Showing papers on "Linear elasticity published in 1968"

Seismic Response of Horizontal Soil Layers

Abstract: Methods of analysis of the response of soil deposits during earthquakes are presented. These methods include linear elastic analyses, a bilinear analysis, and an equivalent linear analysis. All these methods require that: (1) The surface of the layer, the interface between any two sublayers and the base of the layer be essentially horizontal, (2) the material properties of the layer be constant along any horizontal plane, and (3) the applied seismic excitation be also horizontal. The closed-form solutions are utilized to evaluate the accuracy and stability of the lumped-mass analysis and criteria for the accuracy and stability of the lumped-mass representation are proposed. A lumped-mass representation is also used for the analysis of soil layers having bilinear stress-strain characteristics. Finally, a procedure is outlined for obtaining equivalent linear parameters for soils with bilinear stress-strain characteristics. The results obtained by this procedure are shown to be in good agreement with those determined by the bilinear solution.

Nonlinear Analysis of Elastic Framed Structures

Abstract: A consistent formulation for the geometrically nonlinear behavior of linear elastic rigid framed structures is developed and the appropriate equations for two solution procedures, successive substitution and Newton-Raphson, are derived. A linearized incremental loading procedure and stability analysis are described. The formulation is applied to two structures and comparison studies illustrating the effect of load increment size and convergence tolerance on the total computational effort are included. The results indicate that the Newton-Raphson method is more efficient, particularly for low convergence tolerance. Also, if only a single solution is desired, it is more efficient to apply the total load in one step rather than in increments.

Static and Dynamic Finite Element Analysis of Sandwich Structures

Abstract: : The finite element method is extended to the refined elastic analysis of multilayer beams and shells with no restriction placed upon the ratios of the layer thicknesses and properties. The method is applicable to structures wherein shearing deformations are significant, including sandwich-type structures. Element stiffnesses are based on polynomial displacement models and are applicable to the linear elastic analysis of beams and thin, axisymmetric shells of arbitrary meridian. Although attention is restricted to three-layered construction with similar facings, the theory may be generalized to any type of flexural element and any arrangement of laminations. Computer programs have been written for both static analysis and free vibration analysis. Inclusion of rotatory as well as translational inertia allows the determination of natural thickness-shear frequencies and mode shapes in addition to flexural vibration characteristics. Examples are presented to illustrate the effectiveness of the method.

Seismic response of horizontal soil lauers

Abstract: METHODS OF ANALYSIS OF THE RESPONSE OF SOIL DEPOSITS DURING EARTHQUAKES ARE PRESENTED. THESE METHODS INCLUDE LINEAR ELASTIC ANALYSES, A BILINEAR ANALYSIS, AND AN EQUIVALENT LINEAR ANALYSIS. ALL THESE METHODS REQUIRE THAT: (1) THE SURFACE OF THE LAYER, THE INTERFACE BETWEEN ANY TWO SUBLAYERS AND THE BASE OF THE LAYER BE ESSENTIALLY HORIZONTAL, (2) THE MATERIAL PROPERTIES OF THE LAYER BE CONSTANT ALONG ANY HORIZONTAL PLANE, AND (3) THE APPLIED SEISMIC EXCITATION BE ALSO HORIZONTAL. THE CLOSED-FORM SOLUTIONS ARE UTILIZED TO EVALUATE THE ACCURACY AND STABILITY OF THE LUMPED-MASS ANALYSIS AND CRITERIA FOR THE ACCURACY AND STABILITY OF THE LUMPED-MASS REPRESENTATION ARE PROPOSED. A LUMPED-MASS REPRESENTATION IS ALSO USED FOR THE ANALYSIS OF SOIL LAYERS HAVING BILINEAR STRESS-STRAIN CHARACTERISTICS. FINALLY, A PROCEDURE IS OUTLINED FOR OBTAINING EQUIVALENT LINEAR PARAMETERS FOR SOILS WITH BILINEAR STRESS-STRAIN CHARACTERISTICS. THE RESULTS OBTAINED BY THIS PROCEDURE ARE SHOWN TO BE IN GOOD AGREEMENT WITH THOSE DETERMINED BY THE BILINEAR SOLUTION. /ASCE/

General Equations of the Theory of Elasticity

Abstract: A body is deformed when forces are applied to it; the distances between points in the body subject to the forces are somewhat different from those between the same points in the absence of the forces . The change in distance is due to displacement of one relative to another during the deformation. Let r0 be the radius vector of some point in the body (relative to some fixed point in space) before the deformation, and let r(xi) be the same after deformation. The difference between these vectors is the displacement vector of the point and is (1.1) or in coordinate form (1.2)

