Uniform Moment Research Articles

Stationary solutions to stochastic 3D Euler equations in Hölder space

We establish the existence of infinitely many global and stationary solutions in C(R,Cϑ) space for some ϑ>0 to the three-dimensional Euler equations driven by an additive stochastic forcing. The result is based on a new stochastic version of the convex integration method, incorporating the stochastic convex integration method developed in Hofmanová et al. (2022) and pathwise estimates to derive uniform moment estimates independent of time.

On the Expansion of Resolvents and the Integrated Density of States for Poisson Distributed Schrödinger Operators

We consider a Schrödinger operator with random potential distributed according to a Poisson process. We show that under a uniform moment bound expectations of matrix elements of the resolvent as well as the integrated density of states can be approximated to arbitrary precision in powers of the coupling constant. The expansion coefficients are given in terms of expectations obtained by Neumann expanding the potential around the free Laplacian. Our results are valid for arbitrary strength of the disorder parameter, including the small disorder regime.

Lateral-Torsional Buckling Modification Factors in Steel I-Shaped Members: Recommendations Using Energy-Based Formulations

Lateral torsional buckling (LTB) is of concern in long-span flexural members, particularly in the negative flexure regions of continuous-span, steel I-shaped members and during construction. While the elastic critical LTB capacity of a simply supported I-shaped member subjected to uniform moment has a closed-form solution, most LTB modification factors for beams subjected to moment gradients in the literature are empirical and work well only for specific loading and boundary conditions. This paper investigates the suitability of the different LTB modification factors in literature and design specifications for various loading and boundary conditions, accomplished via comparisons with analytical solutions using the Rayleigh-Ritz method and numerical solutions from finite element analyses. The analytical LTB modification factors are derived for doubly symmetric I-shaped members with different combinations of ideal flexural and torsional boundary conditions (simply supported and fixed) and subjected to different loading scenarios. The validity of the LTB modification factors determined using the Rayleigh-Ritz method and other formulae in the literature are also assessed for realistic intermediate restraint conditions, which are neither fully pinned nor fixed, by examining laterally continuous beams. Demonstrating that current design specifications for elastic critical LTB modifications are overly conservative

Behaviour, design, and reliability of monosymmetric I-beams at elevated temperature

Steel is the preferred choice among structural designers because of its exceptional properties compared to other available materials. However, steel is vulnerable to fire risk, as its material characteristics deteriorate rapidly at high temperatures. It results in the instability and failure of unsupported flexural components. In the present study, a numerical analysis is conducted to explore how load patterns and temperatures impact the failure characteristics of monosymmetric I-section beams. The numerical model is validated using existing experimental tests conducted at elevated temperatures. Subsequently, the validated model is employed for the numerical study. The analysis uses the ABAQUS software. The study is conducted for beam lengths ranging from 2.5 to 20 m, incorporating different degrees of monosymmetry. The behaviour of the beams is evaluated under steady and transient-state thermal conditions, using ISO 834 fire, considering uniform/non-uniform loading patterns on the beam. The results highlighted the shortcomings in Eurocode 3 provisions related to the failure patterns of monosymmetric I-beams, particularly for cases involving point loads and uniform moments. Subsequently, a proposed new methodology for calculating design capacity for monosymmetric beams is adopted from past literature and evaluated to ensure its reliability and accuracy in predicting monosymmetric beam capacities at elevated temperatures.

Analysis and verification of uniform moment exponential stability for stochastic hybrid systems with Poisson jump

Analysis and verification of uniform moment exponential stability for stochastic hybrid systems with Poisson jump

Flexural capacity and local buckling half-wavelength of high strength steel tubular beams under moment gradients: An experimental study

An experimental investigation into the effect of moment gradients on the flexural behaviour of hot-rolled high strength steel square hollow section (SHS) beams is presented in this paper. In total, 20 beam specimens in steel grades S690 and S770, and with cross-sections spanning from Class 2 to Class 4 based on the Eurocode 3 slenderness limits, were tested under three- and four-point bending. In the three-point bending tests, the beam spans were varied to achieve a range of moment gradients; the influence of different stiffening arrangements at the loading point was also considered. Local geometric imperfections were measured by means of 3D laser-scanning prior to testing and digital image correlation (DIC) was adopted to monitor the displacement and strain fields at critical regions and to assess the local buckling half-wavelengths of the test specimens for which a consistent measurement approach was proposed. The measured local buckling half-wavelengths were compared against the elastic local buckling half-wavelengths calculated using the finite strip method. It was observed that while the measured local buckling half-wavelengths remained approximately constant up to first yield, a significant reduction in half-wavelength was observed with increasing moment due to the non-uniform spread of plasticity. The comparisons also revealed that the local buckling half-wavelengths reduced with both the presence of moment gradients and intermediate stiffeners, with a new parameter proposed to quantify the local moment gradient. It was shown from the tests that the specimens subjected to moment gradients, despite the presence of shear, exhibited higher ultimate moment capacities (up to 10.5% for stiffened specimens and 3.4% for unstiffened specimens) than those subjected to uniform moments. This is attributed to the delay in the local buckling of the critical cross-section of beams under moment gradients due to the restraint provided by the less heavily stressed adjacent cross-sections.

