Saint-Venant Solution Research Articles

Buckling analysis of thin-walled laminated composite or functionally graded sandwich I-beams using a refined beam theory

This study presents the buckling and lateral torsional buckling behaviors of thin-walled laminated composite or functionally graded sandwich I-beams. A refined beam model (RBT) based on the 3D Saint-Venant (SV) solution is used in this study to formulate the problem. This model allows for a realistic analysis of beams with arbitrary cross-sections and is free from the limitations of classical beam theories. The displacement models establish a consistent 1D beam theory that accurately captures the essence of the cross-section’s nature. The governing models consider cross-section deformations, including in and out of plane warpings and distortions derived from the 3D SV solution associated with the cross-section vibrational behavior. Furthermore, a user-friendly numerical tool called CSB (cross-section and beam analysis) is used to facilitate the implementation of this method. The numerical results are examined in detail and compared with previous works to investigate the influence of boundary conditions, angle-ply, shear effects, span-to-height ratio, and material distribution on the critical buckling loads of thin-walled I-beams. The results demonstrate that the RBT models accurately and efficiently analyze thin-walled I-beams under different loads and boundary conditions. Furthermore, some of the new results are presented as reference values for the future.

Analytic solutions for quasi-3D seepage in a shallow unconfined aquifer as a plane composed of a transpiration-inducing park and its hydraulically commingled exterior

Strack’s (1984, 2017) analysis of infiltration from a pond into a shallow aquifer, with a generated groundwater mound, is modified for steady phreatic groundwater flows in perched aquifers, typical for arid/hyperarid environments, with a zone of intensive transpiration and induced groundwater trough. The interior of an urban phreatophytic park is an n-polygon (in an aerial view), which degenerates into a circle at n→∞. Transpiration acts as an areal groundwater sink. The water table drawdown is hydrologically symmetric with Strack’s drawup. In the exterior of the park, the ground surface is phyto-sterile (e.g. a desert in Arabia or a paved megapolis) and evapotranspiration from the water table there is ignored. The travel (residence) time of advected marked particles along the flowpaths (streamlines) towards the park and under it are analytically evaluated. In terms of the Dupuit-Forchheimer model of a vertically averaged hydraulic head, the Strack potential obeys the Laplace and Poisson equations in polygon’s exterior and interior (correspondingly), with a hydraulic commingling along the park boundary. The tracer particles start their topologically converging journey from a remote circular zone of contamination (e.g. sewage from urban septic tanks recharging the water table). Porosity, hydraulic conductivity of the aquifer, evapotranspiration rate and a piezometric head in a remote observational well are given. For a circular park flow is 1-D radial. In the park exterior, the Schwarz-Christoffel formula maps conformally the physical domain (an infinite trigon) onto a half-strip in the complex potential plane. Strack’s potential is expressed via hypergeometric functions, whose parameters depend on n. The flow net is reconstructed with the help of the stream function. For a triangular park interior, there is no stream function and the Saint-Venant solution to the Poisson equation is engaged to find the water table, hydraulic gradients, and discharge streamlines. The total dewatered volume and groundwater storage under the water table and “gross travel time” are evaluated for circular parks. Implications for ecohydrology of parks and desert oases in the Gulf countries and intuitive isoperimetric problems are discussed.

Analysis of FGM Cantilever Beams under Bending-torsional Behavior Using a Refined 1D Beam Theory

The static bending-torsion problem of functionally graded cantilever beams is studied using a refined 1D/3D beam theory (Refined beam theory RBT and Refined beam theory with distortion modes RBT*) built on the 3D Saint-Venant (SV) solution. In these theories, the displacement models include Poisson's effects, out-of-plane deformations and distortions. For a given section, the sectional displacement modes are derived from the computation of the particular 3D Saint-Venant’s solution. These modes, which reflect the mechanical behavior of the cross-section, lead to a beam theory that actually corresponds to the cross-section type in terms of shape and material. In addition, the models take into account edge effects to predict a 3D solution in a larger internal region to better describe the overall behavior of FGM beams. The models examined are implemented on the CSB (Cross-Section and Beam Analysis) tool. It is based on the RBT/SV (Refined Beam Theory based on the 3D SV’s solution) theory of FGM beams. The mechanical and physical characteristics of the FGM beams vary continuously, according to a power-law distribution, through the thickness of the beams. The numerical and 3D results obtained with homogeneous and FGM beams are systematically compared with other models in the literature and those provided by the full Saint-Venant beam theory (SVBT) calculations.

