Prandtl Stress Research Articles

Multi-phased solutions of Prandtl's stress function for an orthotropic rectangular bar under Saint-Venant's torsion and the general rule of swapping

Based on the conventional solution of Prandtl's stress function for an orthotropic rectangular bar under Saint-Venant's torsion, one can derive displacements, shear strains, shear stresses and torsional rigidity. The conventional solution of Prandtl's stress function has a hyperbolic function for the coordinate in the bar's thickness direction, and a trigonometric function for the coordinate in the width direction. This paper raises questions about the solution. Why is the solution not arranged in the opposite way? Why is the hyperbolic function not for the coordinate in the width direction and the trigonometric function for the coordinate in the thickness direction? How is it that these obvious questions have never been addressed? This study rearranges the solution of the conventional Prandtl's stress function using the TSAI technique and finds that the solution is multi-phased, indicating that the coordinates in the width and thickness directions and their corresponding parameters are swappable, a phenomenon proposed as the general rule of swapping.

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.

Strength Design and Analysis of the Grouting while Drilling Equipment’s Sleeve

According to structure and working-condition of the grouting while drilling equipment, sleeve’s compressive strength and torsion strength are calculated with elastic theory, which are simulated and verified with FEA Software. The results indicate that its maximum internal stress exceeds the elastic yield limit at 30 MPa external pressures, and that the sleeve is working with risk in 3000 m underground soft coal.With Prandtl stress function method, the calculated shear strength in the sleeve’s small end is far more than the material’s shear strength due to 1500 N·m torque, the small end is stress concentration area and it can’t meet the requirement. Finally, sleeve’s theoretical work external pressure is calculated as 29.8 MPa and its axial deformation is about 0.2 mm, with radial deformation level at 0.03 mm, which can be referenced to tolerance design in the assembly.

Finite Element Analysis Of Linear Elastic Torsion For Regular Polygons

This paper presents an explicit finite element integration scheme to compute the stiffness matrices for linear convex quadrilaterals. Finite element formulationals express stiffness matrices as double integrals of the products of global derivatives. These integrals can be shown to depend on triple products of the geometric properties matrix and the matrix of integrals containing the rational functions with polynomial numerators and linear denominator in bivariates as integrands over a 2-square. These integrals are computed explicitely by using symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be applied to solve boundary value problems in continuum mechanics over convex polygonal domains.We have also developed an automatic all quadrilateral mesh generation technique for convex polygonal domain which provides the nodal coordinates and element connectivity.We have demonstrated the proposed explicit integration scheme to solve the Poisson Boundary Value Problem for a linear elastic torsion of a non-circular bar with cross sections having profiles of equilateral triangle, a square and regular polygons (pentagon(5-gon)to icosagon(20gon)) which are inscribed in a circle of unit radius. Monotonic convergence from below is observed with known analytical solutions for the Prandtl stress function and torsional constant .We have shown the solutions in Tables which list both the FEM and exact solutions. The graphical solutions of contour level curves and the corresponding finite element meshes are also displayed.

On Torsion of Functionally Graded Elastic Beams

The evaluation of tangential stress fields in linearly elastic orthotropic Saint-Venant beams under torsion is based on the solution of Neumann and Dirichlet boundary value problems for the cross-sectional warping and for Prandtl stress function, respectively. A skillful solution method has been recently proposed by Ecsedi for a class of inhomogeneous beams with shear moduli defined in terms of Prandtl stress function of corresponding homogeneous beams. An alternative reasoning is followed in the present paper for orthotropic functionally graded beams with shear moduli tensors defined in terms of the stress function and of the elasticity of reference inhomogeneous beams. An innovative result of invariance on twist centre is also contributed. Examples of functionally graded elliptic cross sections of orthotropic beams are developed, detecting thus new benchmarks for computational mechanics.

