Saint-Venant Theory Research Articles

Classical solution to mixed problems from the theory of longitudinal impact on an elastic semi-infinite rod in the case of separation of the impacting body after the collision

In this work, we consider two coupled initial-boundary value problems, which, based on the Saint-Venant theory, model the longitudinal impact phenomena in a semi-infinite rod. The mathematical formulation of the problem is two mixed problems for the one-dimensional wave equation with conjugation conditions. The Cauchy conditions are specified on the spatial half-line. The initial condition for the partial derivative with respect to the time variable has a discontinuity of the first kind at one point. The boundary condition, which includes the unknown function and its first- and second-order partial derivatives, is specified on the time half-line. The solution is constructed by the method of characteristics in an explicit analytical form. The uniqueness of the solution is proved, and the conditions under which a piecewise-smooth solution exists are established. The classical solution to a mixed problem with matching conditions is considered.

FINITE SHEAR STRAIN BASED ON IMAGE ANALYSIS FOR RECTANGULAR SECTION SHAFT WITH LARGE ASPECT RATIO

This paper describes the shearing strain distribution obtained by using image analysis proposed in this study when large torsion is applied to a rectangular cross-section shaft. As is well known, the shearing strain distribution for rectangular cross-section shaft have already been investigated by Saint-Venant. However, since this theory is based on the assumption that torsional angle is relatively small, it is not necessarily rational under large torsion. Therefore, in our previous study, using the image analysis based on the Natural Strain theory, where the commutative law of strain is applicable and the rigid body rotation can be removed accurately from shearing strain component, the shearing strain distributions have been investigated for the rubber shaft having square cross section. In the present paper, these distributions are investigated for a rectangular cross section with a large aspect ratio rather than a square shape. In particular, these distributions on the long side in rectangular cross-section are compared with those on the short sides. Consequently, the difference between the distribution of shearing strain obtained by the image analysis in this study and measurements based on Saint-Venant's theory increases as torsional deformation increases and this tendency is observed particularly on the short side.

FINITE SHEAR STRAIN BASED ON IMAGE ANALYSIS FOR RECTANGULAR SECTION SHAFT WITH LARGE ASPECT RATIO

This paper describes the shearing strain distribution obtained by using image analysis proposed in this study when large torsion is applied to a rectangular cross-section shaft. As is well known, the shearing strain distribution for rectangular cross-section shaft have already been investigated by Saint-Venant. However, since this theory is based on the assumption that torsional angle is relatively small, it is not necessarily rational under large torsion. Therefore, in our previous study, using the image analysis based on the Natural Strain theory, where the commutative law of strain is applicable and the rigid body rotation can be removed accurately from shearing strain component, the shearing strain distributions have been investigated for the rubber shaft having square cross section. In the present paper, these distributions are investigated for a rectangular cross section with a large aspect ratio rather than a square shape. In particular, these distributions on the long side in rectangular cross-section are compared with those on the short sides. Consequently, the difference between the distribution of shearing strain obtained by the image analysis in this study and measurements based on Saint-Venant's theory increases as torsional deformation increases and this tendency is observed particularly on the short side.

On the Torsional Rigidity of Orthotropic Beams with Rectangular Cross Section

The paper deals with the torsional rigidity of homogenous and orthotropic beam with rectangular crosssection. The torsional rigidity of the considered beam is defined in the framework of the Saint-Venant theory ofuniform torsion. Exact and approximate solutions are given to the determination of the torsional rigidity. The shapeof cross section is determined which gives maximum value of the torsional rigidity for a given cross-sectional area.The dependence of torsional rigidity as a function of the ratio shear moduli of beam is also studied.

Two Theorems on the Torsional Rigidity of Piezoelectric Beams

In this paper two inequalities are presented for the torsional rigidity of homogeneous monoclinic piezoelectric beams. All results of the paper are based on the Saint-Venant theory of uniform torsion. The cross section of the considered elastic and piezoelectric beams may be simply connected or multiply connected two-dimensional bounded plane domain. Examples illustrate the proven inequality relations.