Microstructural topology optimization of periodic beam structures based on the relaxed Saint-Venant solution

Design optimization of beam structures is a significant topic as beams are an efficient load-carrying component in engineering applications. Most of the earlier researches concentrate on the structural optimization of beams with invariant cross-section topology, and design optimization of periodic beam structures, which cross-section topology varies along the axial direction, has not been extensively investigated. This work presents a novel approach based on the relaxed Saint-Venant solution to conduct microstructural topology optimization of periodic beam structures for minimum structural compliance. One benefit of adopting Saint-Venant solution based compliance formulation is that the strain energy induced by transverse shear loading is incorporated in the structural compliance, which is not reflected in that based on first-order homogenization of the asymptotic homogenization (AH) theory, so that a more reasonable objective function is adopted for the optimization problem. In addition, a material connectivity constraint, which is constructed by restraining the ratio of the strain energy calculated from relaxed Saint-Venant solution to that obtained from first-order homogenization of the AH theory, is further proposed to prevent material separation or to strengthen material connection through the thickness direction. The detailed sensitivity analysis of the objective function and the constraints are carried out with the adjoint method, and several numerical examples are given to show the validity of the relaxed Saint-Venant solution based compliance formulation and the effectiveness of the proposed material connectivity constraint.

Parametric finite-volume method for Saint Venant’s torsion of arbitrarily shaped cross sections

We extend our recent finite-volume based approach to the solution of Saint Venant’s torsion problems of bars and beams comprised of rectangular sections to enable analysis of arbitrary cross sections characterized by curved boundaries. This is accomplished by incorporating parametric mapping based on transfinite grid generation to enable discretization of the bar cross section by quadrilateral rather than rectangular subvolumes employed in the original version. The construction of the local stiffness matrix that relates the surface-averaged subvolume warping functions to the corresponding tractions is carried out in the reference plane such that the subvolume equilibrium in the physical plane is satisfied in a surface-averaged sense. This produces explicit expressions for the stiffness matrix elements that may be readily coded. Orthotropic subvolumes are intrinsic in the method’s construction so that bars with heterogeneous and composite microstructures may be analyzed. The convergence and accuracy of the parametric finite-volume method are assessed and verified upon comparison with exact elasticity solutions for cross sections with convex and concave boundaries. Examples involving structural applications of prismatic bars with curved boundaries illustrate the utility of the developed methodology. These include cross sections that resemble biological constructs with homogeneous and graded regions aimed at enhancing torsional rigidities, as well as homogeneous and graded elliptical cross sections with orthotropic shear moduli aimed at reducing and eliminating warping. We demonstrate for the first time that by laminating an elliptical cross section with alternating stiff and soft isotropic layers in a manner that mimics the required orthotropic moduli at the homogenized level, warping can be practically eliminated with sufficient microstructural refinement.

Papkovitch Neuber Type Solutions for Viscoelasticity

For viscoelastic problems, Saint Venant solution can only satisfy the boundary conditions in the mean sense, while the local effect solution describes the stress and deformation in the local area near the load. In many practical problems, we only care about the effect of load on the whole material and structure, and the effect of local effect solution is often ignored. However, for some special cases, such as the cantilever beam with free end under bending moment, the stress at the fixed end is not uniformly distributed due to the restriction of displacement conditions, especially the stress at the upper and lower edges is far greater than the average stress on the cross section, so the local solution plays an important role.

Multiphysics cross-section analysis of smart beams

The cross-section response of smart beams under electric, magnetic and thermal fields is investigated. The method predicts the generalized cross-section stiffness of prismatic smart beams, and allows recovering the three dimensional multiphysics fields variables. The semi-analytical procedure divides the contributions of the non-homogeneous governing equation into a homogeneous solution, which resorts to computing the de Saint-Venant solution, and a set of particular solutions which describe the beam cross-section behaviors when external electric potential, magnetic potential and boundary temperature are imposed. The generalized cross-section stiffness is computed by means of an energy equivalence equation.

Saint Venant solutions for elastic cylinders

Under the Hamilton system, the dual form governing equations of the cylindrical problem are established using displacements and stresses as the basic variables, and the problem is accordingly transformed into finding the eigenvalues and eigensolutions by adopting the method of separation of variables. Furthermore, solutions for zero eigenvalues and non-zero eigenvalues are systematic studied. The study shows that zero eigensolutions are composed of all the overall deformation solutions such as the traditional tension and bending problems, while non-zero eigensolutions include the torsion and bending groups charactered by local deformations.