Some analytical solutions of functionally graded Kirchhoff plates

The elastostatic problem of a functionally graded Kirchhoff plate, with no kinematic constraints on the boundary, under constant distributions of transverse loads per unit area and of boundary bending couples is investigated. Closed-form expressions are provided for displacements, bending–twisting curvatures and moments of an isotropic plate with elastic stiffness and boundary distributed shear forces, assigned respectively in terms of the stress function and of its normal derivative of a corresponding Saint-Venant beam under torsion. The methodology is adopted to solve circular plates with local and Eringen-type elastic constitutive behaviors, providing thus new benchmarks for computational mechanics. The proposed approach can be used to obtain other exact solutions for plates whose planform coincides with the cross-section of beams for which the Prandtl stress function is known in an analytical form.

TORSIONAL ANALYSIS OF PIEZOELECTRIC HOLLOW BARS

Analytical formulations are presented for Saint-Venant's torsion of orthotropic piezoelectric hollow members. It is assumed that the cross-section consists of straight and curved segments with the same thickness. Governing equations are based on Prandtl's stress and electric displacement potential functions. Some examples are solved using the presented formulations and verified by the finite element method solutions. The results show the accuracy of the present method for torsional analysis of thin- to moderately thick-walled hollow bars.

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

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

On the relative position of twist and shear centres in the orthotropic and fiberwise homogeneous Saint–Venant beam theory

The relationship between twist and shear centres in an orthotropic Saint–Venant beam, with fiberwise homogeneous elastic moduli and constant Poisson ratios, is investigated. Arbitrary cross-sections are considered. As a new result the relative position of these points is expressed in terms of the scalar potential whose gradient is the rotated field of twist tangential stresses. Its evaluation requires the solution of n+1 boundary value problems, being n⩾0 the number of holes in the cross-section. In an isotropic and homogeneous beam the potential is Prandtl stress function and known formulae, providing the relative position of twist and shear centres, are recovered. Explicit expressions of sliding-torsional compliance blocks for Timoshenko beams, defined by an energy condition of equivalence with the orthotropic and fiberwise homogeneous Saint–Venant theory, are provided. Coincidence of twist centre and Timoshenko shear centre is proven. Numerical computations on homogeneous and composite orthotropic L-sections are performed.

Torsion of carbon fiber composite pyramidal core sandwich plates

A combined theoretical, experimental and numerical investigation of carbon fiber composite pyramidal core sandwich plates subjected to torsion loading is conducted. Pyramidal core sandwich plates are made from carbon fiber composite material by a hot compression molding method. Based on the Classical Laminate Plate Theory and Shear Deformation Theory, the equivalent mechanical properties of laminated face-sheet are obtained; based on a homogenization concept combined with a mechanical of materials approach, the equivalent in-plane and out-of-plane shear moduli of pyramidal core are obtained. A torsion solution is derived with Prandtl stress function and can be used in the sandwich plate with orthotropic face-sheets and orthotropic core. The influences of material properties and geometrical parameters on the equivalent torsional stiffness are explored. In order to verify the accuracy of the analytical torsion solution, experimental tests of sandwich plate samples with different face-sheet thicknesses are conducted and two types of finite element models are developed. Good agreements among analytical predictions, finite element simulations and experimental evaluations are achieved, which prove the validity of the present derivation and simulation. The proposed method could also be applied in design applications and optimization of the pyramidal core sandwich structures.

Saint-Venant Torsion of Open-Section Members of Uniform Thickness

An efficient analytical method for torsion analysis of open-section members of uniform thickness is presented. The conventional method based on the thin-walled theory for torsion analysis of bars can give good solutions only for members with small thickness and fillets of large radius. In the present method, a better approximation is employed, and much more accurate formulas are derived. The cross-section is considered to consist of some straight and curved (circular arc) segments. The Poisson equation of the torsion problem in terms of Prandtl's stress function is solved in each segment separately. By the present method, closed-form solutions for shear stress and torsional rigidity of members with open cross-section can be found. Some formulas for torsion analysis of tubes with a longitudinal slit and for members with angle and channel cross-section are derived. While these formulas are relatively simple, they give excellent results for thin- to moderately thick-walled members.

