Shear deformable bars of doubly symmetrical cross section under nonlinear nonuniform torsional vibrations—application to torsional postbuckling configurations and primary resonance excitations

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross section, taking into account the effects of geometrical nonlinearity (finite displacement—small strain theory) and secondary twisting moment deformation. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are subjected to the most general axial and torsional (twisting and warping) boundary conditions. The resulting coupling effect between twisting and axial displacement components is also considered and a constant along the bar compressive axial load is induced so as to investigate the dynamic response at the (torsional) postbuckled state. The bar is assumed to be adequately laterally supported so that it does not exhibit any flexural or flexural–torsional behavior. A coupled nonlinear initial boundary value problem with respect to the variable along the bar angle of twist and to an independent warping parameter is formulated. The resulting equations are further combined to yield a single partial differential equation with respect to the angle of twist. The problem is numerically solved employing the Analog Equation Method (AEM), a BEM based method, leading to a system of nonlinear Differential–Algebraic Equations (DAE). The main purpose of the present contribution is twofold: (i) comparison of both the governing differential equations and the numerical results of linear or nonlinear free or forced vibrations of bars ignoring or taking into account the secondary twisting moment deformation effect (STMDE) and (ii) numerical investigation of linear or nonlinear free vibrations of bars at torsional postbuckling configurations. Numerical results are worked out to illustrate the method, demonstrate its efficiency and wherever possible its accuracy.

     

Solution of St. Venant's torsion problem by power series

It is shown how St. Venant's problem of torsion of elastic bars can be formulated so as to fit readily into the procedure for deriving approximate equations of motion of bars by expansion of the displacements in a double series of powers of the transverse coordinates. Applications are exhibited for bars with elliptic, equilateral triangular and rectangular cross-sections.     

Minimum-weight design of hollow cylinders for given lower bounds on torsional and flexural rigidities

An approximate analytical solution is obtained for the problem of minimizing the cross-sectional area of elastic, doubly-connected cylindrical bars for given lower bounds on torsional and flexural rigidities. The bars are designed to withstand either a twisting or a bending moment, but not both, at any given time. The shape of the inner contour of the doubly-connected (hollow) cross-section is specified while that of the outer contour is determined as a result of the optimization. The economy achieved by optimization is estimated.     

Torsin of Functionally Graded Isotropic Linearly Elastic Bars

The purpose of this research is to investigate the effects of material inhomogeneity on the torsional response of linearly elastic isotropic bars. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e. materials with spatially varying properties tailored to satisfy particular engineering applications. The classic approach to the torsion problem for a homogenous isotropic bar of arbitrary simply-connected cross-section in terms of the Prandtl stress function is generalized to the inhomogeneous case. The special case of a circular rod with shear modulus depending on the radial coordinate only is examined. It is shown that the maximum shear stress does not, in general, occur on the boundary of the rod, in contrast to the situation for the homogeneous problem. It is shown that the material inhomogeneity may increase or decrease the torsional rigidity compared to that for the homogeneous rod. Optimal upper and lower bounds for the torsional rigidity for nonhomogeneous bars of arbitrary cross-section are established. A new formulation of the basic boundary-value problem is given. The results are illustrated using specific material models used in the literature on functionally graded elastic materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Torsion of composite laminated bars with a large number of layers

I.IntroductionInmodernengineeringstructures,compositelaminatememberswiththickcross-sectionhavebeenextensivelyusedforundergoinghighload/presurcandanti-detonating.Duetotheinherentinbomogencousandanisotropicofmaterialsand3-Dconstructionalfeatures,classicallaminatetheoriesincludinghighorderplatetheory)areinadequate.Therefore,anewchallengetodevelopthicklaminatetheoriesisraised.ItisextremelydifficultandexDensiveforelementsdiscretizationinanalysisofstressesbyFEMI'j,becausethicklaminatesconsistofhu…     

Some analytical solutions for Saint-Venant torsion of non-homogeneous cylindrical bars

The Saint-Venant torsion problem of linearly elastic cylindrical bars with solid and hollow cross-section is treated. The shear modulus of the non-homogeneous bar is a given function of the Prandtl's stress function of considered cylindrical bar when its material is homogeneous. The solution of the torsional problem of non-homogeneous bar is expressed in terms of the torsional and Prandtl's stress functions of homogeneous bar having the same cross-section as the non-homogeneous bar.     