Saint Venant’s torsion of homogeneous and composite bars by the finite volume method

We propose a new finite-volume based approach to the solution of Saint Venant’s torsion problems of homogeneous and composite prismatic bars subjected to pure torsion. The semi-inverse approach is employed to construct an approximate displacement field characteristic of pure torsion within the subvolumes of the beam’s discretized cross section such that the governing differential equation for the warping function is locally satisfied in each subvolume in an integral sense. Orthotropic subvolumes are admitted in the formulation to enable analysis of bars comprised of heterogeneous microstructures, including composite materials with orthotropic shear moduli. Continuity of both displacements and tractions across subvolumes’ interfaces is satisfied in a surface-average sense, together with traction-free lateral boundary conditions. Closed-form expressions for the stiffness matrix elements that relate surface-averaged displacements to tractions are provided for ease of programming. The convergence and accuracy of the proposed method are assessed and verified upon comparison with the exact elasticity solutions for prismatic homogeneous and composite bars of rectangular cross section and the corresponding finite-difference solution for the homogeneous bar. The utility of the developed solution methodology is first illustrated by analyzing torsion of homogeneous T and box beams, focusing on the singular behavior at the re-entrant corners and the concomitant membrane analogy’s applicability. Then, composite cross sections reinforced by angle braces as well as continuous fiber-reinforced wraps with graded shear moduli are investigated with the aim of mitigating the re-entrant corner shear stress singularities, not easily analyzed using the finite-element or finite-difference approaches.

Closed-Form Saint-Venant Solutions in the Koiter Theory of Shells

In this paper we investigate the deformation of cylindrical linearly elastic shells using the Koiter model. We formulate and solve the relaxed Saint-Venant's problem for thin cylindrical tubes made of isotropic and homogeneous elastic materials. To this aim, we adapt a method established previously in the three-dimensional theory of elasticity. We present a general solution procedure to determine closed-form solutions for the extension, bending, torsion and flexure problems. We remark the analogy and formal resemblance of these solutions to the classical Saint-Venant's solutions for solid cylinders. The special case of circular cylindrical shells is also discussed.

On the solutions to the Saint–Venant problem of heterogeneous beam-like structures with periodic microstructures

This paper presents displacement solutions of the Saint-Venant problem for linear elastic heterogeneous beam structures with arbitrarily shaped periodic microstructures. The solutions are generated by introducing unknown axially periodic displacement functions in place of the axially invariant warping functions in Ieşan's rational scheme, due to the periodic characteristics of the beam structure, while retaining the displacement fields integrated from rigid-body displacements, which represent basic kinematics in beam theories. The governing equations of the unknown functions, defined on a base cell, are then developed from equilibrium equations and boundary conditions of the periodic beam structure. Moreover, an improved FE formulation of the governing equations is presented and its efficient numerical implementation approach, which can be readily realized using FE software as a black box, is proposed with the aid of NIAH (Novel Numerical Implementation of Asymptotic homogenization) approach. To illustrate the validity of the Saint-Venant solution and the effectiveness of the numerical implementation approach, several numerical examples are presented and compared with detailed three-dimensional FE analysis for the heterogeneous beam structures in terms of stress components.

Saint-Venant Problems for Two-dimensional Elastic Solids

Using the symplectic system method, the classical Saint-venant problems in elasticity are studied. Based on the Laplace transformation and integral transformation and the symplectic character, all the Saint-Venant solutions, including the tension, bending and local solutions, are obtained. The final solution of the problem can be directly analysed in the time domain solution space. As applications of the proposed method, some stress concentration boundary condition problems are discussed.

Laminar flow of Newtonian liquids in ducts of rectangular cross-section a model for both physics and mathematics

In this paper, we considered the laminar fully developed flow, of a Newtonian fluid, inducts of rectangular cross-section. Poisson’s partial differential equation Saint-Venant solution was used, to calculate Poiseuille number values whatever is rectangles aspect ratio. From these results, we considered limit cases of square duct and plane Poiseuille flow (infinite parallel plates). We showed there exists a rectangle equivalent to a circular cross-section for energy dissipation through viscous friction. Finally, we gave some mathematical consequences of this approach for odd integers zeta function calculations and Catalan’s constant.

Buckling analysis of homogeneous or composite I-beams using a 1D refined beam theory built on Saint Venant's solution

Linear buckling of homogeneous or composite I-beams are investigated using a refined beam theory built on 3D Saint-Venant's solution. The kinematic model includes sectional Poisson's effects and out-of plane warpings extracted from 3D Saint-Venant's solution and distortions related to the free vibrational behaviour of the cross-section. Available for an arbitrary cross-section and free from the classical beam and shell-like assumptions, such displacement model leads to a consistent 1D beam theory which really matches the cross-section nature (shape and materials). Moreover, in order to easily apply this method, a user friendly numerical tool has been developed. To show the capabilities and the accuracy of the proposed model to deal with thin-to-thick walled beams, homogeneous and laminated I-beam configurations subjected to axial or transversal loads are performed. The numerical results are compared to literature and some of them to 3D-FEM solid model.

