John B. Kosmatka

An improved two-node timoshenko beam finite element

Abstract The stiffness, mass, and consistent force matrices for a simple two-node Timoshenko beam element are developed based upon Hamiltons principle. Cubic and quadratic Lagrangian polynomials are used for the transverse and rotational displacements, respectively, where the polynomials are made interdependent by requiring them to satisfy the two homogeneous differential equations associated with Timoshenkos beam theory. The resulting stiffness matrix, which can be exactly integrated and is free of ‘shear-locking’, is in agreement with the exact Timoshenko beam stiffness matrix. Numerical results are presented to show that the current element exactly predicts the displacement of a short beam subjected to complex distributed loadings using only one element, and the current element predicts shear and moment resultants and natural frequencies better than existing Timoshenko beam elements.

An improved two-node finite element for stability and natural frequencies of axial-loaded Timoshenko beams

The linear flexural stiffness, incremental stiffness, mass, and consistent force matrices for a simple two-node Timoshenko beam element are developed based upon Hamiltons principle, where interdependent cubic and quadratic polynomials are used for the transverse and rotational displacements, respectively. The resulting linear flexural stiffness matrix is in agreement with the exact 2-node Timoshenko beam stiffness matrix. Numerical results are presented to show that the current element can accurately predict the buckling load and natural frequencies of axially-loaded isotropic and composite beams for a wide variety of beam-lengths and boundary conditions. The current element consistently outperforms the existing finite element approaches in studies involving the buckling or vibration behavior of axially-loaded short beams.

On Saint-Venant’s Problem for an Inhomogeneous, Anisotropic Cylinder—Part I: Methodology for Saint-Venant Solutions

In this paper, the first in a series of three, a procedure based on semi-analytical finite elements is presented for constructing Saint-Venant solutions for extension, bending, torsion, and flexure of a prismatic cylinder with inhomogeneous, anisotropic cross-sectional properties. Extension-bending-torsion involve stress fields independent of the axial coordinate and their displacements may be decomposed into two distinct parts which are called the primal field and the cross-sectional warpages herein. The primal field embodies the essence of the kinematic hypotheses of elementary bar and beam theories and that for unrestrained torsion. The cross-sectional warpages are independent of the axial coordinate and they are determined by testing the variationally derived finite element displacement equations of equilibrium with the primal field. For flexure, a restricted three-dimensional stress field is in effect where the stress can vary at most linearly along the axis. Integrating the displacement field based for extension-bending-torsion gives that for the flexure problem. The cross-sectional warpages for flexure are determined by testing the displacement equations of equilibrium with this displacement field. In the next paper, the cross-sectional properties such as the weighted-average centroid, center of twist and shear center are defined based on the Saint-Venant solutions established in the present paper and numerical examples are given. In the third paper, end effects or the quantification of Saint-Venants principle for the inhomogeneous, anisotropic cylinder is considered.

Saint-Venant solutions for prismatic anisotropic beams

Abstract An analytical model is presented for determining the displacement and stress distributions of the Saint-Venant extension, bending, torsion and flexure problems for a homogeneous prismatic beam of arbitrary section and rectilinear anisotropy. The determination of the complete displacement field requires solving a coupled two-dimensional boundary value problem for the local in-plane deformations and warping out of the section plane. The principle of minimum potential energy is applied to a discretized representation of the cross-section (Ritz method) to calculate solutions to this problem. The behavior of an anisolropic beam is studied in detail using the resulting displacement and stress solutions, where definitions are presented for the shear center, center of twist, torsion constant and a new geometric parameter: the line of extension bending centers. Two sets of numerical results are presented to illustrate how section geometry, beam length and material properties affect the behavior of a homogeneous anisotropic cantilever beam.

Damping Performance of Cocured Graphite/Epoxy Composite Laminates with Embedded Damping Materials:

Cocuring viscoelastic damping materials in composites has been shown to be successful in greatly increasing the damping of composite structures. The damping performance, however, is often not as high in cocured composites as in secondarily bonded composites, where the damping material does not undergo the laminate cure cycle. The reason for the discrepancy in damping between the cocured and secondarily bonded samples was found to be resin penetration into the damping material. Samples with a barrier layer between the damping material and the epoxy resin had a 15.7% to 92.3% higher effective loss factor (depending on the frequency) than cocured FasTape™ 1125 samples without the barrier and at least 168% higher effective loss factor than cocured ISD 112 samples without the barrier. These higher damping values are very close to the values achieved by secondarily bonding. Viscoelastic damping materials typically have maximum recommended temperatures below that of the composite cure cycles. The effect of cure temperature on viscoelastic damping materials was also studied and it was determined that most damping materials are marginally affected by cure cycle temperature.