阶梯形闭口薄壁杆的约束扭转振动

用奇异函数建立非单一材质的ｎ级阶梯形闭口薄壁杆约束扭转自由振动和强迫振动的微分方程并求得其通解,用Ｗ算子给出主振型函数的表达式及常见支承条件下杆的频率方程。     

Torsional waves of finite amplitude in elastic rod

We propose mathematical models generalizing the Coulomb and Vlasov equations of torsional vibrations of rods by taking the geometric nonlinearity into account. In the general case, the nonlinearity is taken into account both in the system of displacements (because the displacement vector in the case of rod torsion can be finite even for small strains) and in the relations between displacements and strains. We analyze nonlinear torsional stationary waves and find the effect of splitting of soliton-like unipolar waves in countercollisions. We also show that, in several cases, the existence of nonlinearities can also induce dispersion and that nonlinear stationary waves can also exist in the absence of dispersion in the linear medium.     

DNA纳米管的扭转刚度模型

刚度是衡量材料弹性变形难易程度的一个定量表征参数,与DNA纳米管静动力学特性及其结构生物功能密切相关．本文致力于研究DNA纳米管的扭转刚度．首先,在六角形均匀封装条件下,考虑到单个DNA杆件弯扭组合问题的静不定特点,我们利用平衡方程、变形协调方程和弹性本构方程,合理预测了DNA纳米管扭转实验中单个DNA杆件的弯扭组合变形,由此给出了DNA纳米管扭转刚度预测的解析模型．最后的结果表明：随着DNA杆数的增加,DNA纳米管的弯曲刚度显著增加,而其扭转刚度却几乎不变,合理解释了扭转实验中发现的现象．有关结论为DNA折叠结构的设计和应用提供了参考．     

On the torsional rigidity for bars with L- and +-cross-section

This paper is a continuation of the senior author’s previous papers^[1-3]. using the harmonic continuation technique, the torsional rigidity for bars with L- and +-cross-section can be easily found.Numerical results are shown in Tables 1-3 respectively.     

Torsional dynamics of the Birnboim-Schrag multiple lump resonator studied using TV-holography

We have studied the torsional dynamics of the Birnboim-Schrag multiple lump resonator (MLR) using TV holography. The relative torsional displacement amplitudes have been determined for all resonator lumps at their five resonance modes. Due to the low vibration amplitudes for the highest frequency mode, we only give a rough estimate of the relative torsional displacements for this resonance. For some lumps, the observed relative displacement amplitudes deviate by as much as a factor of 2 from the predictions of the dumbbell theory commonly used to analyze MLR dynamics. The experimental data agree well with the recent theoretical analysis carried out by Mikkelsen et al. (1992).     

Secondary torsional moment deformation effect in inelastic nonuniform torsion of bars of doubly symmetric cross section by BEM

In this paper a boundary element method is developed for the inelastic nonuniform torsional problem of simply or multiply connected prismatic bars of arbitrarily shaped doubly symmetric cross section, taking into account the secondary torsional moment deformation effect. The bar is subjected to arbitrarily distributed or concentrated torsional loading along its length, while its edges are subjected to the most general torsional boundary conditions. A displacement based formulation is developed and inelastic redistribution is modeled through a distributed plasticity model exploiting three dimensional material constitutive laws and numerical integration over the cross sections. An incremental–iterative solution strategy is adopted to resolve the elastic and plastic part of stress resultants along with an efficient iterative process to integrate the inelastic rate equations. The one dimensional primary angle of twist per unit length, a two dimensional secondary warping function and a scalar torsional shear correction factor are employed to account for the secondary torsional moment deformation effect. The latter is computed employing an energy approach under elastic conditions. Three boundary value problems with respect to (i) the primary warping function, (ii) the secondary warping one and (iii) the total angle of twist coupled with its primary part per unit length are formulated and numerically solved employing the boundary element method. Domain discretization is required only for the third problem, while shear locking is avoided through the developed numerical technique. Numerical results are worked out to illustrate the method, demonstrate its efficiency and wherever possible its accuracy.     

