rings.

Finite beam element with 26 DOFs for curved composite box girders considering constrained torsion, distortion, shear lag and biaxial slip

Curved steel–concrete composite girders have been widely used in curved bridges of overpasses in urban areas as they have a lower self-weight and require less construction time. These curved beams have been often numerically simulated with the use of a more elaborate three-dimensional (3D) finite element (FE) model to predict their mechanical behavior with high accuracy. However, a more elaborate FE model requires a more complex modeling process and higher computational cost, which significantly reduce its efficiency. In response to these problems, a one-dimensional model of curved composite box girders is proposed as a high-efficiency numerical simulation method, which therefore warrants further investigation. This one-dimensional theoretical model can account for constrained warping, distortion, and the shear lag in concrete slabs and steel bottom plates, biaxial slip at the slab-girder interface and curvature differences along the width of the beam. Although there are numerous studies on curved beam theories, an accurate and efficient one-dimensional theoretical model of curved composite box girders is still lacking. Additionally, experimental studies on curved composite box beams that are subjected to coupled bending and torsion are few compared to those on the mechanical behavior of straight composite girders, especially curved composite box beams with wide flanges. This study proposes a one-dimensional model of curved composite box girders. Then, using an FE discretization method, a high-efficiency finite beam element with 26 degrees of freedom (DOFs) is developed for curved composite box girders which accounts for constrained torsion, distortion, shear lag, biaxial slip at the interface and curvature differences along the width of the beam. Subsequently, two large-scale tests of a curved composite box girder are conducted and the findings are reported. A comparison among the experimental test results, elaborate FE model with shell elements and developed beam element shows good agreement. The force transfer behavior of curved box beams is further analyzed based on the proposed beam element, and the influence of the key parameters on the mechanical performance is investigated. In summary, this study contributes to the current literature with the development of a one-dimensional theoretical model for curved composite box beams and tests on curved box beams with a large curvature.

Distributed fibre optic monitoring of damaged lining in double-arch tunnel and analysis of its deformation mode

Cracking and crushing were observed in the first tunnel lining of the Kaidagu double-arch tunnel (China), excavated in weathered surrounding rock. This concrete damage occurred during excavation of the second tunnel. It was necessary to monitor the continuous strain distribution along the entire cross section of the lining, so distributed fibre optic monitoring was carried out for nearly 1 year to study the deformation and stress of the damaged lining of the first tunnel. Optical fibres were attached to the inner surface of the concrete secondary lining and on opposite sides of the temporary arch support. Crack development was detected on the concrete lining, and the strain distribution indicated that vertical ovalisation and oblique ovalisation deformation was generated on the basis of the original deformation. Overall compression deformation was observed as this deformation mode developed over time. Ignoring the effects of axial force, the radial displacement of the concrete lining caused by the bending moment was calculated from the fibre optic strain data. The results were compared with those obtained using total stations, and obvious errors were found. According to the curved beam theory, a more accurate inversion analysis scheme based on the fibre optic strain data was proposed to determine the deformation mode of arch structures. The combined effect of the axial force and the bending moment was considered for the first time. This scheme was used to calculate the displacements, forces and loads of arch structures and was verified by finite-element simulations. The proposed method was applied to analyse the temporary arch support. The calculation results were consistent with those obtained using total stations and conformed with the tunnel deformation mode. The results and the proposed method are useful in explaining the subsequent deformation mode of the damaged lining in the first tunnel and provide valuable information for post-failure reinforcement.

Improved Strategy of Two‐Node Curved Beam Element Based on the Same Beam’s Nodes Information

The curved beam with a great initial curvature is the typical structure and applied widely in real engineering structures. The common practice in the current literature employs two‐node straight beam elements as the elementary members for stress and displacement analysis, which needs a large number of divisions to fit the curved beam shape well and increases computational time greatly. In this paper, we develop an improved accurate two‐node curved beam element (IC2) in 3D problems, combining the curved Timoshenko beam theory and the curvature information calculated from the same beam curve. The strategy of calculating the curvature information from the same bean curve in the IC2 beam element and then transferring the curvature information to the two‐node straight beam element can greatly enhance the accuracy of the mechanical analysis with no extra calculation burden. We then introduce the finite element implementation of the IC2 beam element and verify by the complex curved beam analysis. By comparison with simulation results from the straight two‐node beam element in the MIDAS (S2‐MIDAS) and the three‐node curved beam element adopted in the ANSYS (C3‐ANSYS), the simulation results of the typical quarter arc examples under constant or variable curvature show that the IC2 beam element based on curved beam theory is a combination of efficiency and accuracy. And, it is a good choice for analysis of complex engineering rod structure with large initial curvature.