TORSIONAL SHEAR STRESS IN PRISMATIC BEAMS WITH ARBITRARY CROSS-SECTIONS USING FINITE ELEMENT METHOD

Determining the shear stress of a structural element caused by torsion is a vital problem. The analytical solution of the Saint-Venant torsion is only suitable for simple cross-sections. The numerical method to evaluate the shear stress of complicated cross-sections is indispensable. Many numerical methods have been studied by scientists. Among these studies, Gruttmann proposed an excellent numerical method, which inherited the Saint-Venant theory. However, the use of isoparametric four-noded quadrilateral elements made the method not to reach the best optimization. The objective of this paper is to improve Gruttmann‘s method by using isoparametric eight-noded quadrilateral elements. MATLAB is the language for programming the numerical method. The validated examples have demonstrated that the author’s numerical method is more effective than Gruttmann‘s method.

Estimation of the Torsional Rigidity of Orthotropic Solid Cross Section

In this paper, two inequality relations are proven for the torsional rigidity of orthotropic elastic solid cross sections. By using the derived inequality relations, lower and upper bounds can be obtained for the torsional rigidity. All results of the paper follow from the Saint-Venant theory of uniform torsion. The presented bounding formulae are based on the mean value theorem of integral calculus.

The Study of Heat Conduction in Functionally Graded Plate Based on Hybrid Numerical Method Loading by Linear Heat Source

In this paper, a physical model of functionally graded materials (FGM) with a linear change of volume fraction of ceramic and metal in its thickness direction is established. The thermal conductivity of the FGM under a linear heat source is studied by a hybrid numerical method. Based on the weighted residual method, the heat conduction equation under a third boundary condition is studied using a hybrid numerical method which takes into account both the accuracy of the analytical method and the efficiency of the finite element method. A linear heat source is applied to the FGM, the temperature of the heat source changes linearly with time, and the temperature distribution in the space-time domain is obtained by finite element discretization in one direction and Fourier transform in the other. The results show that the closer to the heat source, the greater influence of the heat source it is. The influence of the heat source is local, which is similar to the influence of the force field on structure described by Saint Venant theory. In the heat transfer process, the heat transfer efficiency of each surface is different, that is closely related to the material properties. Using this proposed physical model, it is demonstrated that FGM can avoid the sudden change of temperature and relieve the thermal stress.

Analytical solutions of multiple cracks and cavities in a rectangular cross-section bar coated by a functionally graded layer under torsion

A dislocation solution to evaluate the mode III stress intensity factors (SIFs) of the crack tips, torsional rigidity and the hoop stress around the cavities in an infinite rectangular cross-section bar coated by an FG layer under torsion is developed based on the Saint–Venant theory. The solution of the rectangular cross section with the FG coating weakened by a screw dislocation under torsion was first obtained in terms of the dislocation density function. Then, the problem is transformed into a group of Cauchy integral equations in the rectangular cross-section bar with the FG coating layer via the distributed dislocation approach. The singular integral equations are reduced to a set of algebraic equations to evaluate them numerically. After solving the integral equations, the SIFs of the crack tips, the hoop stresses around the cavities and the torsional rigidity of the entire domain are found. The obtained solution is validated by performing comparison studies in multiple cases. It is shown that the interaction between the cracks, the cracks length, variation of the shear module of the FG coating layer and distance of the crack tips from stress-free surfaces can significantly affect the SIFs of crack tips and torsional rigidity of the rectangular cross-section bar coated by an FG layer.

