Pure Bending Beam Research Articles

New four-point bending test protocol for the strength grading of timber: relationship between local and global modulus of elasticity (MOE) of maritime pine

ABSTRACT This study focuses on the characterisation of maritime pine according to standards EN408 and EN384 required for timber grading. A comparison of the EN408 bending test setup is performed with a new four-point bending protocol, reducing as much as possible, the effects of the shear force in the span where the deflection is measured. The study highlights difficulties to estimate accurately, the elasticity modulus of wood by using EN408 bending test setup. This is probably due to inaccuracies in the measurements of small curvatures and due to the local heterogeneities of wood. An original four-point bending test is proposed with 75% of the span of the wood beams in pure bending. This configuration clearly exhibits the convergence of measurements, in comparison to EN408 setup. The deformations generated by this new loading protocol allow a better evaluation of the bending modulus E 0. Moreover, the study reveals that the proposed setup supplies an increase of elasticity modulus close to 20%. In addition, dynamic tests are also performed to characterise the Young’s modulus. The experimental results reveal a better agreement between static test, using the new four-point bending configuration and dynamic method, than considering EN408 as a reference.

Axially Non-Uniform Properties Effected Electro-Elastic Fields in Bended Piezoelectric Semiconductor Beams

This paper studied the effects of axially non-uniform material and non-uniform doping on the electro-elastic fields in bent piezoelectric semiconductor beams. The mathematical model is established based on the phenomenological theory of piezoelectric semiconductor and Euler’s beam theory. To solve the governing equations with variable coefficients, the differential quadrature method is adopted. As two representative structures, the pure bending beam and cantilever beam are studied, respectively. From the calculated results, it can be found the varied material properties changed the effective stiffness, as a result, all the field quantities are altered. Meanwhile, the symmetry of all field quantities in a pure bending beam is broken. Considering different Gaussian doping parameters [Formula: see text] and [Formula: see text], it can be observed that the distribution rule of perturbation carrier density becomes complex. The increment of [Formula: see text] broadens the width of optimum range-producing carriers, however, the increment of [Formula: see text] describes the moving of the optimum range from left to right. Importantly, the introduction of non-uniform doping breaks the limitation of perturbation carrier density in a cantilever beam with uniform material properties, the maximum value of perturbation carrier density does not appear at the fixed end only. The obtained results could be the guidance in designing high-performance electric devices.

Free vibration and buckling of functional graded box beams and the effect of the cross-sectional warping shape

Box beams have been used widely in engineering fields. This paper proposes a higher-order shear deformation beam theory to investigate the transverse vibration and stability of box beams in pure bending, especially free vibration and buckling of functionally graded box beams. A warping shape function is constructed for the first time to take shear deformation and rotary inertia of the hollow-core rectangular cross section into account simultaneously during bending. The governing equation and boundary conditions are derived. The frequency equations for typical end supports are given. The characteristic equations for buckling and critical loads are determined. Numerical results are evaluated. Besides its simplicity and easy-to-implement, the proposed theory has sufficient accuracy on the mechanical behavior of the box beams. By comparing the obtained results with the simulation results based on three-dimensional finite element method, the accuracy of the numerical results obtained is verified. The effects of parameters such as aspect ratio, gradient index, and wall thickness on the natural frequencies and critical loads are explored in detail. When box beams become rectangular beams, our model reduces to the well-known Levinson beam model.

On Strain Gradient Theory and Its Application in Bending of Beam

The general strain gradient theory of Mindlin is re-visited on the basis of a new set of higher-order metrics, which includes dilatation gradient, deviatoric stretch gradient, symmetric rotation gradient and curvature. A strain gradient bending theory for plane-strain beams is proposed based on the present strain gradient theory. The stress resultants are re-defined and the corresponding equilibrium equations and boundary conditions are derived for beams. The semi-inverse solution for a pure bending beam is obtained and the influence of the Poisson’s effect and strain gradient components on bending rigidity is investigated. As a contrast, the solution of the Bernoulli–Euler beam is also presented. The results demonstrate that when Poisson’s effect is ignored, the result of the plane-strain beam is consistent with that of the Bernoulli–Euler beam in the couple stress theory. While for the strain gradient theory, the bending rigidity of a plane-strain beam ignoring the Poisson’s effect is smaller than that of the Bernoulli–Euler beam due to the influence of the dilatation gradient and the deviatoric stretch gradient along the thickness direction of the beam. In addition, the influence of a strain gradient along the length direction on a bending rigidity is negligible.

