Showing papers on "Bending moment published in 1981"

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

The response of flexible piles to lateral loading

Abstract: The majority of piles encountered in practice may be regarded as ‘flexible’ under lateral loading. That is, the induced deformations and bending moments are confined to the upper part of the pile and the overall length of the pile does not significantly affect the response of the pile. The results of a parametric study, conducted using the finite element method and treating the soil as an elastic continuum with a linearly varying soil modulus, are presented as simple algebraic expressions. These expressions are similar in form to those which arise from a Winkler idealization of the soil, enabling immediate estimates to be made of: the active length of the pile; the ground level deformations; and the maximum bending moment down the pile. In addition, expressions are presented quantifying the effect of interaction between neighbouring piles. from which the behaviour of groups of piles subjected to lateral loading may be deduced. Application of the solutions is demonstrated by the analysis of lateral loading...

Peel Analysis Using the Finite Element Method

Abstract: Large displacement finite element analysis and subsequent experimental work has been used to investigate the adhesive peel test; at this stage, only elastic behaviour has been considered. Both non-cracked and cracked configurations have been analysed, representing initial and continuous failure of the peel test. Analysis of the former indicated that initial failure was caused by the adhesive principal stresses driving a crack towards the interface with the flexible adherend. Investigation of the cracked configuration has shown that the amount of mode II loading at the crack tip is significant and is essentially independent of peel angle, load and adhesive or adherend modulus, only decreasing as the adhesive becomes incompressible. Failure (propagation) has been shown to occur at a critical applied bending moment for a particular adherend and adhesive, independent of peel angle. Further, the strength (load)'measured by the peel test is not proportional to the actual strength of the adhesive, a sma...

An improved transverse shear deformation theory for laminated antisotropic plates

Abstract: An improved transverse shear deformation theory for laminated anisotropic plates under bending is presented. The theory eliminates the need for an arbitrarily chosen shear correction factor. For a general laminate with coupled bending and stretching, the constitutive equations connecting resultants with average displacements and rotations are derived. Simplified forms of these relations are also obtained for the special case of a symmetric laminate with uncoupled bending. The governing equation for this special case is obtained as a sixth-order equation for the normal displacement requiring prescription of the three physically natural bounday conditions along each edge. For the limiting case of isotropy, the present theory reduces to an improved version of Mindlin's theory. Numerical results are obtained from the present theory for an example of a laminated plate under cylindrical bending. Comparison with results from exact three-dimensional analysis shows that the present theory is more accurate than other theories of equivalent order.

Plastic Methods for Steel and Concrete Structures

Abstract: Preface - Notation - Some General Concepts - Plastic Bending - Collapse of Simple Frames - Limit Analysis - Deflections and Stability - Design Using Plastic Theory - Application of Plastic Methods to Reinforced Concrete Structures - Yield Line Analysis and the Hillerborg Strip Method for Reinforced Concrete Slabs - Appendices: Yield Criteria - Redundancy Test - Bending Moment Diagrams - Solutions to Problems - Inde x

Force-indicating dummy legs

Abstract: Force-indicating legs adapted to be fitted to the pelvic assembly of an anthropomorphic dummy for use in the dynamic testing of vehicle passenger restraint systems. The legs permit the measurement of most of the skeletal forces that are likely to be experienced during dummy leg contact with a vehicle instrument panel or a knee bolster in dynamic restraint system testing. Each dummy leg has a knee joint connecting the simulated femur and tibia bones and an ankle joint connecting the simulated tibia bone with a simulated foot structure. Forces acting on the simulated femur and tibia bones are measured by load cells capable of measuring axial (Z axis) compression and tension loads, shear loads on both orthogonal (X and Y) axes, lateral and longitudinal (or vertical) bending moments, and torsional moments. Knee shear load in the knee joint and upward angle bending load in the ankle joint are measured by load cells capable of measuring at least the axial loads resulting in the respective knee shear or ankle bending loads. The knee joint preferably includes biasing means to allow biomechanically correct tibia-to-femur movement during knee shear load measurements.