DISTRIBUTION OF SHEAR STRAIN IN SQUARE CROSS-SECTION SHAFT MEASURED USING IMAGE ANALYSIS UNDER LARGE DEFORMATION

In the present study, the distribution of shear strain and warping in a square cross-section shaft generated under a large torsion is investigated using the image analysis based on the Natural Strain theory. The scribe lines are drawn in a grid on a surface of the test pieces made of natural rubber, and the image data of each element in the horizontal direction along the cross-section at the center and upper positions are taken by using a high-pixel camera equipped with macro lens. The distributions for shear strain and warping along the cross-section are obtained from that image data. These measured distributions under large deformation are compared with the distributions based on the conventional torsional theory of a square cross-section shaft, i.e., Saint Venant's theory for torsion. Moreover, taking the effect due to the elongation on the surface of specimens into consideration, the distributions modified elongation of gauge length are compared with experiments, and the validity of the present experimental results are confirmed in this paper.

A BEM approach to the evaluation of warping functions in the Saint Venant theory

The paper illustrates the numerical procedure, based upon a Boundary Element (BE) approach, developed to efficiently evaluate the warping functions in the Saint Venant theory of beam-like solids having both compact and thin-walled sections. Specifically, Chebyshev nodes are selected as collocation points of the BE formulation associated with the relevant pure Neumann problem and the entries of the resulting linear system of equations are evaluated analytically by invoking recursive formulas.Assuming a polynomial interpolation for the unknown function over each boundary element, we show that a reduction in the numerical accuracy of the solution is achieved if the polynomial degree exceeds a given order strictly related to the strategy adopted to discretize the boundary. For this reason, in order to automatically cope both with compact and thin-walled domains, a general criterion has been established for properly selecting the best combination of polynomial degree and edge discretization capable of reducing the numerical error of the procedure below a given tolerance.

Displacement and Stress Function-based Linear and Quadratic Triangular Elements for Saint-Venant Torsional Problems

Torsional problems commonly arise in frame structural members subjected to unsym­metrical loading. Saint-Venant proposed a semi inverse method to develop the exact theory of torsional bars of general cross sections. However, the solution to the problem using an analytical method for a complicated cross section is cumbersome. This paper presents the adoption of the Saint-Venant theory to develop a simple finite element program based on the displacement and stress function approaches using the standard linear and quadratic triangular elements. The displacement based approach is capable of evaluating torsional rigidity and shear stress distribution of homogeneous and nonhomogeneous; isotropic, orthotropic, and anisotropic materials; in singly and multiply-connected sections. On the other hand, applications of the stress function approach are limited to the case of singly-connected isotropic sections only, due to the complexity on the boundary conditions. The results show that both approaches converge to exact solutions with high degree of accuracy.

Nonhydrostatic Dam Break Flows. II: One-Dimensional Depth-Averaged Modeling for Movable Bed Flows

AbstractThe dam break flow over a movable bed is an important problem in fluvial flow processes. These flows are usually predicted by a one-dimensional (1D) approach based on a hydrostatic pressure distribution. Recent three-dimensional (3D) nonhydrostatic simulations of dam break waves over movable beds based on the Reynolds-averaged Navier-Stokes (RANS) equations revealed that the Saint Venant theory is not accurate in predicting the flow dynamics within the scour hole developed. In this work, a generalized 1D nonhydrostatic model for flow over movable beds is proposed assuming a linear, nonhydrostatic, pressure distribution. The new set of 1D equations accounts for the vertical acceleration, which is important in dam break waves over movable beds given the instantaneous curved beds generated over the erodible terrain. These equations account for both the bed and suspended-load transport modes. A high-resolution finite volume numerical scheme with a semi-implicit treatment of nonhydrostatic terms is dev...