On necessary precautions when measuring solid polymer linear viscoelasticity with dynamic analysis in torsion

Solid polymer linear viscoelasticity in shear is often characterized by applying torsion and using the Saint-Venant solution when rectangular prismatic specimens are considered. It is shown that experimental dynamic torsion tests can show a dependency of the storage modulus and damping factor on the dimensions of the rectangular prismatic specimen when linear temperature ramps are applied. While the discrepancy of damping factor is explained by temperature heterogeneities and can be corrected easily by applying temperature steps, the inconsistency of storage modulus is due to an invalid application of the Saint-Venant solution. Finite element simulations allowed definition of the sample dimensions for which the Saint-Venant solution provides a good approximation, and a coefficient is given to correct the results obtained with commercial instruments when other sample dimensions are used.

Justification of the Asymptotic Expansion Method for Homogeneous Isotropic Beams by Comparison with De Saint-Venant’s Solutions

The formal asymptotic expansion method is an attractive mean to derive simplified models for problems exhibiting a small parameter, such as the elastic analysis of beam-like structures. Usually this method is rigorously justified using convergence theorems Yu and Hodges, 2004. In this paper it is illustrated how the Saint-Venant’s solution naturally arises from the lowest order terms of an asymptotic expansion of the elastic state for the case of homogeneous isotropic beams. It is also highlighted that the Saint-Venant solutions corresponding to pure traction, bending and torsion involve the solution of the first-order microscopic problems, while for the simple bending problem, the solution of the second-order microscopic problems is needed. The second-order problems provide therefore a way to characterize the transverse shear behavior and the cross-sectional warping of the beam.

Buckling analysis through a generalized beam model including section distortions

A geometrically nonlinear beam model suitable for describing complex 3D effects due to non-uniform warpings including non-standard in-plane distortions of the cross-section is presented and applied to the buckling analysis of beams. Each section is endowed with a corotational frame where statics and kinematics are described using a refined linear elastic model which exploits a semi-analytical solution of the Cauchy continuum problem based on a FEM discretization of the cross-section. The stress field in this way is fully 3D, allowing both the exact recovery of the standard Saint Venant solution and the consideration of some additional relevant strain modes of the cross-section that are evaluated in a simple and effective way. Numerical results are presented and compared with 3D shell reference solutions obtained by using the commercial code ABAQUS.

A geometrically exact beam model with non-uniform warping coherently derived from the Saint Venant rod

A new geometrically nonlinear model for homogeneous and isotropic beams with generic section including non-uniform warping due to torsion and shear is derived. Each section is endowed with a corotational frame where statics and kinematics are described using a 3D linear elastic model which extends the Saint-Venant solution to non-uniform warping cases. The algebra of change of observer and a mixed variational principle give the model in terms of generalized parameters. Using a mixed interpolation the model is implemented within a FEM Koiter analysis highly sensitive to the geometrical coherence of the formulation.

A composite beam model including variable warping effects derived from a generalized Saint Venant solution

A model for beams with heterogeneous and anisotropic cross-section is presented. A semi-analytical approach which exploits a FEM discretization of the cross-section is used to derive a Ritz–Galerkin approximation of the stress and displacement fields. In this way the Saint-Venant solution is easily generalized to generic composite sections and additional variable warping effects are also included in the model. Numerical results relative to composite beams with compact or thin walled section are presented and compared with full 3D analyses.

A higher-order beam model for tubes

Abstract A refined beam model is proposed for tubes to improve the prediction of the transverse shear deformation and study its influence on the mechanical response of tubes. The current beam model adopts a Laurent series expansion form for the displacement field, instead of the conventional Taylor series expansion form used by most higher-order beam models. This new form can capture the geometric features of tubes very well and, more importantly, shear stress can be vanished readily on the inner and outer surfaces. Both a generalized theory and a simplified third-order model are presented. It is found that the transverse shear stress distribution of the third-order model is very close to that predicted by the Saint-Venant solution. Moreover, the static bending, wave propagation, and free vibration problems are studied, whose results are also compared with those obtained from other beam models and three-dimensional elasticity solutions to show the advantage of the current model.