Local buckling behaviour of high strength steel tubular beams under moment gradients

AbstractThis paper presents an experimental study into the effect of moment gradients on the local buckling behaviour of hot‐rolled high strength steel square hollow section (SHS) beams. The influence of moment gradients was investigated by testing 20 beam specimens, ranging from Class 2 to Class 4 according to the cross‐section classification limits given in prEN 1993‐1‐1:2020. Both four‐ and three‐point bending loading configurations were considered with varying shear span lengths. The influence of intermediate stiffeners at the loading point was examined by employing two different loading configurations in the three‐point bending tests, achieving fully stiffened or unstiffened conditions. Local geometric imperfections were measured by means of 3D laser scanning, and the displacement and strain fields at critical regions were monitored by digital image correlation (DIC) during testing. It was shown from the tests that specimens subjected to moment gradients exhibited higher ultimate moment capacities than those subjected to uniform moments.

Deflection of a Partially Composite Beam Considering the Effect of Shear Deformation

In this research, the deflection in the interface of a partially composite beam considering the effect of shear deformation is deter-mined. The system of beams is structured by two beams of prismatic sections, connected by an adhesive very thin and rigid, subjected to a uniform bending moment and a uniformly distributed load. The governing differential equation of the partially composite beam is obtained from the total functional energy that takes into consideration the shear deformation. The extreme moments creating second moments, shear forces and normal forces are applied to each beam. The differential equation is derived and then, compared to the one found in partial composite beams where the shear deformation is neglected. It is shown that the theoretical results of deflection with and without shear deformation are compared to each other and also with those found in the Timoshenko’s beam theory.

Multiple equilibrium states of a curved-sided hexagram: Part I—stability of states

The stability of the multiple equilibrium states of a hexagram ring with six curved sides is investigated. Each of the six segments is a rod having the same length and uniform natural curvature. These rods are bent uniformly in the plane of the hexagram into equal arcs of 120o or 240o and joined at a cusp where their ends meet to form a 1-loop planar ring. The 1-loop rings formed from 120o or 240o arcs are inversions of one another and they, in turn, can be folded into a 3-loop straight line configuration or a 3-loop ring with each loop in an “8” shape. Each of these four equilibrium states has a uniform bending moment. Two additional intriguing planar shapes, 6-circle hexagrams, with equilibrium states that are also uniform bending, are identified and analyzed for stability. Stability is lost when the natural curvature falls outside the upper and lower limits in the form of a bifurcation mode involving coupled out-of-plane deflection and torsion of the rod segments. Conditions for stability, or lack thereof, depend on the geometry of the rod cross-section as well as its natural curvature. Rods with circular and rectangular cross-sections will be analyzed using a specialized form of Kirchhoff rod theory, and properties will be detailed such that all four of the states of interest are mutually stable. Experimental demonstrations of the various states and their stability are presented. Part II presents numerical simulations of transitions between states using both rod theory and a three-dimensional finite element formulation, includes confirmation of the stability limits established in Part I, and presents additional experimental demonstrations and verifications.

Multiple equilibrium states of a curved-sided hexagram: Part II—Transitions between states

Curved-sided hexagrams with multiple equilibrium states have great potential in engineering applications such as foldable architectures, deployable aerospace structures, and shape-morphing soft robots. In Part I, the classical stability criterion based on energy variation was used to study the elastic stability of the curved-sided hexagram and identify the natural curvature range for the stability of each state with circular and rectangular rod cross-sections. Here, we combine a multi-segment Kirchhoff rod model, finite element simulations, and experiments to investigate the transitions between four basic equilibrium states of the curved-sided hexagram. The four equilibrium states, namely the star hexagram, the daisy hexagram, the 3-loop line, and the 3-loop "8", carry uniform bending moments in their initial states, and the magnitudes of these moments depend on the natural curvatures and their initial curvatures. Transitions between these equilibrium states are triggered by applying bending loads at their corners or edges. It is found that transitions between the stable equilibrium states of the curved-sided hexagram are influenced by both the natural curvature and the loading position. Within a specific natural curvature range, the star hexagram, the daisy hexagram, and the 3-loop "8" can transform among one another by bending at different positions. Based on these findings, we identify the natural curvature range and loading conditions to achieve transition among these three equilibrium states plus a folded 3-loop line state for one specific ring having a rectangular cross-section with a height to thickness ratio of 4. The results obtained in this part also validate the elastic stability range of the four equilibrium states of the curved-sided hexagram in Part I. We envision that the present work could provide a new perspective for the design of multi-functional deployable and foldable structures.