Dressler’s theory for curved topography flows: iterative derivation, transcritical flow solutions and higher-order wave-type equations

The Dressler equations are a system of two non-linear partial differential equations for shallow fluid flows over curved topography. The theory originated from an asymptotic stretching method formulating the equations of motion in terrain-fitted curvilinear coordinates. Apparently, these equations failed to produce a transcritical flow profile changing from sub- to supercritical flow conditions. Further, wave-like motions over a flat bottom are excluded because the bed-normal velocity component is not accounted for. However, the theory was found relevant for several environmental flow problems including density currents over mountains and valleys, seepage flow in hillslope hydrology, the development of antidunes, the formation of geological deposits from hyper-concentrated flows, and shallow-water flow modeling in hydraulics. In this work, Dressler’s theory is developed in an alternative way by a systematic iteration of the stream and potential functions in terrain-fitted coordinates. The first iteration was found to be the former Dressler’s theory, whereas a second iteration of the governing equations results in velocity components generalizing Dressler’s theory to wave-like motion. Dressler’s first-order theory produces a transcritical flow solution over topography only if the total head is fixed by a minimum value of the specific energy at the transition point. However, the theory deviates from measurements under subcritical flow conditions, given that the bed-normal velocity component is significant. A second iteration to the velocity field was used to produce a second-order differential equation that resembles the cnoidal-wave theory. It accurately produces flow over an obstacle including the critical point and the minimum specific energy as part of the numerical solution. The new cnoidal-wave model compares well with the theory of a Cosserat surface for directed fluid sheets, whereas the Saint-Venant theory appears to be poor.

Three-Dimensional Beam Theory for Flexible Multibody Dynamics

In multibody systems, it is common practice to approximate flexible components as beams or shells. More often than not, classical beam theories, such as the Euler–Bernoulli beam theory, form the basis of the analytical development for beam dynamics. The advantage of this approach is that it leads to simple kinematic representations of the problem: the beam's section is assumed to remain plane and its displacement field is fully defined by three displacement and three rotation components. While such an approach is capable of accurately capturing the kinetic energy of the system, it cannot adequately represent the strain energy. For instance, it is well known from Saint-Venant's theory for torsion that the cross-section will warp under torque, leading to a three-dimensional deformation state that generates a complex stress state. To overcome this problem, sectional stiffnesses are computed based on sophisticated mechanics of material theories that evaluate the complete state of deformation. These sectional stiffnesses are then used within the framework of a Euler–Bernoulli beam theory based on far simpler kinematic assumptions. While this approach works well for simple cross-sections made of homogeneous material, inaccurate predictions may result for realistic configurations, such as thin-walled sections, or sections comprising anisotropic materials. This paper presents a different approach to the problem. Based on a finite element discretization of the cross-section, an exact solution of the theory of three-dimensional elasticity is developed. The only approximation is that inherent to the finite element discretization. The proposed approach is based on the Hamiltonian formalism and leads to an expansion of the solution in terms of extremity and central solutions, as expected from Saint-Venant's principle.

Design of Flexible Propellers with Optimized Load-Distribution Characteristics

The mathematical model and experimental verification of flexible propeller blades are presented in this paper. The propeller aerodynamics model is based on an extended blade-element momentum model, while the Euler–Bernoulli beam theory and Saint–Venant theory of torsion are used to account for bending and torsional deformations of the blades, respectively. The proposed blade-element momentum model extends the standard blade-element momentum theory with the aim of providing a quick and robust model of propeller action capable of treating high-aspect-ratio propeller blades with a blade axis of arbitrary geometry. Based on the proposed mathematical model, a static flexible propeller blade design procedure and its associated analysis algorithm are established. Dynamic aeroelastic phenomena like propeller flutter and divergence are not covered by the presented mathematical model, design, and analysis algorithm. Experimental validation was carried out with an objective of evaluating the performance of the developed mathematical model and the design strategy. Both theoretical and experimental results are presented along with pertinent concluding remarks.

