Deflection (engineering)

About: Deflection (engineering) is a research topic. Over the lifetime, 30862 publications have been published within this topic receiving 298849 citations.

Vibrations of shells and plates

Abstract: Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Historical Development of Vibration Analysis of Continuous Structural Elements References Deep Shell Equations Shell Coordinates and Infinitesimal Distances in Shell Layers Stress-Strain Relationships Strain-Displacement Relationships Love Simplifications Membrane Forces and Bending Moments Energy Expressions Love's Equations by Way of Hamilton's Principle Boundary Conditions Hamilton's Principle Other Deep Shell Theories Shells of Nonuniform Thickness References Radii of Curvature References Equations of Motion for Commonly Occurring Geometries Shells of Revolution Circular Conical Shell Circular Cylindrical Shell Spherical Shell Other Geometries References Nonshell Structures Arch Beam and Rod Circular Ring Plate Torsional Vibration of Circular Cylindrical Shell and Reduction to a Torsion Bar References Natural Frequencies and Modes General Approach Transversely Vibrating Beams Circular Ring Rectangular Plates That are Simply Supported Along Two Opposing Edges Circular Cylindrical Shell Simply Supported Circular Plates Vibrating Transversely Examples: Plate Clamped at Boundary Orthogonality Property of Natural Modes Superposition Modes Orthogonal Modes from Nonorthogonal Superposition Modes Distortion of Experimental Modes Because of Damping Separating Time Formally Uncoupling of Equations of Motion In-Plane Vibrations of Rectangular Plates In-Plane Vibration of Circular Plates Deep Circular Cylindrical Panel Simply Supported at All Edges Natural Mode Solutions by Power Series On Regularities Concerning Nodelines References Simplified Shell Equations Membrane Approximations Axisymmetric Eigenvalues of a Spherical Shell Bending Approximation Circular Cylindrical Shell Zero In-Plane Deflection Approximation Example: Curved Fan Blade Donnell-Mushtari-Vlasov Equations Natural Frequencies and Modes Circular Cylindrical Shell Circular Duct Clamped at Both Ends Vibrations of a Freestanding Smokestack Special Cases of the Simply Supported Closed Shell and Curved Panel Barrel-Shaped Shell Spherical Cap Inextensional Approximation: Ring Toroidal Shell The Barrel-Shaped Shell Using Modified Love Equations Doubly Curved Rectangular Plate References Approximate Solution Techniques Approximate Solutions by Way of the Variational Integral Use of Beam Functions Galerkin's Method Applied to Shell Equations Rayleigh-Ritz Method Southwell's Principle Dunkerley's Principle Strain Energy Expressions References Forced Vibrations of Shells by Modal Expansion Model Participation Factor Initial Conditions Solution of the Modal Participation Factor Equation Reduced Systems Steady-State Harmonic Response Step and Impulse Response Influence of Load Distribution Point Loads Line Loads Point Impact Impulsive Forces and Point Forces Described by Dirac Delta Functions Definitions and Integration Property of the Dirac Delta Function Selection of Mode Phase Angles for Shells of Revolution Steady-State Circular Cylindrical Shell Response to Harmonic Point Load with All Mode Components Considered Initial Velocity Excitation of a Simply Supported Cylindrical Shell Static Deflections Rectangular Plate Response to Initial Displacement Caused by Static Sag The Concept of Modal Mass, Stiffness Damping, and Forcing Steady State Response of Shells to Periodic Forcing Plate Response to a Periodic Square Wave Forcing Beating Response to Steady State Harmonic Forcing References Dynamic Influence (Green's) Function Formulation of the Influence Function Solution to General Forcing Using the Dynamic Influence Function Reduced Systems Dynamic Influence Function for the Simply Supported Shell Dynamic Influence Function for the Closed Circular Ring Traveling Point Load on a Simply Supported Cylindrical Shell Point Load Traveling Around a Closed Circular Cylindrical Shell in Circumferential Direction Steady-State Harmonic Green's Function Rectangular Plate Examples Floating Ring Impacted by a Point Mass References Moment Loading Formulation of Shell Equations That Include Moment Loading Modal Expansion Solution Rotating Point Moment on a Plate Rotating Point Moment on a Shell Rectangular Plate Excited by a Line Moment Response of a Ring on an Elastic Foundation to a Harmonic Point Moment Moment Green's Function References Vibration of Shells and Membranes Under the Influence of Initial Stresses Strain-Displacement Relationships Equations of Motion Pure Membranes Example: The Circular Membrane Spinning Saw Blade Donnell-Mushtari-Vlasov Equations Extended to Include Initial Stresses References Shell Equations with Shear Deformation and Rotary Inertia Equations of Motion Beams with Shear Deflection and Rotary Inertia Plates with Transverse Shear Deflection and Rotary Inertia Circular Cylindrical Shells with Transverse Shear Deflection and Rotary Inertia References Combinations of Structures Receptance Method Mass Attached to Cylindrical Panel Spring Attached to Shallow Cylindrical Panel Harmonic Response of a System in Terms of Its Component Receptances Dynamic Absorber Harmonic Force Applied Through a Spring Steady-State Response to Harmonic Displacement Excitation Complex Receptances Stiffening of Shells Two Systems Joined by Two or More Displacement Suspension of an Instrument Package in a Shell Subtracting Structural Subsystems Three and More Systems Connected Examples of Three Systems Connected to Each Other References Hysteresis Damping Equivalent Viscous Damping Coefficient Hysteresis Damping Direct Utilization of Hysteresis Model in Analysis Hysteretically Damped Plate Excited by Shaker Steady State Response to Periodic Forcing References Shells Made of Composite Material Nature of Composites Lamina-Constitutive Relationship Laminated Composite Equation of Motion Orthotropic Plate Circular Cylindrical Shell Orthotropic Nets or Textiles Under Tension Hanging Net or Curtain Shells Made of Homogeneous and Isotropic Lamina Simply Supported Sandwich Plates and Beams Composed of Three Homogeneous and Isotropic Lamina References Rotating Structures String Parallel to Axis of Rotation Beam Parallel to Axis of Rotation Rotating Ring Rotating Ring Using Inextensional Approximation Cylindrical Shell Rotating with Constant Spin About Its Axis General Rotations of Elastic Systems Shells of Revolution with Constant Spin About Their Axes of Rotation Spinning Disk References Thermal Effects Stress Resultants Equations of Motion Plate Arch, Ring, Beam, and Rod Limitations Elastic Foundations Equations of Motion for Shells on Elastic Foundations Natural Frequencies and Modes Plates on Elastic Foundations Ring on Elastic Foundation Donnell-Mushtari-Vlasov Equations with Transverse Elastic Foundation Forces Transmitted Into the Base of the Elastic Foundation Vertical Force Transmission Through the Elastic Foundation of a Ring on a Rigid Wheel Response of a Shell on an Elastic Foundation to Base Excitation Plate Examples of Base Excitation and Force Transmission Natural Frequencies and Modes of a Ring on an Elastic Foundation in Ground Contact at a Point Response of a Ring on an Elastic Foundation to a Harmonic Point Displacement References Similitude General Similitude Derivation of Exact Similitude Relationships for Natural Frequencies of Thin Shells Plates Shallow Spherical Panels of Arbitrary Contours (Influence of Curvature) Forced Response Approximate Scaling of Shells Controlled by Membrane Stiffness Approximate Scaling of Shells Controlled by Bending Stiffness References Interactions with Liquids and Gases Fundamental Form in Three-Dimensional Curvilinear Coordinates Stress-Strain-Displacement Relationships Energy Expressions Equations of Motion of Vibroelasticity with Shear Example: Cylindrical Coordinates Example: Cartesian Coordinates One-Dimensional Wave Equations for Solids Three-Dimensional Wave Equations for Solids Three-Dimensional Wave Equations for Inviscid Compressible Liquids and Gases (Acoustics) Interface Boundary Conditions Example: Acoustic Radiation Incompressible Liquids Example: Liquid on a Plate Orthogonality of Natural Modes for Three-Dimensional Solids, Liquids, and Gases References Discretizing Approaches Finite Differences Finite Elements Free and Forced Vibration Solutions References Index

