Torsion Problem Research Articles

The asymptotic solution of the elasticity theory problem for a radially inhomogeneous cylinder

The elasticity theory problem for a radially inhomogeneous cylinder of small thickness, whose elastic moduli are arbitrary continuous functions depending on the radius of the cylinder, is considered. It is assumed that the side surface of the cylinder is stress-free, and the boundary conditions that keep the cylinder in equilibrium are given at its seats. Asymptotic solutions are constructed by the asymptotic integration method. It is shown that the asymptotic solution consists of the sum of the penetrating solution, simple boundary effect and boundary layer solutions. The character of the stress-strain state corresponding to the penetrating solution, simple boundary effect and boundary layer solutions is determined. Asymptotic formulas are obtained for displacements and stresses, which allow to calculate the stress-strain state of a cylinder.The problem of torsion of a radially inhomogeneous cylinder is studied, with its lateral surface free from stress and the boundary conditions keeping it in equilibrium at its seats. By applying the asymptotic integration method, it is determined that the asymptotic solution for the torsion problem consists of the sum of the penetrating solution and boundary layer solutions.Numerical analysis is performed and the effect of material inhomogeneity on the stress-strain state of the cylinder is evaluated.

Construction of an inhomogeneous solution for the problem of torsion of a radially inhomogeneous transversal isotropic cylinder

Using the method of asymptotic integration of the equations of elasticity theory, inhomogeneous solutions are constructed for the problem of torsion of a radially inhomogeneous transversally isotropic cylinder of small thickness. It is considered that the elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. An algorithm is given for constructing exact partial solutions of equilibrium equations for special types of loads, the lateral surface of the cylinder is loaded with forces that depend polynomially on the axial coordinate. Based on the solutions obtained, it is possible to evaluate the areas of applicability of applied theories for radially inhomogeneous cylindrical shells.

Triaxial Load Cell for Ergonomic Risk Assessment: A Study Case of Applied Force of Thumb

To assess the ergonomic risk level in work systems involving tasks performed with hands or fingers, it is necessary to know the exerted triaxial forces. To address this need, a prototype of a triaxial load cell based on principles of linear elasticity theory and mechanical problems of torsion, bending and axial load is presented. This work includes an analytical strain model for each instrumented point and its solution regarding the applied force to a triaxial load cell. The proposed load cell was calibrated and validated by performing different static experimental tests. As a case study, the applied force in three directions while the thumb activates a cigarette lighter was measured. Triaxial forces and resultant forces were obtained and compared with the parameter of 10 N established by the ergonomic standards as reference values for pressing down with the thumb, finding that the applied forces in eight tests were 23.73 N, 43.51 N, 12.69 N, 14.50 N 20.35 N, 21.67 N, 39.74 N and 46.02 N, exceeding the reference values and establishing a direct relationship with Quervain syndrome. In conclusion, the developed load cell is a valid and reliable alternative to measure many forces that cannot be obtained with commercial devices, allowing the level of ergonomic risk to be determined with great precision.

On the homogeneous torsion problem for heterogeneous and orthotropic cross-sections: Theoretical and numerical aspects

For many years, torsion of arbitrary cross-sections has been a subject of numerous investigations from theoretical and numerical points of view. As it is well known, the resulting boundary value problem (BVP) governing such phenomenon happens to be a pure Neumann BVP and, therefore, its solutions are determined up to a constant. Among a large plethora of finite element method (FEM) techniques that can be used in this context, most of FEM practitioners resolve this uniqueness issue by fixing the candidate solution to a node of the domain. Although such popular and pinpointing technique is widely spread and works well for practical purposes, it does not have a continuous counterpart and therefore its justification remains a matter of debate. Hence, this self-contained work aims to address the modeling of arbitrary heterogeneous and orthotropic cross-sections as well as the theoretical and numerical aspects of their solutions. In particular, we discuss the existence of weak solutions, well-posedness, regularity of solutions, and convergence of Galerkin’s method for different variational settings (with special focus on a regularized variational approach). Moreover, we establish a connection, at a discrete level, between the convergence of solutions of well-posed variational settings and those solutions coming from the usual practice of fixing a datum at a node. Finally, we discuss some numerical aspects of all the FEM discrete formulations proposed here by performing convergence analysis in L2 and H1 norms. The section of numerical results is closed by presenting a series of study cases ranging from a square cross-section composed of two different materials to an isotropic bridge cross-section for which no analytical solution exists.