Application of holographic interferometry to predict long-time torsional relaxation

The objective of the work described in this paper was to develop techniques to predict long-term relaxation in torsional springs based on short-time data. The torsional springs examined in this study were torsion bars fabricated from 300M steel (280–300-ksi tensile strength). A holographic technique was utilized to perform very precise measurements of relaxation in torsion bars. The technique utilizes real-time holographic interferometry and was capable of resolving relaxation (torque losses) as small as. 01 in.-lb for a bar initially torqued to 825 in.-lb. The holographically determined torque-logs data were used to develop a model to estimate the relaxation behavior of the torsion bars. The model determined to best fit the data is described by: ΔT=0.32(t+50)^0.4?qt ^?1 wheret = time in hours ΔT = total change in torque (in.-lb) The model was developed to fit the holographic data from 0 to 5000 h. Excellent agreement between the torque-loss rate predicted by the model and that actually measured holographically at 10,000 h was obtained. This further indicates that long-time relaxation behavior of the torsion bars can be accurately predicted from short-time tests. Using the model, it is now a simple matter to holographically measure and evaluate the relaxation of other lots of torsion bars to predict their long-time relaxation behavior. The measurement period can be as short as 100 h to show the comparison with baseline data. The long-time prediction can be compared with the acceptable engineering-energy requirements to determine suitability for service.     

Nonlinear analysis of underconstrained structures

Underconstrained (kinematically indeterminate) assemblies of bars and pin-joints possess a specific kind of nonlinearity which appears even for small displacements and linear elasticity. Various approaches to nonlinear analysis based on the Newton–Raphson procedure are considered. The subspace Newton–Raphson technique is proposed. Theoretical considerations are accompanied by numerical examples of plane and space underconstrained assemblies.     

The applications of generalized variational principles in nonlinear structural analysis

This paper discusses the generalized variational principles founded by thetechnique of Lagrangian multipliers in structural mechanics and analyzes the nonlinearstatically indeterminate structures.It is assumed that the stress-strain relationship ofthe materials of structures has the form ofσ=Bε~(1/m)orτ=Cγ~(1/m),namely,thephysical equations of structures have the shape of exponential functions.Severalexamples are given to illustrate the statically indeterminate structures such as thetrusses,beams,frames and torsional bars.     

Minimum-weight design of multi-purpose cylindrical bars

We solve the problem of minimizing the weight of elastic cylindrical bars under given constraints on their torsional and bending rigidities. The possible types of solutions are classed in the space of the design parameters. The necessary optimality conditions are derived and then used in the formulation of a closed-form boundary-value problem for a region with an unspecified boundary—the unknown cross-sectional shape of the optimal bar. An analytical solution of this and other related problems in terms of the given design parameters is obtained.     

A new reusable quadriaxial strain transducer

The reusable quadriaxial strain transducer consists of an eight-arm (at 45 degrees from one another) star-shaped plane-sensitive element. Two strain gages are mounted to the inner and outer sides of each arm in the neighborhood of the extremity where hard oblique pins are fixed. The pins are impressed into the surface of the material so the strains on the surface induce bending and torsional displacements of the arms. The paper describes the transducer design and the calibration procedure. Experimental results obtained on wood materials are analyzed.     

Optimization of elastic bars in torsion

The paper considers the problem of optimization of mechanical systems described by partial differential equations. The shape of the region of integration of these equations is not specified beforehand but is to determined from the condition that a certain integral functional attains an extremal value. The mathematical optimization problem is reduced to a variational one having no differential constraints and the necessary optimality conditions are derived. The latter are used for seeking the cross-sectional shape of elastic bars of maximum torsional rigidity. Exact and approximate analytical solutions are given and the effectiveness of the optimal solutions is estimated.     

The application of graph theory to the limit analysis for torsion of thin-walled bars with complex cross-sections

Graph theory is employed in this paper as a means to establish the topological model of complex thin-walled cross-sections. On this basis, the upper and lower bound theorems of the plastic limit analysis are applied to the analysis of the plastic limit shear flows on the cross-section of thin-walled bars under St. Venant torsion. Corresponding mathematical programming problems are formulated and their duality is shown. After solving the linear programming problem corresponding to the lower bound theorem, the limit torsional moment of a thin-walled cross-section can be calculated according to the shear stress distribution in the limit state. The formula for calculating the limit torsional moment is given. Furthermore, the limit state of thin-walled cross-sections under St. Venant torsion is also discussed and the concept of the limit tree is introduced. A computer program has been developed by the author. Results calculated by the program for typical complex cross-sections are given.