A design approach for the ring girder in elevated steel silos

Silos in the form of the thin-walled cylindrical steel shells are commonly supported on a ring girder, which rests on a limited number of columns to discharge the content by gravity flow into transportation systems. Local supports in the elevated steel silos produce a significant circumferential non-uniformity in the axial membrane stresses in the silo shell. One way of reducing the non-uniformity of these stresses is to use a very stiff ring girder which partially or fully redistributes the stresses from the local support into uniform stresses in the shell. Previous works that attempt to produce design expressions only address isolated ring girders and ignore the key role of the interaction between the cylindrical shell and the supporting ring girder. The aim of this study is to achieve a more efficient and realistic design of ring girders resting either on four supports, or on four supports with secondary beams, or on eight supports. Pursuant to this goal, variations of the stress resultants and displacements in the ring girder which rests on different support conditions are derived for the first time using Vlasov's curved beam theory. These expressions have been evaluated with finite element analyses for an isolated ring girder. Subsequently, a complementary finite element parametric study was conducted to investigate the variation of the stress resultants and displacements caused by the connection of the shell and ring girder resting on different supporting conditions. These variations were plotted as a function of the shell-ring girder stiffness ratio (ψ) for each support condition. Finally, considering lower bound limits for the stress resultants and displacements, design equations were proposed for ring girders which rest on different support conditions. These expressions may easily be placed in a practical spreadsheet, which could be used with different data to give realistic predictions.

Generalized Spiral Spring: A Bioinspired Tunable Stiffness Mechanism for Linear Response With High Resolution

Abstract Compliant mechanisms are typically designed for varying stiffness from nearly zero to rigid. However, targeted design for fine-tuning within an application's sensitive range of stiffness remains more desirable for practical implementation in accurate loading or positioning systems. To achieve various competing objectives, a “generalized spiral spring” (GSS) is proposed which achieves small size and other objectives by using a reduced number of parameters as provided by the spiral shape description of the components. An analytical model based on virtual work and curved beam theory is developed for accurate prediction of the stiffness. Moreover, finite element (FE) models are also developed for verification of the proposed designs. Multiobjective design optimization (MDO) is conducted to maximize the linearity in the stiffness versus control parameter (CP) response and improve resolution. The proposed analytical model is validated experimentally and computationally. This approach may be used to achieve finesse by accurate positioning with force control for industrial robots and elegant prostheses.

Theoretical model for variable helical angle of tensile armour wires in bent flexible pipes

This paper systematically addresses a theoretical model to account for the variation of the initial helical lay angle of the tensile armour wires in flexible pipes during bending condition. Based on differential geometrical discussion and curved beam theory, a non-linear differential equation system which contains the unknown variable helical angle is established. Moreover, different expressions of wires’ strain before and after wires’ slip are included in the paper to underline the discrepancy between scenarios in non-slippage and slippage regions. The variation of the armour wires helical angles, which are usually implicit in previous work, can be obtained numerically, revealing potential application in global bending problem. Case studies compared with results obtained through analytical and numerical models available in the literature as well as parametric analyses corresponding to original helical angle, initial tensile force, friction coefficient and bending curvature are performed to verify the feasibility of the proposed theoretical method.

Exact solutions for bending of Timoshenko curved nanobeams made of functionally graded materials based on stress-driven nonlocal integral model

Size-dependent static bending analysis of functionally graded (FG) curved nanobeams based on the Timoshenko beam theory is performed with the application of a stress-driven nonlocal integral model. The governing equations and corresponding boundary conditions are derived via Hamilton’s principle. Through detailed derivation, the integral constitutive equation is equivalent to a differential form equipped with two extra boundary conditions. By using the Laplace transform technique and its inversion, the coupling equations are solved analytically for different boundary and loading conditions. According to the numerical results, it is revealed that for FG Timoshenko curved beam theory, increasing the nonlocal parameter has a consistently stiffening effect on the bending of the curved nanobeams subjected to different boundary and loading conditions. The non-dimensional deflections decrease with the increase of the FG gradient index for all boundary cases, significantly when FG power-law index p < 4. Moreover, through comparing with the results of the Euler-Bernoulli theory, an interesting finding is that for Simply-Clamped, Clamped-Clamped and Clamped-Free nanobeams, the shear-deformable effect becomes more significant with the increase of nonlocal parameter, which cannot be ignored even for larger length-to-height ratio.