Strong convergence for an explicit fully‐discrete finite element approximation of the Cahn‐Hillard‐Cook equation with additive noise

AbstractIn this paper, we consider an explicit fully‐discrete approximation of the Cahn–Hilliard–Cook (CHC) equation with additive noise, performed by a standard finite element method in space and a kind of nonlinearity‐tamed Euler scheme in time. The main result in this paper establishes strong convergence rates of the proposed scheme. The key ingredient in the proof of our main result is to employ uniform moment bounds for the numerical approximations. To the best of our knowledge, the main contribution of this work is the first result in the literature which establishes strong convergence for an explicit fully‐discrete finite element approximation of the CHC equation. Finally, numerical results are finally reported to confirm the previous theoretical findings.

Computational model for the semi‐analytical assessment of the free‐edge effect in composite laminated shells

AbstractThe development of efficient computational models for the accurate prediction of the state variables in general composite laminated shells undergoing uniform edge loadings is a major challenge, especially when stress concentration phenomena such as the free‐edge effect have to be considered. This paper addresses this issue by introducing a higher‐order semi‐analytical approach for the assessment of the three‐dimensional stress field in circular cylindrical composite shells with arbitrary layups subjected to a uniform bending moment. The presented semi‐analytical approach combines a closed‐form analytical plane‐strain solution with a higher‐order layerwise approach, and the governing equations are derived by virtue of the principle of minimum elastic potential. The resulting system of coupled ordinary differential equations is then solved by means of the state‐space approach and the free constants are determined by evaluating the boundary conditions at the traction‐free edges. A comparison of the numerical results of the presented semi‐analytical method with finite element simulations for various composite laminates indicates that the developed method works with high accuracy, although being extremely efficient in terms of computational resources.

Ferromagnetic Crossover within the Ferromagnetic Order of U7Te12

We investigate the physical properties of a single crystal of uranium telluride U$_{7}$Te$_{12}$. We have confirmed that U$_{7}$Te$_{12}$ crystallizes in the hexagonal structure with three nonequivalent crystallographic uranium sites. The paramagnetic moments are estimated to be approximately 1 $\mu_{\rm B}$ per the uranium site, assuming a uniform moment on all the sites. A ferromagnetic phase transition occurs at $T_{\rm C}=48$ K, where the in-plane magnetization increases sharply, whereas the out-of-plane component does not increase significantly. With decreasing temperature further below $T_{\rm C}$ under field-cooling conditions, the out-of-plane component increases rapidly around $T^{\star}=26$ K. In contrast, the in-plane component hardly changes at $T^{\star}$. Specific heat measurement indicates no $\lambda$-type anomaly around $T^{\star}$, so this is a cross-over suggesting a reorientation of the ordering moments or successive magnetic ordering on the part of the multiple uranium sites.

An adaptive splitting method for the Cox-Ingersoll-Ross process

We propose a new splitting method for strong numerical solution of the Cox-Ingersoll-Ross model. For this method, applied over both deterministic and adaptive random meshes, we prove a uniform moment bound and strong error results of order 1/4 in L1 and L2 for the parameter regime κθ>σ2. We then extend the new method to cover all parameter values by introducing a soft zero region (where the deterministic flow determines the approximation) giving a hybrid type method to deal with the reflecting boundary. From numerical simulations we observe a rate of order 1 when κθ>σ2 rather than 1/4. Asymptotically, for large noise, we observe that the rates of convergence decrease similarly to those of other schemes but that the proposed method making use of adaptive timestepping displays smaller error constants.