The torsion problem of the [formula omitted]-Bilaplacian

For each bounded and open set Ω⊂RN (N≥2) with smooth boundary denoted by ∂Ω and each real number p∈(1,∞) we analyze the torsion problem of the p-Bilaplacian, namely Δ(|Δu|p−2Δu)=1 in Ω with u=Δu=0 on ∂Ω. Firstly, we show that for each p∈(1,∞) the problem has a unique weak solution up. Secondly, we prove that up converges uniformly, as p→∞, in C1(Ω¯) to a certain function, say v2, which is exactly the unique solution of the problem −Δu=1 in Ω with u=0 on ∂Ω. Moreover, for each real number q∈[1,∞), Δup converges strongly to Δv2 in Lq(Ω), as p→∞. Next, we show that each solution up is also a solution for the minimization problem T(p;Ω)≔infu∈Xp(Ω)∖{0}1|Ω|∫Ω|Δu|pdx1|Ω|∫Ωudxp where Xp(Ω)≔{u∈W2,p(Ω)∩W01,p(Ω):u(x)≥0,a.e.x∈Ω}. Further, we show that the function (1,∞)∋p↦T(p;Ω) is strictly increasing provided that Ω is a convex and bounded open set for which |Ω|−1∫Ωv2dx is small. Finally, using this monotonicity result, we give an alternative variational characterization of the constant T(p;Ω) when |Ω|−1∫Ωv2dx is small. That last variational characterization fails to hold true when |Ω|−1∫Ωv2dx>1.

Construction of homogeneous solutions of the torsion problem for a radially inhomogeneous transversely isotropic cylinder

The torsion problem for a radially inhomogeneous transversely isotropic cylinder of small thickness was investigated by the method of asymptotic integration of elasticity theory equations. It is assumed that the side part of the cylinder is stress-free, and boundary conditions are set at the ends of the cylinder, leaving the cylinder in equilibrium. The elastic moduli are thought to be arbitrary continuous functions of the variable along the cylinder radius. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thin-walledness of the cylinder. Homogeneous solutions are built, i.e. any solutions of the equilibrium equation satisfying the condition of no stresses on the side surfaces. It is shown that the solution of the torsion problem consists of a penetrating solution and a boundary layer character solution similar to Saint-Venant's edge effect in the theory of inhomogeneous plates. The penetrating solution determines the internal stress-strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the torsional moments of stresses acting in the cross-section perpendicular to the cylinder axis. Solutions having the boundary layer character are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in applied shell theories. Asymptotic formulas for displacement and stresses are built, which make it possible to calculate the three-dimensional stress-strain state of a radially inhomogeneous transversely isotropic cylinder of small thickness. Based on the obtained asymptotic expansions, it is possible to assess the applicability of applied theories and build a refined applied theory for radially inhomogeneous cylindrical shells

Galerkin-Kantorovich variational method for solving saint venant torsion problems of rectangular bars

The unrestrained torsional analysis of bars is an important theme in elasticity theory, first solved by Saint-Venant using semi-inverse methods. It has been considered and solved by several others using analytical methods and numerical procedures due to the importance in the design of machine parts under torsional moments. In this paper, the Saint Venant torsion problem is solved for rectangular prismatic bars using Galerkin-Kantorovich variational method (GKVM). The work presents a detailed theoretical framework of the problem, deriving using first principles considerations the stress compatibility equation in terms of the Prandtl stress function ϕ(x,y). The derived domain equation which is required to be satisfied over the rectangular cross-sectional domain is a partial differential equation of the Poisson type. GKVM is adopted as the solution method for finding the solution to the domain equation. The unknown Prandtl stress function ϕ(x,y) is assumed, following Kantorovich method to be a product of an unknown function for fx sought to minimize the Galerkin-Kantorovich variational functional (integral) (GKVF) and a known function (y2-b2) which satisfies the boundary conditions at all boundary points in the y-direction, that is, at y=±b. The resulting GKVF is a simplified functional whose integral is a second order inhomogeneous ordinary differential equation (ODE) in fx. The integrand is solved to find fx leading in a full determination of the Prandtl stress function. The expression for stresses, torsional moments and torsional parameters are then found and they satisfy the boundary conditions and the domain equation. The results for the torsional moments and torsional parameters are identical to previous results obtained using double finite sine transform method (DFSTM), and analytical methods. The merit of GKVM is that it has led to the exact solution of the unrestrained torsion problems.

