Lando Mentrasti

Torsion of closed cross-section thin-walled beams: The influence of shearing strain

Abstract In this study a theory of rectangular closed cross-section thin-walled beams subject to torsion is presented, with express consideration being given to the shearing strain. The equations of equilibrium (in weak form) of the elastic system subjected to twisting distributed loads and to bimomental loads, in the unknown θ (angle of rotation by torsion) and βw (average shearing strain in the webs)—or ψw (distorsional angle)—are derived from the variational method of Minimum Potential Energy. The problem of Shear Lag in torsion is posed when non-canonical torsional contraint, which does not distribute the torque uniformly between the elements of the beam, is present. Particular emphasis was given to the case in which the geometry of the section is such that pure torsion warping is identically zero ( w = 0 and J w = 0) , an aspect which is not usually considered in classic Non Uniform Torsion (NUT). The differential equation in θ derived from the proposed theory is formally identical to the NUT equation; however, the singularity in the coefficient of θiv for ψ → 1 is no longer present. It is shown that the present theory, including the known results, determines, depending on the actual constraints conditions, the shearing stress which can differ considerably from the results predicted by the NUT. The paper concludes with a numerical study of a bimomental distorsion at the ends and also suggests possible improvements of the model.

Shear-torsion of large curvature beams-Part I. General theory

Abstract The shear-torsion problem for an elastic circular beam with large curvature and multiconnected cross section is analysed using warping function and potential stress function approaches. The resultant forces and the shear-torsion modulus are computed. The circulation of the shear stress along a curve is defined and consequently the compatibility equations for solving the problem in multi-connected cross section are presented. The positivity of the potential function is established as a consequence of a general theorem on a Boundary Value Problem (BVP). The maximum shear stress is proved to be attained at the boundary and a lower bound of its value is also given.

Bending of large curvature beams.: I. Stress method approach

Abstract The paper presents a coherent theory of the uniform bending problem in a circular curved beam, with multi-connected cross-section, having a large radius of curvature with respect to its width. The three-dimensional elastic problem is solved, in the case of linear homogeneous isotropic body, assuming the stress tensor as the unknown and by exactly satisfying the field compatibility equations. The mathematical structure of the governing boundary value problem (BVP), enlightened here for the first time, is unexpectedly complicated: a fourth-order elliptic (variable coefficients) partial differential equation with two degenerate unstable boundary conditions (i.e. involving second and third order partial derivatives in a direction that becomes tangent at several points of the boundary). Such a kind of BVP seems to be typical of the curved beam bending problem since it also appears in the displacement approach ( Mentrasti, 2001 . part II, Int. J. Solids Struct. 38, 5727–5745). As a final point, it is reduced to a simpler problem by an ad hoc integral representation, assuming the ρ -convexity of the cross-section domain.

Distortion (and Torsion) of rectangular thin-walled beams

Abstract A study of rectangular cross-section thin-walled beams under torsional, distortional and bimomental loads is presented. The assumed displacement field describes both torsional and distorsional behaviour, the shearing strain in the walls being taken into account. By a variational principle the equilibrium and boundary equations are derived and a physical interpretation of the natural conditions is given. A closed form solution is given with reference to the matrix form of the governing differential system together with a discussion of a sixth-order ‘uncoupled’ differential equation generalizing the BEF analogy. The present formulation shows that, in general, torsion and distortion are pair of coupled problems and a study of distortion without considering torsion would therefore not be legitimate. The comparison of the results predicted by the theory is in good agreement with experimental evidence.

Bending of large curvature beams.: II. Displacement method approach

Abstract A large curvature circular beam subject to uniform bending in its plane is investigated by a displacement method . This is the companion of a previous paper in which the same subject has been investigated by a stress method approach (Part I, Int. J. Solids Struct. 38 (2001) 5703–5726). The functional form of the three-dimensional displacement field is determined exactly: the cross-section remains plane and rotates about the Z -axis, while each longitudinal fiber remains circular. This result is independent of the constitutive equation of the material, provided it is compatible with the uniform bending hypothesis. The equilibrium equation (in several form) and the boundary conditions are derived for the linear elastic and homogeneous beam: they constitute an unstable degenerate boundary value problem with a structure similar to that found in Part I. The non-variational nature of this kind of problem is also detected. Finally, the relationship with the potential stress function ψ is derived, governed by a hyperbolic second order partial differential equation (another unusual occurrence appearing in bending problem).

Shear-torsion of large curvature beams—Part II. Applications to thin bodies

The shear-torsion state of stress in a curved beam, whose cross sections is a thin rectangle with sides not parallel to the plane of the beam, are determined in closed form. The maximum value of the shear stress is attained at the concave boundary of the beam. The shear-torsion moment of inertia Jst, in multi-connected thin-walled cross sections is evaluated. Several examples of cross sections are discussed.

New test rig for creased paperboard investigation to confectionery industry reconfigurable folders

In packaging industry, the duration of the carton folding plays a fundamental role in the production process; in particular in the erection process when each panel rotates around the die-pressed lines called creases. Their bending response can be very complex, depending on forming and environment conditions. The crease mechanical properties, such as geometrical parameters, temperature, moisture and folding speed, influence the overall production. It is therefore necessary to control all of these parameters, from both a theoretical and experimental point of view. About this, an experimental setup is expressly designed and built w.r.t. the rotation angle for a paper around the respective crease. The results of this research allow demonstrating the reliability of the experimental setup, the substantial negligibility of the geometrical errors and determining the number of sample so that the dispersion is compatible with the experimental errors. Then the test-ring is suitable for carton folding investigation.

Paradoxes in rigid-body kinematics of structures

The paper discusses two paradoxes appearing in the kinematic analysis of interconnected rigid bodies: there are structures that formally satisfy the classical First and Second Theorem on kinematic chains, but do not have any motion. This can arise when some centers of instantaneous rotation (CIR) relevant to two bodies coincide with each other (first kind paradox) or when the CIRs relevant to three bodies lie on a straight line (second kind paradox). In these cases two sets of new theorems on the CIRs can be applied, pointing out sufficient conditions for the nonexistence of a rigid-body motion. The question is clarified by applying the presented theory to several examples.