The convergent condition and united formula of step reduction method

The step reduction method was first suggested by Prof. Yeh Kai-yuan^[1]. This method has more advantages than other numerical methods. By this method, the analytic expression of solution can he obtained for solving nonuniform elastic mechanics. At the same time, its calculating time is very short and convergent speed very fast. In this paper. the convergent condition and united formula of step reduction method are given by mathematical method. It is proved that the solution of displacement and stress resultants obtained by this method can converge to exact solution uniformly. when the convergent condition is satisfied. By united formula, the analytic solution can be expressed as matrix form, and therefore the former complicated expression can be avoided. Two numerical examples are given at the end of this paper which indicate that by the theory in this paper, a right model can be obtained for step reduction method.Project Supported by Science and Technic Fund of the National Education Committee.The author would like to thank Prof. Yeh Kai-yuan for his directing.     

The general solution on nonlinear axial symmetrical deformation of nonhomogeneous cylindrical shells

In this paper, the general solution on nonlinear axial symmetrical deformation of nonhomogeneous cylindrical shells is obtained by step reduction method⁽¹⁾. The general formula of displacements and stress resultants, which is used to solve the bending problems of nonhomogeneous cylindrical shells under arbitrary axial symmetric loads, is derived. Its uniform convergence is proved. Finally, it is only necessary to solve one set of binary linear algebraic equations. A numerical example is given at the end of the paper which indicates satisfactory results of displacement and stress resultants can be obtained and converge to the exact solution.     

The general solution for axial symmetrical bending of nonhomogeneous circular plates resting on an elastic foundation

In this paper,a new method,the exact analytic method,is presented on the basis of stepreduction method.By this method,the general solution for the bending of nonhomogenouscircular plates and circular plates with a circular hole at the center resting,on an elastfcfoundation is obtained under arbitrary axial symmetrical loads and boundary conditions.The uniform convergence of the solution is proved.This general solution can also be applieddirectly to the bending of circular plates without elastic foundation.Finally,it is onlynecessary to solve a set of binary linear algebraic equation.Numerical examples are givenat the end of this paper which indicate satisfactory results of stress resultants anddisplacements can be obtained by the present method.     

Incompatible curved quadrilateral plate bending element with 12-degrees of freedom

This paper presents a new curved quadrilateral plate element with12-degree freedom by the exact element method.The method can be used to arbitrary non-positive and positive definite partial differential equations without variation principle.Using this method,the compatibility conditions between element can be treated very easily,if displacements and stress resultants are continuous at nodes between elements.The displacements and stress resultants obtained by the present method can converge to exact solution and have the second order convergence speed.Numerical examples are given at the end of this paper,which show the excellent precision and efficiency of the new element.     

Accurate equations for laminated composite deep thick shells

This paper derives accurate equations of elastic deformation for laminated composite deep, thick shells. The equations include shells with a pre-twist and accurate force and moment resultants which are considerably different than those used for plates. This is due to the fact that the stresses over the thickness of the shell have to be integrated on a trapezoidal-like cross-section of a shell element to obtain the stress resultants. Numerical results are obtained and showed that accurate stress resultants are needed for laminated composite deep thick shells, especially if the curvature is not spherical. A consistent set of equations of motion, energy functionals and boundary conditions are also derived. These may be used in obtaining exact solutions or approximate ones like the Ritz or finite element methods.     

Exact analytic method for solving variable coefficient differential equation

Many engineering problems can be reduced to the solution of a variable coefficient differential equation. In this paper, the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition. By this method, the general computation format is obtained. Its convergence is proved. We can get analytic expressions which converge to exact solution and its higher order derivatives unifornuy. Four numerical examples are given, which indicate that satisfactory results can be obtained by this method.     

An exact element method for bending of nonhomogeneous thin plates

In this paper, based on the step reduction method, a new method, the exact element method for constructing finite element, is presented. Since the new method doesn't need the variational principle, it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficient. By this method, a triangle noncompatible element with 6 degrees of freedom is derived to solve the bending of nonhomogeneous plate. The convergence of displacements and stress resultants which have satisfactory numerical precision is proved. Numerical examples are given at the end of this paper, which indicate satisfactory results of stress resultants and displacements can be obtained by the present method.     

An exact element method for plane problem

In this paper, based on the step reduction method^[1] and exact analytic method^[2] anew method-exact element method for constructing finite element, is presented. Since the new method doesn ’t need the variational principle, it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficient. By this method, a quadrilateral noncompatible element with 8 degrees of freedom is derived for the solution of plane problem. Since Jacobi ’s transformation is not applied, the present element may degenerate into a triangle element. It is convenient to use the element in engineering. In this paper, the convergence is proved. Numerical examples are given at the endof this paper, which indicate satisfactory results of stress and displacements can be obtained and have higher numerical precision in nodes.     