Generalized plastic hinge concepts for 3D beam-column elements

Abstract: Two basic procedures may be used for modeling the inelastic behavior of beams and columns. In the "fiber" type of model, the element cross section is divided into a number of small areas (fibers), and the behavior is governed by the stress-strain characteristics of the fiber material. Detailed and accurate results can be obtained, but the computational cost is high. In the "section" type of model, inelastic behavior is defined for the cross section as a whole, not for individual fibers. Actiondeformation relationships for the cross section must be devised, considering the stress-strain characteristics of the cross section material. Models of this type are less accurate than fiber models, but more efficient computationally. The purpose of the research has been to explore in depth the theory and computational techniques for the "section" type of model. In developing the model, inelastic interaction between bending moments, torque and axial force has been considered by means of yield interaction surfaces and a flow-rule type of plasticity theory. Emphasis has been placed on the ability to consider arbitrary loading-unloading cycles of the type likely to be induced by an earthquake. The study has considered both stable hysteretic action-deformation characteristics and relationships involving stiffness degradation. Three separate inelastic beam-column elements, which share similar concepts, have been developed, as follows. (a) An element with distributed plasticity and nondegrading stiffness, for the computer programs ANSR and WIPS. This element is most suitable for modeling inelastic behavior in piping systems. (b) An element with lumped plasticity and nondegrading stiffness, for the ANSR program. This element is most suitable for modeling inelastic steel beams and columns in buildings. (c) An element with lumped plasticity and degrading stiffness, for the ANSR program. This element is most suitable for modeling inelastic reinforced concrete beams and columns in buildings. The theory and computational procedure are described in detail for each element. Five example structures have been analyzed to test the elements and to assess their acceptability for different applications. The examples include a steel tubular beam-column; a steel tubular braced frame; a reinforced concrete cantilever beam under biaxial bending; a reinforced concrete frame subjected to earthquake excitation; and a pipe undergoing large displacements following pipe rupture.

Collapse resistance of submarine lines studied

Abstract: Theoretical and experimental studies on line-pipe bearing capacity and collapse pressures indicate that bending the pipe during deepwater pipe laying operations can initiate pipe collapse; stringent curvature control of pipelines is therefore of extreme importance. During installation, a pipe section can become unstable and collapse or buckle because of high hydrostatic pressures and certain combinations of axial forces, bending moments, and line-pipe imperfections (such as out-of-roundness). Researchers have studied these parameters, determining their qualitative and quantitative influence on the formation of a collapse mechanism.

Method of reducing springback in mechanically pressed sheet materials

Abstract: A method of pressing sheet material utilizing counterpad pressure to eliminate springback is disclosed. The sheet material is sequentially bent at pairs of bend radii during first and second increments of striking. The first increment of striking is carried out with little or no counterpad pressure to permit a curvilinear section to form immediately beneath the male punch member due to the bending moment. The second increment of striking is carried out with a positive counterpad pressure effective to flatten the curvilinear section and thereby space the second pair of bend radii from the first pair. Residual springback from the first bending action subtracts from the springback of the second bending to reduce the resultant springback in the product.

Shaft alignment using strain gages

Abstract: A detailed description of the strain gage shaft alignment procedure is presented including a comparison between bearing reactions obtained using strain gages and hydraulic jacking. The various gage configurations available for measuring bending moments are discussed and estimates of the resulting error in bending moment are given. A simplified procedure is presented to calculate bearing reactions from the measured moments. Gage site requirements to produce a determinate system are established and a method is outlined to establish bearing reaction error bounds for a combination of gage configurations and sites.

Tank container with mounting means

Abstract: A tank container comprising a framework (1-4) and a tank (7) disposed therein. The tank (7) is secured in the framework (1-4) by means of members (8, 11) transmitting tensile forces in the axial and radial directions of the tank wall so that no bending moment is exerted on the tank wall. These members are preferably radial suspension plates (8) between the ends of the tank and the corners of the end frames and axial fastening plates (11) between the cylindrical portion of the tank and the corners of the end frames.

Behavior of Concrete Columns UnderControlled Lateral Deformation

Abstract: This paper reports tests of nine model cantilever columns. The objectives of this phase of a study of the deformation capacity of columns in structural systems were to experimentally determine and analytically predict the complete axial load-moment-curvature (PM-4) relationships for columns with large axial loads and minimal ties. The controlled lateral deformation loading method used maintained stability after the maximum moment resistance was reached and allowed the descending branch of the P-M-+ relationships to be measured. This bracing against instability is similar to the lateral resistance which stronger members in an unbraced frame can provide. The measured P-M-+ relationships were not as brittle as previously indicated, but were nearly flat-topped and exhibited significant ductility. An analytical procedure was developed which closely agreed with the relationships observed. Concrete crushing strains ranged between 0.0096 and 0.0160 in./in. Predictions of an ultimate ltrain equation indicated that the values greater than 0.003 in./in. ,here due to the appreciable moment gradients. The results indicate ;,at the ACI Building Code limiting strain criteria can be safely inn-eased to 0.004 in./in. for columns with moment gradients which produce reversal of moments in the column length. Such an increase could have substantial benefits by allowing economical usage of higher strength reinforcement (f, of 80 to 90 ksi) in many columns.