Solving Saint Venant torsion problems for rectangular beams using single finite Fourier sine transform method

This research presents the single Fourier sine transform method (SFSTM) for solving the Saint Venant torsion problem of rectangular prismatic bars. The problem is a common theme in the theory of elasticity of unrestrained torsion which was previously expressed by Prandtl using Prandtl stress functions ϕ(x,y) as a Poisson type nonhomogeneous partial differential equation (PDE) called the stress compatibility equation. In this work the SFSTM was applied to the stress compatibility equation, converting the PDE to an easier to solve ordinary differential equation (ODE) in the transformed domain. The boundary conditions were used to find the integration constant and inversion was used to find the solution in the physical domain. The non vanishing stresses and torsional moments were thus found as a single series of infinite terms with rapid convergence. The maximum stresses and moments were found in standard form in terms of torsional parameters which were tabulated for various ratios of the cross-sectional dimensions. A comparison of the torsional parameters with previous results show that the present results are identical with previous results illustrating the accuracy of the SFSTM used. The sine kernel of the SFSTM satisfies the boundary conditions of the problem and contributed to the exact solution obtained. The SFSTM simplified the PDE to an ODE which is simpler to solve.

Torsional vibration behavior of a restrained non-circular nanowire in an elastic matrix

This study introduces an approach to analyze torsional vibration in non-circular nanowires within magnetic fields, considering various boundary conditions on an elastic foundation. Analytical formulas for natural angular frequencies are obtained using nonlocal strain gradient theory. The analysis covers three non-circular cross-sections, incorporating the warping effect. Elastic springs at the wire ends simulate support conditions, restricting rotation around the wire’s axis. The torsion function around the axis is represented by a Fourier series, discretized at the spring points and linked using the Stokes’ transform alongside the boundary values. This leads to an eigenvalue problem that includes higher-order material size parameters (strain gradient, nonlocal), spring parameters, and the warping function. The study’s novelty lies in effectively solving torsional vibration for non-circular sections, addressing warping function, elastic medium, and size effects under both deformable and non-deformable boundary conditions. The presented solution is capable of determining vibration frequencies for both rigid and deformable boundary conditions. This is accomplished by specifying the torsional spring stiffness values, thereby obviating the necessity for additional recalculations. In order to verify the obtained results and compare them with the existing literature, nanowires with free and clamped boundary conditions are solved numerically by changing the spring parameters in the eigenvalue problem. In the formulation of natural angular frequency; length scale parameters, warping, magnetic field, elastic medium effects are included. In addition, since the support conditions are modeled with elastic springs, the resulting formulas are quite general and can be used to solve different types of torsional vibration problems.

SEISMIC BEHAVIOR OF MULTI-STORIED RC BUILDING WITH RE-ENTRANT CORNER

Nowadays, buildings often feature irregular floor plans for functional, aesthetic, or economic reasons. Constructing earthquake-resistant structures in seismic areas, especially with irregular shapes like re-entrant corners, poses challenges. Such corners are common when maximizing limited space is a priority. In earthquakes, re-entrant corners in structures pose major vulnerabilities, causing stress concentration and torsion problems. The main aim of this paper is to develop finite element structural analysis models of 10, 12, and 15-storied L-shaped RC buildings with a re-entrant corner under different seismic zones using equivalent static analysis (ESA) and response spectrum analysis (RSA). This research also focuses on the overall behavior of analysis results (story drift, overturning moment, base shear, etc.) with the influence of re-entrant corner. It also investigates the performance of columns and beams near re-entrant corners as the number of stories increases in different seismic zones. For the current study, ETABS V19 is used. Models consider seismic, dead, and live loads. The Bangladesh National Building Code (BNBC) 2020 is used to examine the equivalent static analysis (ESA) and the response spectrum analysis (RSA) methods. It has been concluded from the study that re-entrant corner beams consistently exhibit the highest bending moments and torsion levels across all seismic zones. Similarly, re-entrant corner columns consistently demand the most axial force and rebar when compared to similar column types. Furthermore, the study identifies maximum stress levels in re-entrant corner slabs across all seismic zones.