Influence of moment modification factor in lateral torsional buckling of doubly and monosymmetric sections

Simply supported doubly symmetric prismatic beam subjected to uniform moment along the unsupported length is the simplest and basic case in the buckling analysis. The closed form solution for the elastic critical moment was derived based on the assumptions of perfect beam with standard restraint conditions at both ends of the beam. A perfect beam will be having linear-elastic behaviour without any type of geometrical or material imperfections. The standard conditions of restraint at each end of the beam in basic case are restrained against lateral movement, restrained against rotation about the longitudinal axis and free to rotate on plan. But in reality, loading conditions and conditions of restraint will vary widely from the basic case. An equivalent uniform moment factor is typically applied to elastic critical moment to account for the effects of variable moment and end restraints along an unbraced girder length. The values are derived numerically and varies with specifications. In this article equivalent uniform moment factor for IS 800-2007 and Eurocode 3 provisions are compared. The study is also extended to monosymmetric I sections.

Limit behaviors of regression function estimator with martingale difference errors

In the present paper, we consider the nonparametric regression model Y i = g ( x i ) + ε i , 1 ≤ i ≤ n , and study the strong consistency and uniform strong consistency of the estimator of Priestley and Chao based on the martingale difference errors. These results partially extend and improve some known works. Furthermore, the moment convergency and uniform moment convergency of the estimator are established also.

Process of Straightening by Three-Point and Four-Point Bending for Curved Brass Rack

Racks are typically curved after cutting their gear teeth, and a straightening process is required to correct the distortion due to machining. In this study, key factors in the straightening of curved racks by three-point and four-point bending are examined with to automate the correction. The relationship between load and deformation is plotted in real time to determine the unloading point to correct the target deflection for straightening. The parameters constituting the above-mentioned relationship are important for achieving precise correction. The load and deflection at the central loading point are known to be suitable parameters for three-point bending. The smaller the deflection required for correction, the higher is the precision of the displacement sensor required for three-point bending. In the case of four-point bending, the bending moment and bending angle should be selected. In addition, a four-point bending jig is required to load the uniform bending moment during the correction. A modified four-point bending jig is made and the effectiveness was examined.

Estimating the characteristics of stochastic damping Hamiltonian systems from continuous observations

We consider nonparametric invariant density and drift estimation for a class of multidimensional degenerate resp. hypoelliptic diffusion processes, so-called stochastic damping Hamiltonian systems or kinetic diffusions, under anisotropic smoothness assumptions on the unknown functions. The analysis is based on continuous observations of the process, and the estimators’ performance is measured in terms of the sup-norm loss. Regarding invariant density estimation, we obtain highly nonclassical results for the rate of convergence, which reflect the inhomogeneous variance structure of the process. Concerning estimation of the drift vector, we suggest both non-adaptive and fully data-driven procedures. All of the aforementioned results strongly rely on tight uniform moment bounds for empirical processes associated to deterministic and stochastic integrals of the investigated process, which are also proven in this paper.

Elastic lateral-torsional buckling of beams with warping restraints at supports

Analytical formulations for the evaluation of the critical bending moment for lateral-torsional buckling of a beam taking into account warping restraints at supports are proposed in this paper. A theoretically exact solution is firstly derived from differential equilibrium equations using infinite power series for the twist rotation. Next, the deflected shape is approximated using a single or several base functions for the displacement and rotation fields. The energy method is then used to derive approximate expressions of the critical bending moment of beams subjected to various loading conditions: constant/linear bending moment distributions, mid-span pointwise transverse force or uniformly distributed load. The warping coefficient kw does not depend on the bending moment diagram. The equivalent uniform moment factor C1 can be conveniently expressed as a product of two coefficients, one related to the bending moment distribution and the other to the warping restraints stiffness. Finite Element analyses of beams with warping restraints at supports have been performed considering a shell elements model created in ANSYS and a beam elements model developed in LTBeamN.

A three‐phase laminated anisotropic elliptical inhomogeneity with internal uniform stress resultants

AbstractWe use the Stroh octet formalism for the coupled stretching and bending deformations of Kirchhoff laminated anisotropic thin plates together with the complex variable formulations of the decoupled stretching and bending deformations of Kirchhoff homogeneous isotropic thin plates to study the internal stress resultants inside a laminated anisotropic elliptical inhomogeneity bonded to a surrounding infinite laminated anisotropic matrix through a confocal interphase layer composed of a homogeneous isotropic plate when the matrix is subjected to uniform remote membrane stress resultants and bending moments. We prove that for given geometric and material parameters of the composite, the internal stress resultants inside the elliptical inhomogeneity are uniform when four specific conditions on the remote loading are satisfied. Furthermore, the internal uniform elastic field is determined in real‐form in terms of the two 8 × 8 fundamental elastic plate matrices for both the inhomogeneity and the matrix and the three 4 × 4 real matrices S, H, and L for the matrix.