Seismic Response of Reinforced Concrete Frames

Abstract: Analytical studies of the elastic and inelastic response of reinforced concrete frames subjected to earthquake motion are presented. The effect of stiffness degradation on the seismic response and in particular on ductility demand is examined. Several single-story frames are analyzed; stiffness degradation significantly increases ductility demand in short period systems. Studies of multistory frames are also presented; stiffness degradation again increases ductility demand but the effect is pronounced only in the upper stories. The bending moments and forces induced in an inelastic multistory frame are smaller than those in a purely elastic frame. However, if the columns are designed to remain elastic while the girders yield, the reduction in the column moments and axial forces is not as large as that in the girder moments.

Force transducer flexure with conductors on surfaces in the neutral bending plane

Abstract: A flexure in a force transducer, as an inertial guidance accelerometer, for securing a force sensitive element to a mounting base includes one or more flexure sections having one or more recessed surfaces which are substantially coincident with the neutral bending plane of the flexure Electrically conductive coating on the recessed surfaces provide electrical connections to components located on the force sensitive element The conductive coatings on or near the neutral bending plane of the flexure sections minimizes bending moments caused by stresses set up between the conductive coatings and the flexure which may in turn lead to bias errors At the same time, the flexure configuration provides for the desired strength and spring rate for the force sensitive element

Construction of a door beam of a motor vehicle

Abstract: A door beam of a motor vehicle is provided in a vehicle door. Section modulus of the door beam is progressively decreased from a center section toward opposite ends of the door beam and the contour of cross-section of the door beam is of a continuously formed round wave shape, whereby the door beam absorbs an impact of a collision by virtue of imparting the force of the impact to both bending and tensile forces so that the bending moment acting on the center of the door beam decreases.

Cantilevered roll shaft bearing bracket

Abstract: Apparatus for limiting ruinous rotary deflection and bending moment of roll shafts of a cantilevered type rolling mill. The present invention is a dual bearing bracket which connects the cantilevered ends of roll shafts in bearings to maintain constant parallel rotary alignment of the roll shafts. Thus, roll shaft service life is increased by limiting roll shaft deflection and bending moment.

Bending and springback theory of metal-polymer sandwich laminates

Abstract: In this report an analysis is made of the behavior of sandwich beams in which the core polymer is laminated on both sides with surface metal sheets, each of which has a different thickness and mechanical properties when they are loaded with a uniform bending moment which is then released resulting in springback of the bent sandwich beam. It is assumed that the polymer behaves elastically because the bending strain in the core is small and its elastic limit is much larger than that of metals. Sandwich beams have various elastoplstic stess distributions when bent depending on the mutual relationships between their dimensions, the mechanical moduli, and the applied bending moment. Further, residual curvatures, shifting position of neutral axis, and residual stress distributions in sandwich beams variously elastoplastically stressed initially on the decrease of applied bending moment are analyzed.

A Study on Longitudinal Motions and Bending Moment of a Container Ship in Following Sea

Abstract: The vertical bending moment in following sea to be calculated by Strip Methods has an important problem as obviously shown in the results of the international joint calculation under the Seakeeping Committee of I.T.T.C.In this paper, for the first time, the following Transient Water Wave was adopted in experiments. As the results suggested, good data was obtained effectively, and the fact that Transient Water Wave is useful even in following sea conditions was proved. Next, a practical method of calculation taking into account three-dimensional correction is presented. By applying this method, experimental values of vertical bending moment in following waves at near -zero encounter frequency can be explained satisfactorily.

Shrinkage and tension stiffening in negative moment regions of composite beams

Abstract: Results are given of bending tests in the elastic range on seven composite beams with the concrete slab in tension. They show that, when cracks in the concrete first occur, they are wider than would be predicted by the existing theory for cracking in reinforced concrete members. The reason is that cracking releases the locked-in stresses due to the restraint of shrinkage of the slab by the steel beam. When account is taken of this, the accuracy of the methods of BS 5400: part 5 for the prediction of hogging curvature and mean tensile strain at a given bending moment, is much improved. It is shown in another paper that these results also enable the widths and spacings of cracks in the concrete slab to be predicted more accurately. (Author/TRRL)