Extensional Waves in Elastic Bars of Rectangular Cross Section

Abstract: A one‐dimensional approximate theory is derived that accounts for the first eight modes of propagation of extensional waves in uniform, isotropic, linear elastic bars of rectangular cross section. The theory provides an optimal quadratic approximation to the lateral spatial dependence of the three‐dimensional bar response. Applicable to motions in bars of arbitrary rectangular section, it is possible to predict bar responses that are neither plane stress nor plane strain, but are transitional between the two. The relation of bar to plate theory is established, and an identification of bar and plate modes is proposed. Discussion of dispersion of harmonic waves is limited to bars of square section. Typical spectra are given, and their variation with Poisson's ratio is established.

Finite element analyses of slopes in soil.

Abstract: : The study described in this report was conducted to investigate the behavior of excavated slopes and embankments using the finite element method of analysis. The aspects of slope behavior considered include stresses, strains, displacements, pore pressures, and progressive development of failure zones. Analyses have been performed using nonlinear stress-strain characteristics as well as linear elastic stress-strain behavior. A method has been developed for simulating analytically the excavation of a slope in a clay layer with arbitrary initial stress conditions. (Author)

Symbolic Approach to Dynamical Problems in Plates

Abstract: A formal approach is used to obtain two‐dimensional differential equations (of infinite order) for dynamical problems in plates. It is assumed that the displacements may be expanded in power series in z, the thickness coordinate. These power series are substituted into the three‐dimensional dynamical equations of linear elasticity. The coefficients of powers of z are equated to zero leading to an infinite sequence of differential equations which by formal manipulation are reduced to three differential equations of infinite order in which the midplane displacements are the dependent variables and x, y, t are the independent variables. It is shown how various special theories including the classical theories may be obtained from the general equations by making certain assumptions on the frequency and wavelength of the expected solutions. A short discussion of the solution of an initial value problem by means of the superposition of solutions of the various special theories is also given.

Three-dimensional fields due to a Volterra dislocation imbedded in a layered half-space: Analytical representation of a seismic mechanism

Abstract: : A three-dimensional boundary value problem in static linear elasticity is presented as a restricted formal characterization of a conceptual model for the crust-mantle system of the earth. The system is considered as a two-layered half-space, where elastic and inelastic properties are attributed to both media. It is assumed that both materials are incompressible. A Volterra dislocation, representing a fault plane, is imbedded in the upper layer, lying in a vertical plane with slip parallel to the free surface. The problem is solved by means of a modified form of the Galerkin vector and use of double Fourier transforms. Solutions are obtained in quadrature for the surface displacements and some stresses at the interface. Time dependent properties are introduced into the system by means of Biot's correspondence principle of linear viscoelasticity. This investigation is a segment of a comprehensive study in geomechanics, based upon a more complete conceptual and ontological model of the crust and mantle, which may prove essential for implementing an effective shallow-earthquake prediction program. (Author)

Stress Concentration in Strain-Gradient Bodies

Abstract: An analytical solution is presented to a boundary-value problem in the linear elasticity in which potential energy depends on strains and strain gradients. The problem considered is that of an infinite medium, containing a spherical cavity, subjected to uniaxial tension at infinity. For an isotropic, centrosymmetric material, the strain-gradient theory contains five additional material constants (micro-constants), which may be related to Lame' constants by means of two material lengths. The range of the micro-constants is determined on the basis of positive definiteness of the potential energy, and under this condition no difficulty is encountered in reducing the strain-gradient solution to either the couple-stress or the classical solution. A reduction to the classical result does not have to employ the couple-stress solution as an intermediate step. Numerical calculations are performed for various values of nondimensionalized, independent parameters. The results show that, in contrast to predictions of couple-stress theory, the stress concentration factor can exceed the classical value.

Strain-Gradient Effects on Force Transfer Between Embedded Microflakes:

Abstract: In this paper the plane strain problem concerned with shearing stresses along the fiber-matrix interface due to a fracture of the adjacent fiber is solved anew based on Mindlin's strain-gradient theory of linear elasticity. Classical method of selecting proper stress functions and determining superposition constants to satisfy all homogeneous boundary conditions is used to obtain the auxiliary solution for the problem. Applying a Fourier integral transformation to the auxiliary solution both homogeneous and non-homogeneous boundary conditions are simultaneously satisfied. The exact solu tion is thus obtained in integral form. Numerical results of the solution for some selected elastic constants have been worked out and compared with results previously obtained based on classical theory and couple-stress theory.