Torsion and flexure of a prismatic isotropic beam using the boundary element method

Abstract A boundary element solution procedure is developed for studying the coupled torsion and flexure problem of an isotropic beam having an arbitrary cross section. The torsion and flexure problems are formulated using the St. Venant semi-inverse method. The development of the boundary element procedure for the solution of the coupled flexure–torsion problem is presented. A three-node isoparametric boundary element is used to allow an accurate geometric representation of cross-sections having curved boundaries. A formulation for calculation of the cross-sectional geometrical properties (centroid, area, moments of inertia, etc.) and the elastic properties (shear center location, shear correction factor, torsional rigidity) is shown. A formulation is developed for shear stress calculation at any point in the cross section. Numerical results for cross-sectional geometric and elastic properties of six different cross-sectional shapes and shear stress calculations are presented to demonstrate the efficiency and accuracy of the method.

Exact stiffness matrix of a nonuniform beam—I. Extension, torsion, and bending of a bernoulli-euler beam

Abstract The exact axial, bending, and torsion stiffness matrices have been developed for an arbitrary nonuniform beam element. The coefficients of the bending stiffness matrix require the evaluation of three integrals, while the axial and torsion stiffness matrices require only one integral. These coefficients are evaluated for a uniform beam (verification) and a nonuniform beam with either linearly- or parabolically-varying cross-section dimensions. Two sets of numerical results are presented to provide a comparison of the current exact approach with a commonly used displacement-based approach and an approximate approach found in most commercial finite element programs. The two existing approaches produced acceptable results for an extremely small range of tapers. As more elements are used with the two existing approaches, their solutions will converge to the current exact solution which requires only one element.

On Saint-Venant’s Problem for an Inhomogeneous, Anisotropic Cylinder—Part II: Cross-Sectional Properties

Cross-sectional properties of a prismatic inhomogeneous, anisotropic cylinder are determined from Saint-Venant solutions for extension-bending-torsion and flexure, whose method of construction was presented in a previous paper. The coupling of extensional, bending and twisting deformations due to anisotropy and inhomogeneity leads to some very interesting features. Herein it is shown that far an inhomogeneous, anisotropic cylinder whose cross-sectional plane is not a material symmetry plane, distinct modulus-weighted and compliance-weighted centroids and distinct principal bending axes are possible. A line of extension-bending centers is given on which an axial force causes extension and bending only but no twist. Two shear centers are given, one using the Griffith-Taylor definition that ignores cross-sectional warpages and the other by stipulating a zero mean rotation over the cross section. The center of twist is discussed, and this property depends on root end fixity conditions that are prescribed in terms of their mean values based on integrals over the cross section rather than by a pointwise specification. While these shear center and center of twist definitions have some rational bases, it is recognized that other definitions are possible, for example those based on modulus or compliance-weighted integrals. Two examples, an angle and a channel, both composed of a two-layer ±30 deg angle-ply composite material, illustrate the procedures for determining these cross-sectional properties.

Exact stiffness matrix of a nonuniform beam—II. Bending of a timoshenko beam

Abstract The exact bending stiffness matrix is developed for an arbitrary nonuniform beam including shear deformation. The current development is based upon Timoshenkos beam theory. The coefficients of the bending stiffness matrix require the evaluation of only three integrals. These coefficients are evaluated for a uniform beam (verification) and a nonuniform beam with either linearly or quadratically varying cross-section dimensions. Numerical results are presented to assess the interaction of taper rate and shear deformation on the behavior of short beams having a circular cross-section with linear taper and also short beams having a rectangular cross-section with either linear taper or two different types of simple quadratic taper.

On Saint-Venant’s Problem for an Inhomogeneous, Anisotropic Cylinder—Part III: End Effects

End effects or displacements and stresses of a self-equilibrated state in an inhomogeneous, anisotropic cylinder are represented by eigendata extracted from an algebraic eigensystem. Such states are typical of traction and/or displacement boundary conditions that do not abide by the distributions according to Saint-Venants solutions, whose construction were discussed in the first paper of this series of three. This type of analysis of end effects quantitifies Saint-Venants principle, and the algebraic eigensystem providing the eigendata is based on homogeneous displacement equations of equilibrium with an exponential decaying displacement form. The real parts of the eigenvalues convey information on the inverse decay lengths and their corresponding eigenvectors are displacement distributions of self-equilibrated states. Stress eigenvetors can be formed by appropriate differentiation of the displacement eigenvectors. The eigensystem and its adjoint system provide complete sets of right and left-handed eigenvectors that are interrelated by two bi-orthogonality relations. Displacement and stress end effects can be represented by means of an expansion theorem based on these bi-orthogonality relations or by a least-squares solution. Two examples, a beam with a homogeneous, isotropic cross section and the other of a two layer beam with a ±30 deg angle-ply composite cross section, are given to illustrate the representation of various end effects.