Plieninio Rėmo PlastinėS Deformacijos: Statika Ir Dinamika

When all deformations of a column are elastic, transverse deflections of the column depend on transverse force and axial displacements depend on axial force only. These classical dependences are unsuitable for elastic-plastic deformations. Plastic deformations develop in columns when steel frame is influenced by extreme action. When a steel column is in the elastic-plastic state, the distribution of elastic and plastic deformations in the cross-section depends on both the bending moment and compressing force. The ideal elastic-plastic material is assumed in this investigation (Prandtl stress – strain diagram). If the shape of the column section is double tee, flange width is neglected with respect to web height, but the area of the flange cross-section is assumed a constant. Single-sided or double-sided yield depends on the moment and force, and therefore curvature and the axial strain of the column can be calculated when yielding dependences are determined. Transverse and axial displacements of the highest point of the column are deduced by integration and depend on two arguments: bending force and axial force. These dependences are essentially non-linear, so linear approximations can be assessed for some vicinity of axial force and bending moment values. When axial force is a constant and transverse force increases, both axial and transverse displacements tend to increase. If transverse force is a constant and axial force increases, both displacements increases but dependence lines remain different and depend on cross-section shape parameter equal to the ratio of the flange area and the area of the whole cross-section. A distinguished feature of plastic deformations is dependence on the history of loading a frame of which can be selected in an arbitrary way by an investigator if a quasi-static solution is under examination. The loading of a frame and inertia forces have to be deduced if dynamic analysis is studied. Not only the ultimate result but also the way of approaching a plastic piston – plastic hinge is important. The bended and compressed column is the structure when inelastic dynamic analysis is really important.

Closed-Form Approximate Formulas for Torsional Analysis of Hollow Tubes with Straight and Circular Edges

ABSTRACTSome analytical formulas are presented for torsional analysis of homogeneous hollow tubes. The cross section is supposed to consist of straight and circular segments. Thicknesses of segments of the cross section can be different. The problem is formulated in terms of Prandtl's stress function. The derived approximate formulas are so simple that computations can be carried out by a simple calculator. Several examples are presented to validate the formulation. The accuracy of formulas is verified by accurate finite element method solutions. It is seen that the error of the formulation is small and the formulas can be used for analysis of thin to moderately thick-walled hollow tubes.

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.

Torsion of functionally graded hollow tubes

An analytical formulation is presented for torsional analysis of functionally graded hollow tubes of arbitrary shape. Moreover, relatively simpler formulas are presented for torsional analysis of functionally graded hollow tubes of polygonal shape. Thicknesses of all segments of the cross section are the same. Shear modulus of rigidity changes continuously across the thickness between two phases according to a power law distribution. Governing equations in terms of Prandtl's stress function are used to derive the formulas. Several examples are presented to show the accuracy and efficiency of the formulation. The obtained results are verified by accurate finite element solutions. Effects of changing thickness and volume fraction of constituent materials are also investigated. Derived formulas can be useful for analysis of thin to moderately thick-walled hollow tubes.

Torsion of prismatic elastic bodies containing screw dislocations

The problem of the stressed state of a prismatic anisotropic rod containing screw dislocations, the axes of which are parallel to the rod axis, is considered. Such defects may arise during the growth of filamentary crystals (metal “whiskers”), and may also exist in multiply connected cylindrical structures. The torsion of an anisotropic elastic bar with a multiply connected cross-section is investigated initially, assuming that the stresses and strains are single-valued but dispensing with the requirement that the warping function should be single-valued. The boundary-value problem is formulated in terms of the Prandtl stress function, which, unlike the warping function, is single-valued in a multiply connected region. A variational formulation of the boundary-value problem for the stress function is given. From the variational principle obtained a torsion boundary-value problem is formulated when there are lumped or continuously distributed dislocations. A modification of the membrane analogy for the torsion problem is proposed which takes into account the presence of dislocations. General theorems of the theory of the torsion of a rod containing dislocations are formulated. An effective formula is derived for the angle of torsion of a bar due to a specified dislocation distribution. Problems on dislocations in a thin-walled rod and a rectangular anisotropic bar are solved.