The problem of axisymmetric torsion of a rigid disk parallel to an internally penny crack and interface in a bilayer medium

The aim of this work was to investigate an axisymmetric problem involving a penny-shaped crack embedded in a bi-material with finite thickness under torsion. This torsion is caused by a circular rigid disk attached to the upper layer and fixed by an undeformable support at the lower layer. The location of the crack can be found in the upper layer (case 1) or the lower layer (case 2), both cases are considered. The problem was solved by applying the Hankel transform to reduce it to a system of dual integral equations, which further simplified into Fredholm integral equations of the second-kind, These equations were then solved numerically using the Gaussian quadrature rule. Dimensionless plots were presented to demonstrate the influence of layers thickness, the distance between the support and the crack, and the shear modulus ratio between layers on the stress intensity factor of the crack. The obtained results illustrated the agreement between the analytical solution and the numerical results.

Harmonic analysis of the torque of autotractor engines by speed and load modes of operation

When assessing the cause of torsional vibrations on the crankshaft of tractor engines, forced vibrations are taken into account, which are provoked by torques arising from each cylinder and transmitted to the connecting rod necks, taking into account the order of operation of the cylinders. Researchers have only calculated or experimental values of the torque for different angular positions of the crankshaft with a certain sampling step. The algorithm for solving the torsional vibration problem involves the decomposition of discrete values of the torque of one cylinder into a harmonic series in order to determine its amplitudes and phase shifts. The adequacy of the resulting decomposition is determined by the number of harmonics taken into account and, as a rule, includes 10-12 harmonics. At the same time, it is necessary to perform such decompositions for each speed and load mode, which increases the amount of calculations. An algorithm is proposed that considers these modes separately, and depending on the values of the gas and inertial forces, allows using the characteristics of the nominal operating mode and the decomposition with a smaller number of terms. In this case, the results of the expansions have a satisfactory convergence. The proposed method for determining the parameters of motor harmonics of exciting moments under changing speed and load modes of the internal combustion engine, can be useful in the design of new and fine-tuning dampers of torsional vibrations of crankshafts.

Investigation on damping mechanism of machine‐side dynamics of permanent magnet synchronous generator‐based wind generation system

AbstractA permanent magnet synchronous generator‐based wind generation system has been predominantly applied in wind farms. With the wide application of wind generation, mechanical shafting attracts wide attention as torsional vibration problems may occur in its mechanical rotational system, which can further affect the power system. The paper first intentionally designs a two‐open‐loop two‐mass shaft subsystem model to investigate the interactions among wind turbine mass, generator mass, and machine‐side converter, that is, the machine‐side dynamics. Then, a bilateral damping contribution analysis is proposed to investigate the damping mechanism of these machine‐side dynamics. The impact mechanism of one dynamic on another through the damping contribution channel can be revealed by modal analysis, indicating the coupling of different oscillation modes and the complex interactions of machine‐side dynamics. The established two‐open‐loop two‐mass shaft subsystem model and the proposed bilateral damping contribution analysis with the identified damping contribution channel of the permanent magnet synchronous generator‐based wind generation system are validated.

A Gauss curvature flow approach to the torsional Minkowski problem

This paper is devoted to reconsidering the Minkowski problem for torsional rigidity introduced by Colesanti-Fimiani [9]. Employing a Gauss curvature flow, we establish the new existence of smooth solutions of the torsional Minkowski problem under the premise of giving an initial convex body.

A perturbation analysis of pure torsion of incompressible hyperelastic cylinders

The stretch-based formulation of basic problems for incompressible isotropic hyperelastic materials has been the subject of renewed recent attention largely motivated by application to modelling the mechanical response of soft tissues. Here we are concerned with the classical problem of simple torsion of a circular cylinder which has in the past been primarily investigated by employing the classical approach of Rivlin in terms of invariants of the Cauchy–Green deformation tensors. Here we propose a perturbation approach to the pure torsion problem for thin (but not necessarily infinitesimally thin) cylindrical specimens motivated by potential applications to rubber-like or biological filaments and for the infinitesimal twisting of thick cylinders. The advantage of this approach is that the results are valid for all strain–energy functions whether based in terms of the principal stretches or in terms of the classical deformation invariants. The asymptotic results obtained for the resultant force and moment are applied to the special case of the one-term stretch-based Ogden model. In particular, a prediction of a transition in the Poynting effect for values of the Ogden parameter α>6 is obtained. Other illustrative examples for both invariant and stretch-based strain energies are described. In view of the complexity of a general analysis of torsion for arbitrary strain energies, the perturbation approach provides a viable alternative for assessing the resultant force and moment.

