Showing papers on "Timoshenko beam theory published in 1997"

Mechanics of laminated composite plates : theory and analysis

Abstract: Introduction and Mathematical Preliminaries Fiber-Reinforced Composite Materials. Vectors and Tensors. Matrices. Transformation of Vector and Tensor Components. Integral Relations. Equations of Anisotropic Elasticity Classification of Equations. Kinematics. Kinetics. Constitutive Equations. Equations of Thermoelasticity and Electroelasticity. Summary. Virtual Work Principles and Variational Methods Virtual Work. The Variational Operator and Functionals. Extrema of Functionals. Virtual Work Principles. Variational Methods. Summary. Introduction to Composite Materials Basic Concepts and Terminology. Constitutive Equations of a Lamina. Transformation of Stresses and Strains. Plane Stress Constitutive Relations. Classical and First-Order Theories of Laminated Composite Plates Introduction. An Overview of ESL Laminate Theories. The Classical Laminated Plate Theory. The First-Order Laminated Plate Theory. Stiffness Characteristics for Selected Laminates. One-Dimensional Analysis of Laminated Plates Introduction. Analysis of Laminated Beams Using CLPT. Analysis of Laminated Beams Using FSDT. Cylindrical Bending Using CLPT. Cylindrical Bending Using FSDT. Closing Remarks. Analysis of Specially Orthotropic Plates Using CLPT Introduction. Bending of Simply Supported Plates. Bending of Plates with Two Opposite Edges Simply Supported. Bending of Rectangular Plates with Various Boundary Conditions. Buckling of Simply Supported Plates Under Compressive Loads. Buckling of Rectangular Plates Under Inplane Shear Load. Vibration of Simply Supported Plates. Buckling and Vibration of Plates with Two Parallel Edges Simply Supported. Closure. Analytical Solutions of Rectangular Laminates Using CLPT Governing Equations in Terms of Displacements. Admissible Boundary Conditions for the Navier Solutions. Navier Solutions of Antisymmetric Cross-Ply Laminates. The Navier Solutions of Antisymmetric Angle-Ply Laminates. The LTvy Solutions. Analysis of Midplane Symmetric Laminates. Transient Analysis. Summary. Analytical Solutions of Rectangular Laminates Using FSDT Introduction. Simply Supported Antisymmetric Cross-Ply Laminates. Simply Supported Antisymmetric Angle-Ply Laminates. Antisymmetric Cross-Ply Laminates with Two Opposite Edges Simply Supported. Antisymmetric Angle-Ply Laminates with Two Opposite Edges Simply Supported. Transient Solutions. Summary. Finite Element Analysis of Composite Laminates Introduction. Laminated Beams and Plate Strips by CLPT. Timoshenko Beam/Plate Theory. Numerical Results for Beams and Plate Strips. Finite Element Models of Laminated Plates (CLPT). Finite Element Models of Laminated Plates (FSDT). Summary. Refined Theories of Laminated Composite Plates Introduction. A Third-Order Plate Theory. Higher-Order Laminate Stiffness Characteristics. The Navier Solutions. LTvy Solutions of Cross-Ply Laminates. Displacement Finite Element Model. Layerwise Theories and Variable Kinematic Models In troduction. Development of the Theory. Finite Element Model. Variable Kinematic Formulations. Nonlinear Analysis of Composite Laminates Introduction. Nonlinear Stiffness Coefficients. Solution Methods for Nonlinear Algebraic Equations. Computational Aspects and Numerical Examples. Closure. Index Most chapters include Exercises and References for Additional Reading

A Unified Beam Finite Element Model for Extension and Shear Piezoelectric Actuation Mechanisms

Abstract: This paper presents a finite element model for adaptive sandwich beams to deal with either extension or shear actuation mechanism The former corresponds to an elastic core sandwiched beam between two transversely polarized active surface layers; whereas, the latter consists of an axially polarized core, sandwiched between two elastic surface layers For both configurations, an electric field is applied through thickness of the piezoelectric layers The mechanical model is based on Bernoulli-Euler theory for the surface layers and Timoshenko beam theory for the core It uses three variables, through-thickness constant deflection, and the mean and relative axial displacements of the core's upper and lower surfaces Augmented by the bending rotation, these are the only nodal degrees of freedom of the proposed two-node adaptive sandwich beam finite element The piezoelectric effect is handled through modification of the constitutive equation, when induced electric potential is taken into account, and additio

A four-point bending technique for studying subcritical crack growth in thin films and at interfaces