Differential quadrature method for magneto-hygrothermal bending of functionally graded graphene/Al sandwich-curved beams with honeycomb core via a new higher-order theory

Based on the differential quadrature method (DQM), the bending of sandwich-curved beams with graphene platelets/aluminum (GPLs/Al) nanocomposite face sheets and aluminum honeycomb core is investigated using a new shear and normal deformations curved beam theory. The present new model is resting on elastic foundation and subjected to a circumferential magnetic field, thermal load, and humid conditions. The face sheets are made of several bonded composite layers with randomly oriented and uniformly distributed graphene platelets in each layer. The mechanical and hygrothermal properties of the faces are assumed to be functionally graded (FG) using a piece-wise law by varying the weight fraction of the GPLs in the face thickness direction. Four governing differential equations are derived based on a novel four-variable curved beam theory taking into account the thickness stretching effect. The governing equations are solved for various boundary conditions on the basis of the DQM. The displacements presented by the DQM are compared with those obtained by Navier solution. Impacts of various parameters such as geometric shape parameters, magnetic parameter, temperature, moisture, elastic foundation parameters, core thickness, boundary conditions, and graphene weight fraction on the displacements and stresses of the functionally graded graphene/aluminum sandwich-curved beams are illustrated.

Buckling and vibration of FG graphene platelets/aluminum sandwich curved nanobeams considering the thickness stretching effect and exposed to a magnetic field

This paper studies the buckling and vibration analyses of a new model composed of a homogeneous ceramic curved nanobeam sandwiched between two reinforced composite layers. The face layers are made of an aluminum matrix reinforced with functionally graded (FG) graphene nanosheets (nanoplatelets) (GNShs). The present model is assumed to be embedded in an elastic substrate, exposed to an axial magnetic field and axial compressive external loads. Moreover, various boundary conditions are considered. The faces contain multi-composite nanolayres. Each nanolayer is composed of an aluminum matrix reinforced with uniformly distributed GNShs. The material properties of each face layer as a whole are graded through the thickness according to a piecewise micromechanical model. A refined higher-order curved beam theory is introduced in the polar coordinates considering the thickness stretching effect. Based on the proposed theory, four motion equations are presented containing Lorentz magnetic force and beam-foundation interaction. These equations are analytically solved to obtain the mechanical buckling load as well as the natural frequencies of the sandwich curved nanobeam. Since the general form of the strain-displacement relations is considered in the polar coordinates, the present formulation gives precise results for both shallow and deep curved beams. The effects of the different parameters such as arc angle, magnetic field parameter, elastic substrate parameters, boundary conditions, graphene wight fraction and core thickness on the buckling load and free vibration of the sandwich curved composite beams are investigated.

Effect of temperature difference on the mechanical responses of ribbon kirigami: Toward the highly stretchable conductors

Kirigami-patterned designs through photolithography technology present a promising set of strategy for highly stretchable conductors, which have been investigated to open up a wide range of novel technological solutions for various applications, such as stretchable bioprobe and wearable thermotherapy. The kirigami-patterned conductors usually suffer steady temperature difference and consequent thermal load, because of the Joule heating effect under input voltages. The concealed relationships between thermal effects, geometric effects and mechanical responses are of great significance for practical applications. Here, a closed-form analytical solution, considering the thermal effect and large curvature curved beam theory, is developed to study the stretchability and stiffness for a class of ribbon kirigami structures. Both of finite element method and experiments are performed to validate the accuracy and scalability of model. Comparisons of the developed closed-form stretchability and stiffness to the model with thermal effect absent present quantitative characteristic towards the thermal effect, the remarkable underestimate of stretchability (e.g., >8% relatively) and overestimate of stiffness (e.g., >5% relatively) can be induced by the thermal-effect-absent model for many representative ribbon kirigami patterns. Moreover, several demonstrations present the capability of developed model in optimization of ribbon-kirigami-patterned conductors under the thermal effect and practical geometry constraint conditions, achieving the most stretchable devices. This study provides the theoretical guide for kirigami-based conductor designs in applications.

Delamination R-curve behavior of curved composite laminates

The effect of curvature on the delamination R-curve behavior of composite unidirectional laminates is investigated in both the experimental and numerical manners. The flat and curved double cantilever beam specimens with different radii of curvatures are manufactured and tested subjected to mode I loading. A new data reduction method is developed for the curved specimens by adopting the Timoshenko curved beam theory. Experimental R-curves indicate that curvature has no effect on the initiation toughness, while it significantly affects the steady-state toughness and fiber bridging length. Variation of the steady-state toughness and fiber bridging length vs. The different curvatures is formulated for the curved specimens with R/h > 25 (R/h: the ratio of the radius of curvature to thickness). Finally, delamination propagation is simulated in the curved double cantilever beam specimens in commercial finite element software, ABAQUS, by applying both the virtual crack closure technique and cohesive zone model.