Torsion of the Truncated Hollow Orthotropic Elastic Body of Rotation

This paper deals with the torsion of the body of rotation. The meridian section of the body is bounded by two ellipses and two straight lines which are perpendicular to the axis of rotation of the body. The material of the body is elastic and cylindrical orthotropic. To solve the torsion problem, the theory of the torsion of shafts of varying circular cross-sections is used, which was developed by Mitchell and Töppl. An analytical solution is given for the shearing stresses and circumferential displacement. A numerical example illustrates the application of the presented analytical solution. The results of this paper can be used as a benchmark solution to verify the accuracy of the results computed by finite element simulations and finite different methods.

A New Method for Correcting the Deviation of a Middle Pier Tower of a Long-Span Intermediate Arch Bridge

To control the deviation of a long-span concrete-filled steel tube (CFST) arch bridge during construction monitoring, a practical method for controlling tower deviation is studied and established. The form of construction of this bridge is an intermediate double-arch bridge, which differs from conventional bridges, thus requiring the urgent resolution of the issue of unbalanced middle piers. Therefore, the mechanical characteristics and construction process of an intermediate long-span, dual-coupled steel pipe arch bridge are meticulously examined by using a 1:10 scale model, with particular focus on discussing the deflection of the buckle tower during the installation of the arch rib segments. Construction control is implemented using a novel tower deflection control method that addresses unilateral torsion problems and difficulties in controlling the deflection of the tower. The model results are compared with the finite element analysis output, demonstrating that this new approach can resolve unbalanced tower deviations by maintaining absolute values within 0.5 mm. After correcting these deviations, the measured results from the model bridge tower align with the calculated analytical results and even surpass the theoretical expectations for tower deviation. This remarkable new method accurately resolves real-world bridge tower deviations.

Some Unresolved Problems of High-Pressure Torsion

This overview highlights some salient features of one of the most popular severe plastic deformation techniques: high-pressure torsion (HPT). It focuses on the unresolved challenging problems of HPT. The problems selected touch upon some fundamental questions of mechanics of plasticity, fracture, and friction that are at the core of the HPT process. The scientific significance of these problems and the proposed pathways to resolving them are discussed. The article is meant to promote the use of HPT as a potent tool for studying plasticity at large strains theoretically and also as a practical method enabling novel micromanufacturing routes.

Application of sensitivity analysis in extension, inflation, and torsion of residually stressed circular cylindrical tubes

This paper deals with applying two main sensitivity analysis (SA) methods, namely, the Sobol method and the Fourier Amplitude Sensitivity Test (FAST) method on the problem of mixed extension, inflation, and torsion of a circular cylindrical tube in the presence of residual stress. The mechanical side of the problem was previously proposed by Merodio & Ogden (2016). The input parameters in the form of the initial cylinder geometry, the amount of the residual stress, the azimuthal stretch, the axial elongation, and the torsional strain are distributed according to three probability distribution methods, namely the uniform, the gamma, and the normal distribution. In the present work, through applying Sobol and FAST methods, the most influential factors among input parameters on the output variable have been determined. The assessment of our results is then determined by the computation of bias and standard deviation of Sobol and FAST indices for each input parameter in the model.

Pure torsion for incompressible hyperelastic materials of Valanis–Landel form

The stretch-based formulation of basic problems for incompressible isotropic hyperelastic materials has been the subject of renewed recent attention largely motivated by application to modelling the mechanical response of soft tissues. Here, we are concerned with this formulation of the classic problem of simple torsion of a circular cylinder that has been primarily in the past investigated by employing the classical approach of Rivlin in terms of invariants of the Cauchy–Green strain tensor. Attention is focused on examining the resultant axial force necessary to sustain pure torsion, thereby investigating the Poynting effect, and the resultant moment needed to generate the twist. General results for the resultant force and moment are obtained for all hyperelastic materials that can be modelled in the celebrated separable Valanis–Landel form. These expressions are specialized for the one-term Ogden model to provide explicit results valid for all strain-stiffening exponents in this model. Some particular examples are provided, mainly for even values of the exponent. A notable result obtained for the Ogden model is that for values of the hardening exponent n in that model for which n ≤ 6 , one obtains the classical Poynting effect, whereas for n > 6 , a reverse Poynting effect always occurs for sufficiently small twist.