A survey of A. Yu. Ishlinsky’s papers on the occasion of his 100th birthday

A. Yu. Ishlinsky’s scientific publications impress one by the extremely wide scope of his scientific interests. Most of his results lie in two fundamental fields of research, mechanics of deformable bodies and mechanics of gyroscope and navigation systems; chronologically, they appeared mainly in this order. One of his first important results that became widely known was the construction of an exact theory of the Brinell test (1942). This theory dealt with determining the stress state under a rigid punch of spherical or plane shape penetrating into a perfectly plastic medium. The problem was solved in the rigorous statement in the framework of the Haar–von Karman complete plasticity hypothesis containing two relations between the principal stresses rather than one relation as in the Saint-Venant theory. The subsequent development of this problem led (1954) to construction of a general theory of plasticity which, in particular, allowed one to overcome the difficulties related to the derivation of the 3D strain equations beyond the elasticity range. A series of Ishlinsky’s papers (1940–1946) deals with imperfect elasticity, fracture theory, and plasticity theory. In this series, he constructed a description of aftereffect and relaxation generalizing the Maxwell and Thomson concepts on the basis of a linear differential relation between the stress and the strain. Plastic fracture models were also studied in these works. In particular, one of the fracture models was constructed for the example of a body whose one-dimensional strain obeys the law dσ/dt+ rσ= bde/dt+ bne, where σ is the stress and e is the strain. This model also admits a simple and illustrative interpretation as a damper with springs, which permits readily understanding the introduced fracture hypotheses and analyzing specific examples for various loading laws. Ishlinsky returned to plasticity and fracture theories in 1965–1968. The problem on the impact of a viscoplastic rod on a rigid obstacle is of interest. In the model accepted by Ishlinsky, the motion of the rod cross-sections is governed by heat equations with complex boundary conditions. It turned out unexpectedly that on impact the viscoplastic state does not instantly extend to the entire rod; instead, it runs away from the impact point and then returns without reaching the free end, thus leaving a rigid segment of the rod. In fracture theory of this time period, one should mention Ishlinsky’s paper of 1968, where two crack development models are compared. One model is the Griffith–Irvin model, and the other, which was developed later, is due to S. A. Khristianovich and G. I. Barenblatt. Ishlinsky showed that the latter model tends to the former as the range of the adhesion forces postulated in the latter tends to zero. Ishlinsky’s papers on the stability of elastic systems are of extreme importance. In a joint 1949 paper with M. A. Lavrent’ev, he theoretically explained the spectacular experiment in which a tube subjected to an underwater explosion loses its shape in such a way that the number of corrugations is the greater the closer they are to the explosion site. It turned out that the relationship between the characteristic exponents, the equilibrium mode number, and the applied external load constructed when studying the equilibrium modes permits one to maximize the buckling rate with respect to the mode number. The greater the applied force, the higher the mode number maximizing the characteristic exponent is. This result has numerous practical applications. The buckling problem for elastic systems, which goes back to Euler, is usually solved by methods of the theory of strength of materials for simplified models of rod or plate type. Ishlinsky studied this

Calculation of the dynamic deflection of a beam on inelastic impact by the Cox and Saint-Venant theories

A comparison of the results of a calculation of dynamicity coefficient for deflections, obtained by the Cox and Saint-Venant theories, showed them to agree well in a wide range of mass ratios of colliding bodies. Therefore, cumbersome calculations by the Saint-Venant theory can be replaced by simplified calculations from the compact Cox formula.

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.

On torsion and shear of Saint-Venant beams

Torsion and shear stress fields of a Saint-Venant beam and the relative location of shear and twist centres are investigated for sections of any degree of connectedness. The sliding-torsional compliance tensor of a Timoshenko beam is evaluated by an energy equivalence with Saint-Venant theory. Accordingly, the mutual sliding-torsional term is shown to depend linearly on the relative position of shear and twist centres and the standard definition of shear centre in a Timoshenko beam is found to be coincident with Saint-Venant twist centre. Coincidence of shear and twist centres is assessed for sections with vanishing Poisson ratio and for open, closed and multi-cell thin-walled cross sections. The eigenvalues of the shear factors tensor and the torsion factor are shown to be greater than unity, with the principal directions of shearing and bending compliances non necessarily coincident for non-symmetric cross sections. Numerical examples are developed to provide evidences of the location of the centres and of the principal shearing directions, for non-symmetric L-shaped cross sections with various thickness ratios.