Reinforced Concrete response to simulated earthquakes

Abstract: A series of reinforced concrete specimens has been subjected to static tests as well as periodic and simulated earthquake motions to develop realistic analytical models for the earthquake response of the elements and materials involved. During some of the dynamic tests the specimen responded with a displacement of the order of six times the initial yield deflection. The stiffness and energy absorbing capacity of the specimens changed considerably and, at times, very rapidly during the dynamic tests. A realistic conceptual model for predicting the dynamic response of a reinforced concrete member should be based on a static force-displacement relationship which reflects the changes in stiffness for loading and unloading as a function of the previous loading history. The dynamic response calculated on the basis of the proposed force-displacement relationship resulted in satisfactory agreement with the measured response. With the hysteresis loops defined by the proposed force-displacement relationship, it was not necessary to invoke additional sources of energy absorption for a satisfactory prediction of the dynamic response.

A microstructure-dependent Timoshenko beam model based on a modified couple stress theory

Abstract: A microstructure-dependent Timoshenko beam model is developed using a variational formulation. It is based on a modified couple stress theory and Hamilton's principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Timoshenko beam theory. Moreover, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, which differ from existing Timoshenko beam models. The newly developed non-classical beam model recovers the classical Timoshenko beam model when the material length scale parameter and Poisson's ratio are both set to be zero. In addition, the current Timoshenko beam model reduces to a microstructure-dependent Bernoulli–Euler beam model when the normality assumption is reinstated, which also incorporates the Poisson effect and can be further reduced to the classical Bernoulli–Euler beam model. To illustrate the new Timoshenko beam model, the static bending and free vibration problems of a simply supported beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that by the classical model, with the difference between them being significantly large only for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally. Finally, the Poisson effect on the beam deflection, rotation and natural frequency is found to be significant, which is especially true when the classical Timoshenko beam model is used. This indicates that the assumption of Poisson's effect being negligible, which is commonly used in existing beam theories, is inadequate and should be individually verified or simply abandoned in order to obtain more accurate and reliable results.

Comb-drive actuators for large displacements

Abstract: The design, fabrication and experimental results of lateral-comb-drive actuators for large displacements at low driving voltages is presented. A comparison of several suspension designs is given, and the lateral large deflection behaviour of clamped - clamped beams and a folded flexure design is modelled. An expression for the axial spring constant of folded flexure designs including bending effects from lateral displacements, which reduce the axial stiffness, is also derived. The maximum deflection that can be obtained by comb-drive actuators is bounded by electromechanical side instability. Expressions for the side-instability voltage and the resulting displacement at side instability are given. The electromechanical behaviour around the resonance frequency is described by an equivalent electric circuit. Devices are fabricated by polysilicon surface micromachining techniques using a one-mask fabrication process. Static and dynamic properties are determined experimentally and are compared with theory. Static properties are determined by displacement-to-voltage, capacitance-to-voltage and pull-in voltage measurements. Using a one-port approach, dynamic properties are extracted from measured admittance plots. Typical actuator characteristics are deflections of about at driving voltages around 20 V, a resonance frequency around 1.6 kHz and a quality factor of approximately 3.