Abstract: A technique was developed to obtain the subcritical crack growth velocity in a 4-point bending sample by analyzing the load-displacement curve. This was based on the observation that the compliance of a beam increases as the crack grows. Beam theory was used to analyze the general configuration where two cracks propagated in the opposite directions. A simple equation relating the crack velocity to the load and displacement was established, taking advantage of the fact that the compliance was linearly proportional to the crack lengths; thus the absolute crack length was not important. Two methods of obtaining crack velocity as a function of load were demonstrated. First, by analyzing a load-displacement curve, a corresponding velocity curve was obtained. Second, by changing the displacement rate and measuring the corresponding plateau load, a velocity value was calculated for each plateau load. While the former was capable of obtaining the dependence of crack velocity versus load from a single test, the latter was found to be simpler and more consistent. Applications were made to a CVD SiO2 system. In both cases of crack propagation either inside the SiO2 layer or along its interface with a TiN layer, the crack growth velocity changed with the stress intensity at the crack tip exponentially. As a result, a small crack will grow larger under essentially any tensile stresses typically existing in devices, provided that chemical agents facilitating stress corrosion mechanisms are also present.

Vibration Studies of Tsing Ma Suspension Bridge

Abstract: A three-dimensional dynamic finite element model is established for the Tsing Ma long suspension Bridge in Hong Kong. The two bridge towers made up of reinforced concrete are modeled by three-dimensional Timoshenko beam elements with rigid arms at the connections between columns and beams. The cables and suspenders are modeled by cable elements accounting for geometric nonlinearity due to cable tension. The hybrid steel deck is represented by a single beam with equivalent cross-sectional properties determined by detailed finite element analyses of sectional models. The modal analysis is then performed to determine natural frequencies and mode shapes of lateral, vertical, torsional, longitudinal, and coupled vibrations of the bridge. The results show that the natural frequencies of the bridge are very closely spaced; the first 40 natural frequencies range from 0.068 to 0.616 Hz only. The computed normal modes indicate interactions between the main span and side span, and between the deck, cables, and tower...

A system for the dynamic characterization of microstructures

Abstract: This paper describes a fully automated measurement system designed to evaluate the dynamic characteristics of micromechanical structures (millimeter dimensions). To validate the system, vibration measurements have been carried on two structures-a micromachined silicon cantilever and bridge-and the results are presented. Out-of-plane measurements show that for the cantilever, both the mode shapes and resonant frequencies agree with beam theory predictions. However, for the bridge structure, tension due to boron doping causes a change from beam-like behavior and a more complex model is required. Mode-shapes natural frequencies and modal damping are determined from data obtained by vibrating the structures using a piezoelectric mounting system and deriving the transfer function between the piezodrive voltage and beam vibrational velocity.

Distributed Control of Laminated Beams: Timoshenko Theory vs. Euler-Bernoulli Theory:

Abstract: In this paper, the governing equations and boundary conditions of laminated beamlike components of smart structures are reviewed. Sensor and actuator layers are included in the beam so as to facilitate vibration suppression. Two mathematical models, namely the shear-deformable (Timoshenko) model and the shear-indeformable (Euler-Bernoulli) model, are presented. The differential equations of the continuous system are approximated by utilizing finite element techniques for both models. A cantilever laminated beam with and without a tip mass is investigated to assess the validity and the accuracy of the two models when used for vibration suppression. Comparison between the two models is presented to show the advantages and the limitations of each of the models. Since the Timoshenko beam theory is higher order than the Euler Bernoulli theory, it is known to be superior in predicting the transient response of the beam. The superiority of the Timoshenko model is more pronounced for beams with a low aspect ratio...

Distributed control of laminated beams: Timoshenko theory vs. Euler-Bernoulli theory

Abstract: In this paper, the governing equations and boundary conditions of laminated beamlike components of smart structures are reviewed. Sensor and actuator layers are included in the beam so as to facilitate vibration suppression. Two mathematical models, namely the shear-deformable (Timoshenko) model and the shear-indeformable (Euler-Bernoulli) model, are presented. The differential equations of the continuous system are approximated by utilizing finite element techniques for both models. A cantilever laminated beam with and without a tip mass is investigated to assess the validity and the accuracy of the two models when used for vibration suppression. Comparison between the two models is presented to show the advantages and the limitations of each of the models. Since the Timoshenko beam theory is higher order than the Euler Bernoulli theory, it is known to be superior in predicting the transient response of the beam. The superiority of the Timoshenko model is more pronounced for beams with a low aspect ratio...