A new phenomenological constitutive model for shape memory alloys

A new thermomechanical constitutive modeling approach for shape memory alloys (SMAs) that undergo a martensite to austenite phase transformation and the associated pseudoelastic and shape-memory responses is presented. The novelty of this new formulation is that a single transformation surface is implemented in order to capture the forward and reverse phase transformations, as well as the reorientation and detwinning of the martensite phase. The framework is akin to the usual flow theory plasticity with kinematic hardening, however in addition to the transformation strain there is also a transformation entropy that is directly related to the martensite volume fraction appearing in prior theories. A transformation surface in effective stress and effective temperature space is introduced and an associated flow rule governs the evolution of the transformation strain and entropy. In order to capture the multitude of SMA behaviors, a transformation potential function is introduced in transformation strain and entropy space for the derivation of the back stresses and back temperatures that define the kinematic hardening behavior. It is this potential function that governs all of the important behaviors within the model. After the description of the general theory, specific forms for the transformation surface and the transformation potential are devised and results for the behaviors captured by the model are provided for a range of thermomechanical loadings. Finally, the model is implemented in the finite element method and applied to the pseudoelastic and shape memory effects of a beam in pure bending.

A review of mechanical properties and constitutive theory of rock mass anisotropy

Rock anisotropy is an intrinsic property of natural rock mass, and layered rock has the most significant effect on the stress distribution and deformation of a rock mass. With the development of rock mechanics theory and constitutive theory, the study of rock anisotropy has become one of the focuses and hotspots in the field of rock mechanics. Based on the rock isotropic self-consistent theory, the rock anisotropy theory of different media such as triclinic anisotropy, monoclinic anisotropy, and orthorhombic anisotropy has been greatly developed. Rock mechanical property test, not only limited to the traditional single-axis, triaxial compression test, Brazilian split test and pure bending beam tensile test, and meso-structural characteristic test (XRD, ultrasonic test, etc.), is also widely used. With the development of numerical analysis theory and computer technology, numerical analysis methods such as finite difference method, discrete element method, finite element method, boundary element method, and interface element method have played an important role in the study of rock anisotropy characteristics. According to the assumptions and reasoning methods of anisotropic failure criterion, it can be classified into two categories: continuous failure criterion and discontinuous piecewise failure criterion. Continuous medium failure criterion can also be classified into mathematical and empirical ones. Because of the complexity of rock anisotropy, the research on yield criterion and constitutive relation of rock anisotropy is still in the stage of theoretical research, and it is seldom applied in actual rock engineering. The physics model of rock has an important influence on the study of the anisotropic characteristics of rock. Therefore, the establishment of an appropriate rock physics model has a direct impact on the accurate determination of rock anisotropy parameters and P-S wave velocity. Finally, the key and difficult points of anisotropic rock research and the direction of future research are discussed and analyzed to provide a reference for engineering practice and related problems.