AN EXACT ELEMENT METHOD FOR THE BENDING OF NONHOMOGENEOUS REISSNER’S PLATE

In this paper,based on the step reduction method and exact analytic method,a new method,the exact element method for constructing finite element,is presented.Since the new method doesn’t need variational principle,it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficients.By this method,a triangle noncompatible element with15 degrees of freedom is derived to solve the bending of nonhomogenous Reissner’s plate.Because the displacement parameters at the nodal point only contain deflection and rotation angle.it is convenient to deal with arbitrary boundary conditions.In this paper,the convergence of displacement and stress resultants is proved.The element obtained by the present method can be used for thin and thick plates as well,Four numerical examples are given at the end of this paper,which indicates that we can obtain satisfactory results and have higher numerical precision.     

Stress singularities resulting from various boundary conditions in angular corners of plates of arbitrary thickness in extension

The stress singularities in angular corners of plates of arbitrary thickness with various boundary conditions subjected to in-plane loading are studied within the first-order plate theory. By adapting an eigenfunction expansion approach a set of characteristic equations for determining the structure and orders of singularities of the stress resultants in the vicinity of the vertex is developed. The characteristic equations derived in this paper incorporate that obtained within the classical plane theory of elasticity (M.L. Williams’ solution) and also describe the possible singular behaviour of the out-of-plane shear stress resultants induced by various boundary conditions.     

The Explicit Expression of Internal Forces in Prismatic Membered Structures*

ABSTRACT

Analytic expressions for member forces in linear elastic redundant trusses have recently been given by the author. It was shown that the internal forces in a truss are the ratios of two multilinear homogeneous polynomials in the longitudinal stiffnesses of the elements of the structure. The order of the polynomials is equal to the number of nodal degrees of freedom of the structure. The number of terms of each polynomial is equal to the number of statically determinate stable substructures that can be derived from the original structure. It was shown that coefficients of the polynomials can be computed through the equilibrium equations and by enforcing global compatibility of deformations. This paper generalizes these results to the case of linear elastic structures, composed of uniform prismatic elements that have extensional, flexural, and torsional stiffness. This is done by replacing each bi-modal bending element with a unimodal moment element and a unimodal shear element. This allows the representation of deformation of the elements by six uncoupled basic deformation patterns in the case of structures in space, thus paving the way for truss-type analysis for general structures that are composed of uniform prismatic elements. As a result, multilinear polynomials that appear in expressions for stress resultants are the products of axial, bending, and torsional stiffnesses of subsets of the original structure. The number of terms appearing in the polynomials renders the exact analytic expressions intractable for practical engineering structures. However, the construct of the analytic equations may constitute a basis for writing approximate expressions for member forces in frames, explicitly in terms of rigidities of components of the structure. This paper describes such an approximate expression, with a reduced number of terms in the polynomials. The theory is illustrated with numerical examples of analysis of planar structures     

Shear correction factors and an energy-consistent beam theory

Presented here is a new derivation of shear correction factors for isotropic beams by matching the exact shear stress resultants and shear strain energy with those of the equivalent first-order shear deformation theory. Moreover, a new method of deriving in-plane and shear warping functions from available elasticity solutions is shown. The derived exact warping functions can be used to check the accuracy of a two-dimensional sectional finite-element analysis of central solutions. The physical meaning of a shear correction factor is shown to be the ratio of the geometric average to the energy average of the transverse shear strain on a cross section. Examples are shown for circular and rectangular cross sections, and the obtained shear correction factors are compared with those of Cowper (1966) . The energy-averaged shear representative is also used to derive Timoshenkos beam theory.     

Bending and flexure of cylindrically monoclinic elastic cylinders

We consider a circular cylinder of linearly elastic material with cylindrically monoclinic material symmetry. This represents a model for a helically wound composite cable or wire rope. The elastic moduli are allowed to be arbitrary functions of the radius r. The cylinder undergoes deformation in which the axis of the cylinder is bent into a plane quartic curve. For the resulting stress field, we obtain exact integrals of the equilibrium equations, and derive simplified expressions for the shear stress resultants and bending moments.     

Stress resultants plasticity with general closest point projection

In this paper, general closest point projection algorithm is derived for the elastoplastic behavior of a cross-section of a beam finite element. For given section deformations, the section forces (stress resultants) and the section tangent stiffness matrix are obtained as the response for the cross-section. Backward Euler time integration rule is used for the solution of the nonlinear evolution equations. The solution yields the general closest projection algorithm for stress resultants plasticity model. Algorithmic consistent tangent stiffness matrix for the section is derived. Numerical verification of the algorithms in a mixed formulation beam finite element proves the accuracy and robustness of the approach in simulating nonlinear behavior.     