Analysis of Controlled Beam Deflections Using SMA Wires

Abstract: In this paper the active control of beam deflection through heating and cooling of Shape Memory Alloy (SMA) wires is examined. A phenomenological constitutive law for SMA wires is coupled with beam theory to provide predictions of beam shape upon temperature change in the SMA actuator. Both the linear and nonlinear beam theory are presented, enabling calculation of large deflections. Examples for a single wire attached at the tip of a uniform beam are given, but the procedure can easily be generalized for other configurations and utilized in control algorithms. Issues of design constraints for shape control with shape memory wires are addressed and the model is qualitatively verified by experiments.

Mechanical properties of thin and thick coatings applied to various substrates. Part I. An elastic analysis of residual stresses within coating materials

Abstract: Internal residual stresses significantly influence the overall mechanical properties of multi-layer systems and consequently affect the coated material's performance. The determination of residual stresses within coatings has been extensively carried out for thin (the coating thickness being less than the substrate thickness) films (Stoney, Roll, etc.) and for thick (the coating thickness being approximately equal to the substrate thickness) films (Timoshenko, Inoue, etc.). This work extends currently existing models to cover cases where the coating thickness approaches that of the sheet substrate. We developed relationships for the determination of the first-order residual stress. The construction of these models was carried out using a one-dimensional analysis based on beam theory (the width-to-thickness ratio of the system being less than 5). As suggested by Timoshenko and later on by Hoffman, we also introduced the bi-axial modulus for isotropic stresses when the thin plate theory (the width-to-thickn...

The design of perforated cold-formed steel sections subject to axial load and bending

Abstract: The uprights in a typical pallet rack are typically singly-symmetrical cold-formed sections subject to axial load together with bending about both axes. They usually contain arrays of holes in order to enable beams to be clipped into position at heights that are not pre-determined prior to manufacture. Their slenderness is such that their behaviour may be influenced by the three generic forms of buckling, namely local, distortional and global (lateral torsional). In practice, these members have generally been designed on the basis of expensive test programmes. This paper addresses the problem of how they might be designed analytically. The basis of the investigation is a comprehensive set of test results on upright sections in compression which embraces both stub column tests, in which the load position was varied along the axis of symmetry, and longer columns. The test results were analysed using both finite elements and a version of “Generalized Beam Theory” (GBT) which incorporated systematic imperfections. Consideration was also given to the design procedures proposed by the “Federation Europeene de la Manutention” (FEM) and recent research into the influence of perforations on the performance of cold formed steel sections. It is shown that GBT can be modified to take account of perforations so that the lower bound results give a sufficiently accurate column design curve, which takes account of local, distortional and global buckling, thus making extensive testing unnecessary.

A method for improving the stress analysis performance of one- and two-dimensional theories for laminated composites

Abstract: This paper proposes a method for substantially improving the performance of the higher-order plate and shell theories in connection with the accurate stress analysis of homogeneous and laminated composite structural elements. The presentation of the method is based on the equations of the “general five-degrees-of-freedom” shear deformable plate theory. Since the method is entirely new, it is initially applied to the solution of the problem of simply supported plates deformed by cylindrical bending, for which there exists an exact elasticity solution [12]. Hence, its reliability is substantially validated by means of appropriate comparisons between numerical results based on the present plate theory and this exact elasticity solution. Moreover, the one-dimensional version of the present plate theory, employed for the cylindrical bending of plates, is considered as a general three-degrees-of-freedom shear deformable beam theory. This advanced beam theory is used for an accurate stress analysis of two-layered composite beams having one of their edges rigidly clamped and the other either rigidly clamped, free of tractions or simply supported. This final set of applications can be thought of alternatively as a stress analysis of two-layered plates deformed in cylindrical bending and subjected to several, different sets of edge boundary conditions.

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

Abstract: 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.

An Efficient Curved Beam Finite Element

Abstract: The plane two-node curved beam finite element with six degrees of freedom is considered. Knowing the set of 18 exact shape functions their approximation is derived using the expansion of the trigonometric functions in the power series. Unlike the ones commonly used in the FEM analysis the functions suggested by the authors have the coefficients dependent on the geometrical and physical properties of the element. From the strain energy formula the stiffness matrix of the element is determined. It is very simple and can be split into components responsible for bending, shear and axial forces influences on the displacements. The proposed element is totally free of the shear and membrane locking effects. It can be referred to the shear-flexible (parameter d) and compressible (parameter e) systems. Neglecting d or e yields the finite elements in all necessary combinations, i.e. curved Euler–Bernoulli beam or curved Timoshenko beam with or without the membrane effect. Applying the elaborated element in the calculations a very good convergence to the analytical results can be obtained even with a very coarse mesh without the commonly adopted corrections as reduced or selective integration or introduction of the stabilization matrices, additional constraints, etc., for the small depth–length ratio. © 1997 John Wiley & Sons, Ltd.