Asymptotic dynamic modeling and response of hysteretic nanostructured beams

The nonlinear dynamic response of carbon nanotube (CNT)/polymer nanocomposite beams to harmonic base excitations is investigated asymptotically via the method of multiple scales. The hysteresis associated with the CNT/polymer interfacial frictional sliding is described by a 3D mesoscopic theory reduced via a uniaxial strain assumption for a beam in pure plane bending. Such reduction leads to a Bouc–Wen-like hysteretic moment–curvature relationship. The generalized memory-dependent constitutive law is developed asymptotically and, subsequently, introduced in two archetypal cases of nonlinear beam models. A beam model is tailored for axially restrained, extensible beams (e.g., hinged–hinged beams) for which the dominant geometric nonlinearity is associated with the multiplicative effect of the tension with the bending curvature. The second model is valid for inextensible beams (e.g., cantilever beams) dominated by inertia and curvature nonlinearities. The piece-wise integration of the moment–curvature relationship yields an exponential law which is treated asymptotically to obtain the quadratic and cubic curvature contributions. The ensuing asymptotic equations of motion in the unknown deflection field are discretized according to the Galerkin method employing the eigenmode directly excited near its primary resonance to thus obtain a piece-wise reduced-order model (ROM). The method of multiple scales applied to the ROM yields the asymptotic response together with the frequency response functions for the lowest mode. A parametric study unfolds rich nonlinear dynamic responses in terms of behavior charts highlighting regions of hardening and softening behavior, regions of single-valued stable behavior and regions of multi-valued multi-stable behavior. Such richness of responses is caused by the unusual and unique combination of material and geometric nonlinearities.

Evaluating the Effect of the Compressive Strength Development Factor on Concrete Creep Deformation in Bridge Construction

Abstract Excessive midspan deflection is often observed in large-span prestressed concrete girder bridges. In particular, the deformation caused by shrinkage and creep of concrete is an important part of deflection that often exceeds expectations. To achieve creep deformation control, extending the curing time and postponing the loading age are frequently adopted during construction. This article proposed an approach based on creep experiments with prismatic and beam specimens and viscoelastic model-based finite element analyses to evaluate the effect of the concrete strength development factor on the ultimate creep coefficient in the Comité Euro-International du Béton-Fédération internationale de la précontrainte (CEB-FIP) model. The concrete strength development factor ξcc(t/tu) is defined as the ratio of the mean compressive strength at the loading time to the maximum mean compressive strength obtained in the test. The strategies for targeted regulation and control of ξcc(t/tu) were discussed by establishing an artificial neural networks model for strength prediction from a database provided in our previous work. Uniaxial compressive prismatic specimens and pure bending beam specimens with a span of 5.0 m were used in the experiment for creep behavior observation. A viscoelastic finite element analysis (FEA) model was established based on solidification theory. The FEA model results were verified with measured data for capturing the creep behavior under ambient conditions to obtain reliable long-term creep deformation predictions. Finally, the proposed method was validated with the Xincheng bridge construction for the optimistic loading age determination, and the results indicated good feasibility in girder deflection control.

Quadrilateral 2D linked-interpolation finite elements for micropolar continuum

Quadrilateral finite elements for linear micropolar continuum theory are developed using linked interpolation. In order to satisfy convergence criteria, the newly presented finite elements are modified using the Petrov–Galerkin method in which different interpolation is used for the test and trial functions. The elements are tested through four numerical examples consisting of a set of patch tests, a cantilever beam in pure bending and a stress concentration problem and compared with the analytical solution and quadrilateral micropolar finite elements with standard Lagrangian interpolation. In the higher-order patch test, the performance of the first-order element is significantly improved. However, since the problems analysed are already describable with quadratic polynomials, the enhancement due to linked interpolation for higher-order elements could not be highlighted. All the presented elements also faithfully reproduce the micropolar effects in the stress concentration analysis, but the enhancement here is negligible with respect to standard Lagrangian elements, since the higher-order polynomials in this example are not needed.

Buckling of beams with finite prebuckling deformation

The prebuckling deformation of structures is usually very small in conventional concepts, and is always neglected in the conventional buckling theory (CBT) and numerical method (CNM). In this paper, we find a class of structures from the emerging field of stretchable electronics, of which the prebuckling deformation becomes large and essential for determining the critical buckling load. Although great progress has been made for the buckling theory in the past hundred years, it is still challenging to analyze the buckling problems with finite prebuckling deformation (FPD buckling) straightforwardly. Here, the experimental stretch of a series of serpentine interconnects was firstly conducted as a representative example to show the FPD buckling behaviors and inapplicability of the CBT and CNM. The CNM can yield a huge error of 50% on the critical buckling load for the case with thickness-to-width ratio of the cross section h/b = 0.6. Most importantly, a systematic and straightforward theory (FPD buckling theory) is developed to analyze the FPD buckling behaviors of beams with the coupling of bending, twist and stretch/compression. As a comparison, various theoretical and numerical methods are applied to three classic problems, including lateral buckling of a three-point-bending beam, lateral buckling of a pure bending beam and Euler buckling. Our FPD buckling theory for beams is able to give a good prediction, while the CBT (by Timoshenko et al.) and CNM (by commercial program packages) yield unacceptable results (with 70% error for a three-point-bending beam with h/b = 0.8, for example). Discussion on the FPD buckling of bulk structures is deferred to a following paper.