An explicit integration scheme based on recursion for the curved triangular finite elements

This paper is concerned with explicit formulae for computing integrals of rational functions of bivariate polynomial numerator with linear denominator over a unit triangle {( ξ, η)|0⩽ ξ, η⩽1, ξ+ η⩽1} in the local parametric space ( ξ, η). These integrals arise in a variety of applications governing the second order linear partial differential equations. Explicit evaluation of these integrals produce algebraic finite element relations, provided that the original element geometry is restricted to a triangle with two straight sides and one curved side. For a rational integral of nth degree bivariate polynomial numerator with a linear denominator there are {( n+1)( n+2)/2} rational integrals of monomial numerators with the same linear denominator. By an expansion it is shown that these {( n+1)( n+2)/2} integrals can be computed in two ways. We have presented a recursive scheme to compute { n( n+1)/2} such integrals efficiently whenever ( n+1) integrals of order 0 (zero) to n in one of the variates are known by explicit integration formulae. The analytical quadrature formulae from zeroth to sextic order due to different element geometry are, for clarity and reference, summarized in tabular forms. An integration formula is also presented for the evaluation of integrals that occur in the calculation of stiffness matrices when straight sided triangular finite elements are considered. Finally we have considered two application examples: First example is to compute the Prandtl stress function values and the torsional constant k for an elliptic cross-section with varying ratios of major and minor distances. The other example is concerned with numerical solution for the Poisson equation in 2D. It is observed in the calculation of components of element matrices by using the Gauss quadrature rule (e.g. 7-point and 13-point) a remarkable discrepancy occurs if the element geometry contains a concave curved side. A computer code based on the present integration scheme to obtain the element stiffness matrices for both the curved and straight sided elements is appended.

Stochastic second-order perturbation approach to the stress-based finite element method

The paper is devoted to the application of the second-order perturbation second probabilistic moment method to the stress-based finite element method (FEM). The approach is introduced for the linear elastic heterogeneous medium – up to the second-order, variational equations of the complementary energy principle are presented together with an additional stochastic finite element discretization based on Airy and Prandtl stress functions. The numerical examples shown in this paper illustrate the probabilistic stress and strain tensors in the cantilever beam under shear loading and torsioned square beam with randomly defined material and geometrical parameters. The results obtained in the tests can be applied in probabilistic analyses of the boundary value problems having any closed form mathematical solutions as well as in the stress-based stochastic FEM analysis of solids and structures.

AN EXPLICIT INTEGRATION SCHEME BASED ON RECURSION AND MATRIX MULTIPLICATION FOR THE LINEAR CONVEX QUADRILATERAL ELEMENTS

In this paper we present some analytical integration formulae for computing integrals of rational functions of bivariate polynomial numerator with linear and bilinear denominator over a 2-square |ξ| = 1, |η| = 1 in the local parametric space (ξ, η). These integrals arise in finite element formulations of second order partial differential equations of plane and axisymmetric problems in continuum mechanics to computer the components of element stiffness matrices. In case of a rational integrals of n-th degree bivariate polynomial numerator with a linear of bilinear denominator there are exactly rational integrals of monomial numerators with the same linear denominator. By an expansion it is shown that these integrals can be computed in two ways, accordingly we have presented an explicit and a recursive scheme. By use of the recursive scheme such integrals can be computed efficiently with less computational effort whenever (n + 1) integrals of order zero to n in one of the variates are known by explicit integration formulae. Integration formulae from zeroth to octic order are, for clarity and reference summarized in tabular forms. Finally to show the application of the derived formulae three application examples to compute the Prandtl stress function values and the torsional constant k are considered. A computer code based on the present integration scheme to obtain the element stiffness matrices for plane problems is also developed.