On the displacement boundary-value problem of static linear elasticity theory

Abstract: Unter der Voraussetzung, dass die Verschiebungsgleichungen der linearen Elastostatik fur isotrope homogene Materialien stark elliptisch sind, haben wir einige globale Extremalprinzipien, ein Reziprozitatstheorem, einen Satz betreffend die dazu verwandte Greensche Abbildung und zwei Mengen von oberen und unteren Grenzen fur die erste Randwertaufgabe hergeleitet.

Embankment analysis and field correlation

Abstract: EXPERIMENTAL RESULTS FROM A PROGRAM TO DEVELOP ACCURATE AND RELIABLE INSTRUMENTATIONS FOR STRESS AND DEFORMATION MEASUREMENTS IN HIGHER EMBANKMENTS ARE NUMERICALLY ANALYZED FOR A PARTICULAR EMBANKMENT TO COMPARE THE RESULTS WITH THE FIELD MEASUREMENTS. THIS COMPARISON PERMITS AN ASSESSMENT OF THE ACCURACY OF THE MATHEMATICAL MODEL AND PREDICTION TECHNIQUE. THE ANALYSIS IS BASED ON A PLANE STRAIN SYSTEM WITH LINEARLY ELASTIC ISOTROPIC MATERIALS THAT IS CONSTRUCTED AND LOADED INCREMENTALLY. THE FINITE ELEMENT METHOD IS EMPLOYED TO OBTAIN STRESSES AND DISPLACEMENTS IN THIS TYPICAL CROSS-SECTION. THE CONTINUOUS SYSTEM IS IDEALIZED AS AN ASSEMBLAGE OF SMALLER, BUT FINITE, ELEMENTS CONNECTED AT A DISCREET NUMBER OF POINTS CALLED NODES. THE BASIS FOR THE METHOD IS A CONSISTENTLY DERIVED STIFFNESS FOR EACH OF THESE ELEMENTS, THE STIFFNESS BEING A RELATIONSHIP BETWEEN THE GENERALIZED FORCES AND THE DISPLACEMENTS AT THE NODES OF EACH ELEMENT. STIFFNESS OF THE ELEMENTS USED IS PREDICTED ON A MATERIAL POSSESSING LINEAR ELASTIC PROPERTIES. EXAMINATION OF THE ANALYTICAL RESULTS AND COMPARISON WITH THE FIELD DATA INDICATE THAT THE MATHEMATICAL MODEL CAN LEAD TO QUALITATIVE PREDICTIONS. THE WEAKEST FEATURE OF THE MATHEMATICAL MODEL IS THE MATERIAL CHARACTERIZATION. IT IS RECOMMENDED THAT: (1) ESTIMATES FOR STRESS AND DISPLACEMENT FIELDS SHOULD BE OBTAINED PRIOR TO PLACEMENT OF INSTRUMENTATION, (2) LATERAL PRESSURES SHOULD BE MEASURED AS WELL AS VERTICAL STRESSES, (3) VERTICAL PRESSURE CELLS MIGHT BE USED TO CALIBRATE THE CELLS MEASURING HORIZONTAL STRESSES, AND (4) AN ACCURATE DETERMINATION OF THE MATERIAL PROPERTIES OF THE EARTH FILL MUST BE MADE WHEN INSTRUMENTING AN EMBANKMENT.

Two-dimensional dynamical problems for incompressible isotropic linear elastic solids with time dependent moduli and variable density

Abstract: This paper describes a new general method of solution of two-dimensional problems involving the dynamical behaviour of incompressible linear elastic solids. As formulated, the solution takes account of density variation with both position and time and a shear modulus which is purely time dependent. Several simple applications are made to problems involving thick cylinders and an infinite plate containing a circular hole.

Creep buckling of plates.

Abstract: : The effect of creep on the load carrying ability of plates was investigated The plates were loaded by compressive forces applied in the plane of the plate It was assumed that the material deforms due to linear elasticity and steady state creep, and creep strain rate is given by the power law When a fixed uniaxial load is applied to the plate in such a way that the load remains uniformly distributed during the buckling process, the governing equations indicate that a finite time exists after which the lateral deformations of the plate become unbounded Simple, closed form expressions for this critical time are derived for various values of the creep exponent, and for the case where primary creep plays an important role (Author)