Evolution of the length scale in strain gradient plasticity

An equivalence is assumed between a microstructural length scale related to dislocation density and the constitutive length scale parameter in phenomenological strain gradient plasticity. An evolution law is formed on an incremental basis for the constitutive length scale parameter. Specific evolution equations are established through interpretations of the relation between changes in dislocation densities and increments in plastic strain and strain gradient. The length scale evolution has been implemented in a 2D-plane strain finite element method (FEM) code, which has been used to study a beam in pure bending. The main effect of the length scale evolution on the response of the beam is a decreased strain hardening, which in cases of small beam thicknesses even leads to a strain softening behavior. An intense plastic strain gradient may develop close to the neutral axis and can be interpreted as a pile-up of dislocations. The effects of the length scale evolution on the mechanical fields are compared with respect to the choice of length evolution equation.

Minimum energy based method to predict the multiple cracking pattern in quasi-brittle beam

A minimum energy based method is proposed to predict the evolution of multiple cracking pattern in a pure bending beam made of quasi-brittle material. The minimum energy is found from the energy landscape of the cracked beam calculated by the Finite Element (FE) Method, and the crack pattern corresponding to the minimum energy is taken as the crack pattern most likely to occur. The fracture behavior of the quasi-brittle material is simulated by cohesive zone model (CZM) using the bilinear tension softening law. After appropriate normalization based on dimensional analysis, the multiple cracking process is found to be governed by only three parameters and their effects are systematically studied. Physically, the ratio between the “crack-tip toughness” and the “bridging toughness” is important for the crack pattern evolution. As a validation, the proposed method is used to predict the crack pattern of fiber reinforced concrete beam and good agreement with experiment is achieved. The goal of this paper is twofold. First, with a systematic parametric study focusing on the constitutive law of quasi-brittle material, this investigation would facilitate material design or selection for crack control; Second, by quantitatively determining the crack pattern development during the whole multiple cracking process based on a sound physical principle, i.e., the commonly adopted minimum energy criterion, the result can serve as a benchmark for checking the reliability of other methods for multiple cracking analysis.

Finite Element Analysis’s Discussion of Steel-Concrete

To analyze the mechanical properties of steel tube filled with steel-reinforced concrete(STSRC), the mechanical models and some related problems of STSRC under different loading ways are proposed for the analysis on the base of finite element software: the concrete plastic damage constitutive model, the contact between steel and the treatment of boundary condition, etc. There are three types of specimen for analysis: short column, long column and pure bending beam. The results indicate that the mechanical models and the relevant technical analysis of STSRC are reasonable, and are beneficial to convergence. The discussed methods can provide a reference for the scholars to study on other composite steel-concrete structures.

Influence of skin wrinkles and resin insertions in maximum stress of transitions of pure bending sandwich beams

AbstractThe sandwich beam has emerged as a part of new engineering applications in recent years. Dimensional and mechanical control tests are used to avoid possible service failures. Sandwich beams are modeled as large deflection pure bending beams with monolithic ends. The manufacturing process of sandwich beams can lead to different dimensional and shape defects such as wrinkles in traction and compression skins, insertions of epoxy resin in foam core or in the skins, shape defects of the foam, etc. In order to study the maximum stress behavior of beams with different types of defects, a specific finite element model (FEM) has been developed. Finally, the results of FEM analysis are validated using experimental tests. The implications of these findings on the structural integrity of glass-fiber-reinforced polymer (GFRP)-foam sandwich composites are discussed.