Timoshenko curved beam bending solutions in terms of Euler-Bernoulli solutions

Summary  This paper presents the exact relationships between the deflections and stress resultants of Timoshenko curved beams and that of the corresponding Euler-Bernoulli curved beams. The curved beams considered are of rectangular cross sections and constant radius of curvature. They may have any combinations of classical boundary conditions, and are subjected to any loading distribution that acts normal to the curved beam centreline. These relationships allow engineering designers to directly obtain the bending solutions of Timoshenko curved beams from the familiar Euler-Bernoulli solutions without having to perform the more complicated shear deformation analysis. Accepted for publication 26 July 1996     

A hybrid stress approach for laminated composite plates within the First-order Shear Deformation Theory

This paper presents a hybrid stress approach for the analysis of laminated composite plates. The plate mechanical model is based on the so called First-order Shear Deformation Theory, rationally deduced from the parent three-dimensional theory. Within this framework, a new quadrilateral four-node finite element is developed from a hybrid stress formulation involving, as primary variables, compatible displacements and elementwise equilibrated stress resultants. The element is designed to be simple, stable and locking-free. The displacement interpolation is enhanced by linking the transverse displacement to the nodal rotations and a suitable approximation for stress resultants is selected, ruled by the minimum number of parameters. The transverse stresses through the laminate thickness are reconstructed a posteriori by simply using three-dimensional equilibrium. To improve the results, the stress resultants entering the reconstruction process are first recovered using a superconvergent patch-based procedure called Recovery by Compatibility in Patches, that is properly extended here for laminated plates. This preliminary recovery is very efficient from the computational point of view and generally useful either to accurately evaluate the stress resultants or to estimate the discretization error. Indeed, in the present context, it plays also a key role in effectively predicting the shear stress profiles, since it guarantees the global convergence of the whole reconstruction strategy, that does not need any correction to accommodate equilibrium defects. Actually, this strategy can be adopted together with any plate finite element. Numerical testing demonstrates the excellent performance of both the finite element and the reconstruction strategy.     

A nonlinear composite shell theory

A general nonlinear theory for the dynamics of elastic anisotropic circular cylindrical shells undergoing small strains and moderate-rotation vibrations is presented. The theory fully accounts for extensionality and geometric nonlinearities by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. Moreover, the linear part of the theory contains, as special cases, most of the classical linear theories when appropriate stress resultants and couples are defined. Parabolic distributions of the transverse shear strains are accounted for by using a third-order theory and hence shear correction factors are not required. Five third-order nonlinear partial differential equations describing the extension, bending, and shear vibrations of shells are obtained using the principle of virtual work and an asymptotic analysis. These equations show that laminated shells display linear elastic and nonlinear geometric couplings among all motions.     

Reconciliation for Monoclinic Materials of the Linear,Elastically Isotropic Plate Theories of Reissner and Ladevèze

Reissner’s well-known sixth-order linear theory for the bending of elastically isotropic plates, here extended to elastically monoclinic materials, is shown to be a compatible precursor to a recent extension to monoclinic materials of Ladevèze’s exact fourth-order plate theory. With a solution of Reissner’s extended equations in hand, Ladevèze’s extended equations reduce to three equations for three unknowns, two additional, previously unknown functions now being determinant. The key link is the requirement that the stress resultants and couples of the two theories agree.     

On the linking up between Bingham fluid and plugged flow

When Bingham fluid is in motion, plugged flow often occurs at places far from the boundary walls. As there is not a decisive formula of constitutive relation for plugged flow, in some problems the solutions obtained may be indefinite. In this paper, annular flow and pipe flow are discussed, and unique solution is obtained in each case by utilizing the analytic property of shear stress. The solutions are identical in form with the commonly used formula for the pressure drop of mud flow in petroleum engineering.     

A general nonlinear theory of elastic shells

This paper presents a general nonlinear theory of elastic shells for large deflections and finite strains in reference to a certain natural state. By expanding the displacement components into power series in the coordinate θ³ normal to the undeformed middle surface of shells, the expansions of the Cauchy-Green strain tensors are expressed in terms of these expanded displacement components. Through the modified Hellinger-Reissner variational principle for a three-dimensional elastic continuum, a set of the fundamental shell equations is derived in terms of the expanded Cauchy-Green strain tensors and Kirchhoff stress resultants. The Love-Kirchhoff hypothesis is not assumed and higher order stretching and bending are taken into consideration. For elastic shells of isotropic materials, assuming the strain-energy to be an analytic function of the strain measures, general nonlinear constitutive equations are then derived. Thus, a complete and consistent two-dimensional shell theory incorporating the geometrical and physical nonlinearities is established. The classical theories of shells are directly derivable from the present results by proper truncations of the series.