Stress resultants for wind girders in open-top cylindrical steel tanks

Empty open-top cylindrical steel tanks are susceptible to buckling when subjected to external pressure due to wind or partial vacuum due to blocked vents. A “wind girder” is commonly used at the top of the tank wall to increase its strength against external pressure instability. The wind pressure varies around the circumference of the tank, but is relatively constant up its height. A series of cosine functions is typically used to describe the variation of wind pressure around the circumference. Expressions for stress resultants, that form the basis of widely used design specifications such as API 650 and SH 3046, are based on simple mechanical models that adopt simplified pressure distributions and ignore interactions between the wind girder and the tank shell. Furthermore, in classical treatments a tributary height is postulated for wind loading on the girder, and this height is taken as independent of the properties of the girder and the shell. The purpose of this study is to develop a rational procedure to determine the stress resultants in a wind girder. Pursuant to this goal, Vlasov’s curved beam theory is used to derive the stress resultants and displacements for an isolated wind girder under a pressure distribution defined in terms of cosine functions. A parametric study employing finite element analysis is conducted to investigate the interaction of the wind girder with the tank shell. The stress resultants and the tributary height are found to be closely related to a shell-girder stiffness ratio that is devised in this study. This stiffness ratio was developed by considering the relative radial stiffness of the cylindrical shell and that of the wind girder. The changes in response quantities are expressed as functions of the shell-girder stiffness ratio. The developed expressions are presented in a form that is immediately useful for adoption into design standards.

Dynamic and instability analyses of FG graphene-reinforced sandwich deep curved nanobeams with viscoelastic core under magnetic field effect

Based on the nonlocal strain gradient theory, the effect of an axial magnetic field on the free vibration and mechanical buckling responses of an FG graphene-reinforced sandwich deep curved nanobeam with viscoelastic core embedded in a viscoelastic medium is elucidated in this paper. The curved beam is also assumed to be exposed to axial external compressions. The volume fraction of the constituents of the composite face layers are presumed to be functionally graded through the thickness according to a layer-wise law. The material properties are calculated in the framework of Halpin-Tsai micromechanical scheme. Lorentz magnetic force is deduced employing electro-dynamic Maxwell's relations for a conducting body. According to a refined shear and normal deformations curved beam theory, the motion equations are introduced in the polar coordinates. Analytical solutions are obtained for the natural frequencies and critical buckling load of the viscoelastic sandwich curved nanobeams. By comparing the present results with the available data in the literature, the developed formulations are validated. Furthermore, other several numerical examples are performed to show the effects of different parameters including viscoelastic core thickness, magnetic field, structural and foundation damping factor, opening angle and graphene concentration on the frequency and buckling load of the viscoelastic sandwich curved beams.

Theoretical and experimental investigation of stress distribution in a crane hook

Abstract Crane hooks are frequently used in material handling systems for grasping loads to be lifted to keep them from dropping, thus preventing catastrophic accidents in such situations. In this study, the response of crane hooks to their payload is investigated theoretically and experimentally. The approximate calculation method and curved beam theory are used for a theoretical calculation of maximum and minimum normal stress values occurring on crane hook. In addition, a computer aided model of a crane hook, adhering to DIN 15401 is used, and a stress analysis is performed by means of finite element simulation. Experimental data acquisition regarding crane hooks and their payload are conducted by means of strain gage sensors. The theoretical method solutions and the experimental results are compared. The results obtained are in good harmony with the experimental measurements, curved beam theory and finite element analysis.

Free vibration of stiffened laminated shells of revolution with a free-form meridian and general boundary conditions

The free vibration of stiffened composite laminated shells of revolution with classical and elastic boundaries is investigated by a semi analytical approach. The energy functionals of the shells and the stiffeners are derived separately. The energy functional of shells of revolution with a free-form meridian and general boundary conditions is formulated based on the first-order shear deformation theory (FSDT). The displacement field of the stiffeners is obtained based on that of the shell without involving extra degrees of freedom in the final mass and stiffness matrixes. Based on the three-dimensional curved beam theory, the energy functionals of the stiffeners in both meridian and circumferential directions are introduced in the same form. A modified Ritz approach is employed in this paper to solve the energy functionals. The results obtained by the proposed method show good agreements with those by the finite element method (FEM). Finally, parametric studies concerning some key parameters of stiffened shells are carried out.