A Large Rotation Matrix for Nonlinear Framed Structures, Part 2: Numerical Verification

In this paper, some numerical verifications would be presented and discussed, mainly including the following three types: (1) the pure bending beam in which the structural stiffness would maintain the original value and not change along with the load; (2) the clamped arc-beam in which the structural stiffness would decrease gradually with the increment of load and the structure would be buckling at some certain load value; and (3) the cantilever beam in which the structural stiffness would increase significantly with the increment of load. For all of the above examples, the present results are in good agreement with the analytical results and numerical results in other literatures, testifying and illustrating the validity of the large rotation matrix for nonlinear framed structure, which is developed in the part 1 of this paper.

Effective Shaping of Cold-Formed Thin-Walled Built-up Beams in Pure Bending

Cold-formed thin-walled beams with an open cross section are widely used in different branches of industry. They tend to buckle in a variety of modes, which can be described as local, distortional, lateral-torsional buckling and bending. In this paper, the investigation on built-up I-sections, formed by assembling the stiffened cold-formed channel sections, with or without edge stiffener is presented. Parametric studies are carried out for five different section geometries with 16 specimens, tested under two point loading, to identify factors affecting the flexural strength of stiffened cold-formed steel built-up sections. The section geometries are chosen such that all types of buckling modes are met with. Critical load obtained from experiments is compared with finite element analysis results. Finally, it discusses the suitability of the current design methods (American Iron and Steel Institute) and provides design recommendations for stiffened cold-formed built-up sections.

Finite Element Analysis of Material Mechanics Experiment

The finite element analysis is used for general experimental mechanics of materials. By analyzing numerical value of finite element in ANSYS software platforms, the corresponding stress, strain data and deformation diagram will be obtained. This lays a foundation for academic research of experiment. This paper compares the conventional electric method with finite element analysis by the normal-stress distribution of pure bending beams experiment. The comparing shows that the experimental results of the two methods are uniformity. In this paper, the study provides an effective way to experimental mechanics of innovative research.

Bending creep of beams in aggressive media

The effect of an aggressive medium on high-temperature creep and creep-rupture characteristics of beams in pure bending is considered. Creep modeling is based on the Rabotnov kinetic theory that involves two structural parameters, i.e., damage and the concentration of the environmental chemical elements in a beam. The problem of beam bending allows for the difference in the material creep processes in tension and compression. The novelty of the problem under consideration is the use of the governing and kinetic equations in the form of singular fractional power dependencies of the creep rate and damage accumulation rate on the stress; instantaneous elastic deformations are neglected because they are low compared to creep deformations. The singularity makes it possible to allow for nonlinear viscosity together with instantaneous fracture characteristics.

The discussion on optimization models of pure bending beam

This article is devoted to shape optimization design of pure bending beams under single loading condition. Compliance minimization with material volume constraint, the maximum stress minimization problem, and the maximum displacement are considered. In the case of trusses, it has been shown that the former two problems have the same optimal topology. The possibility of extending this result for pure bending beam problems is examined in the present work. First, the comparison of the optimum design results between the maximum displacement, the conventional mean compliance, and the maximum stress is carried out by an example of optimal cross-sectional design of a continuous beam. Then, geometric average displacement (GAD) is introduced in optimization models of linearly elastic structures. The elevated accuracy in results achieved with GAD is shown in this article.

The Research of Pure Bending Flanged Beam’s Stress Distribution in Civil Engineering

Experimental method is an important means to research the mechanical properties of materials in civil engineering. In this paper, first, analyzing the section of pure bending beam’s normal-stress distribution, drafting the measure scheme of flanged beam’s pure bending normal-stress, arranging measuring point in different place about the section of pure bending, adopting multi-function composite test-bed, static strain gauge and other testing instrument, by step loading, to finish the measure of each measuring points’ stress and strain, also, analysis and treatment for test data. According to the result of analysis and treatment, pointing out that measured stress and strain of pure bending beam is proportional to loading. The measured normal-stress distribution of the pure bend beam is uniform with theoretical analysis, which is satisfy the engineering demand. Knowing and mastering multipoint measurement technology and method is an effective means for the scientific research.