Stress bounds for twisted bars of strip cross section

Bounds are established for the maximum resultant shear stress, and used to investigate the approximation involved in the value predicted for thin sections on the basis of the membrane analogy.     

Stress bounds for the torsion of tubes of uniform wall thickness

Upper and lower bounds are derived for the torsional rigidity and the maximum shear-stress magnitude. The values provided by Bredt's formulas are characterized as asymptotic approximations. Explicit results on error bounds are included.     

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.     

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.     

On the torsion of chiral bars in gradient elasticity

This paper contains a study of the problem of torsion of chiral bars with arbitrary cross-sections in the context of the linear theory of gradient elasticity. The solution is expressed in terms of solutions of four auxiliary plane problems characterized by loads which depend only on the constitutive coefficients. It is shown that, in general, the torsion produces extension (or contraction) and bending effects. The results are used to investigate the torsion of a homogeneous circular bar. In contrast with the case of achiral circular cylinders, the torsion and extension cannot be treated independently of each other.     

Large strain torsion of axially-constrained solid rubber bars

Large strain fixed-end torsion of circular solid rubber bars is studied semi-analytically. The analyses are based on various non-Gaussian network models for rubber elasticity, some of which were proposed very recently. Results are presented in terms of predicted torque vs. twist curves and axial force vs. twist curves. In some cases, the predicted stress distributions are also given. The sensitivity of the second-order axial force to the employed models is considered. The predicted results are compared with experimental results found in the literature.     

Free-edge stresses and progressive cracking in surface coatings of circular torsion bars

This paper considers the explicit solutions of free-edge stresses near circumferential cracks in surface coatings of circular torsion bars and their application in determining the progressive cracking density in the coating layers. The problem was formulated within the framework of linear elastic fracture mechanics (LEFM). The free-edge stresses near crack tip and the shear stresses in the cross-section of the torsion bar were approached in explicit forms based on the variational principle of complementary strain energy. Criterion for progressive cracking in the coating layer was established in sense of strain energy conservation, and the crack density is thereby estimated. Effects of external torque, aspect ratio, and elastic properties on the density of progressive cracking were examined numerically. The present study shows that, in the sense of inducing a given crack density, compliant coating layer with lower modulus has much higher critical torque than that of a stiffer one with the same geometries and substrate material, i.e., compliant coating layer has greater cracking tolerance. Meanwhile, the study also indicates that thicker surface coating layer is more pliant to cracking than the thinner ones. The present model can be used for analyzing the damage mechanism and cracking tolerance of surface coatings of torsion shafts and for data reduction of torsional fracture test of brittle surface coatings, etc.     

Coupled thermo-mechanical analysis of shape memory alloy circular bars in pure torsion

Pure torsion of shape memory alloy (SMA) bars with circular cross section is studied by considering the effect of temperature gradient in the cross sections as a result of latent heat generation and absorption during forward and reverse phase transformations. The local form of energy balance for SMAs by taking into account the heat flux effect is coupled to a closed-form solution of SMA bars subjected to pure torsion. The resulting coupled thermo-mechanical equations are solved for SMA bars with circular cross sections. Several numerical case studies are presented and the necessity of considering the coupled thermo-mechanical formulation is demonstrated by comparing the results of the proposed model with those obtained by assuming an isothermal process during loading–unloading. Pure torsion of SMA bars in various ambient conditions (free and forced convection of air, and forced convection of water flow) subjected to different loading–unloading rates are studied and it is shown that the isothermal solution is valid only for specific combinations of ambient conditions and loading rates.     

On transient creep bounds and approximate solutions for the torsion of thin strips

Integral equations are derived which govern transient primary and secondary creep in thin rectangular strips subject to torsion. Formal similarity between these equations and others arising in previous work are exploited to obtain bounds, monotonicity and convexity of the stress profile as well as uniform approximations.     

Non-linear elastic non-uniform torsion of bars of arbitrary cross-section by BEM

In this paper an elastic non-uniform torsion analysis of simply or multiply connected cylindrical bars of arbitrary cross-section taking into account the effect of geometric non-linearity is presented employing the boundary-element(BE) method. The torque-rotation relationship is computed based on the finite-displacement (finite-rotation) theory, that is the transverse displacement components are expressed so as to be valid for large rotations and the longitudinal normal strain includes the second-order geometric non-linear term often described as the “Wagner strain”. The proposed formulation does not stand on the assumption of a thin-walled structure and therefore the cross-section's torsional rigidity is evaluated exactly without using the so-called Saint-Venant's torsional constant. The torsional rigidity of the cross-section is evaluated directly employing the primary warping function of the cross-section depending on its shape. Three boundary-value problems with respect to the variable along the beam axis angle of twist, to the primary and to the secondary warping functions are formulated. The first one, employing the Analog Equation Method (a BEM-based method), yields a system of non-linear equations from which the angle of twist is computed by an iterative process. The remaining two problems are solved employing a pure BE method. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The developed procedure retains most of the advantages of a BEM solution over a pure domain discretization method, although it requires domain discretization.     

The warping functions of regular prismatic bars under torsion evaluated by caustics

Reflection of a bundle of coherent light on the warped cross section of a prismatic bar submitted to torsion forms a caustic on a receiver plane. From the mathematical expression of this curve and the theory of reflected caustics, it is possible to evaluate accurately the warping function of the cross section. Using this idea, it was possible to study the torsion problem in prismatic bars with sections which were equilateral triangles and squares. It was observed that the shape of the caustic is an hypocycloid curve with three or four cusps respectively. By evaluating the warping function by using elements from the respective caustics it was possible to find out that, for the triangular cross section, the expression for the warping function coincided exactly with the expression given by the exact solution of the problem. For the square cross section, a closed-form solution for its warping function was readily derived, to which the series approximation solution differed only by a few percent at maximum for the shear stresses. Since the method can be readily extended to any canonical polygonic cross section, it constitutes a general solution for the torsion of prismatic bars, which approximates their exact deformations better than the solutions based on the Saint-Vénant assumptions.     

Plastic deformation and springback of rectangular bars subject to combined torsion and tension

Springback of rectangular bars under combined torsion and tension is investigated. A theoretical model for springback is developed and evaluated by comparing calculated and experimental results. It is concluded that springback is analytically predictable. Both analysis and test data show that an axial tension always reduces angular springback in a twisted bar. The order of plastic deformation is found to be important : twist followed by pull produces smaller angular springback upon release of torque and force than does a deformation in the reverse order.     

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.     

Stress state of a cross-base prism under torsion

The stress state of a cross-base prism under torsion is analyzed. The lower- and upper bounds for stresses are found by solving an infinite system of linear algebraic equations and using Koyalovich’s limitants method. It is proved that the infinite system is regular and that its solution exists and is unique. The asymptotic behavior of the unknowns is established. The convergence of the series for stresses is accelerated Translated from Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 79–93, November 2008.