Dynamic analysis of a beam resting on an elastic half-space with inertial properties

This work gives a semi-analytical approach for the dynamic analysis of beams and plates resting on an elastic half-space with inertial properties. Such calculations have been associated with significant mathematical challenges, often leading to unrealizable computing processes. Therefore, this paper presents a detailed analysis of Green's function defining surface displacements of such a space in the contact zone with structures, which allows determination of reactive forces and other physical magnitudes. The obtained solutions can be applied to (i) study dynamic interaction between soil and structures, (ii) determine transient wave fields caused by a seismic source, and (iii) assess numerical computations with different numerical methods programs. Natural frequencies, natural shapes, and the dynamic response of a beam due to external harmonic excitation are determined. Eigenfrequencies and Eigenshapes are presented. Validation with a Boussinesq problem illustrates the inertia effect on the results of the dynamic analysis.     

Dynamic response of flexible rectangular foundations on an elastic half-space

An approximate method for the analysis of the dynamic interaction between a flexible rectangular foundation and the soil with consideration of the out-of-plane deformation of the foundation is presented. The procedure is based on an extension of the subdivision method developed by Wong and Luco for rigid foundations. Numerical results describing the influence of the flexibility of the foundation on the vertical and rocking impedance functions and on the contact stresses between the foundation and the soil are presented. The possibility of representing a flexible foundation by an equivalent rigid foundation having the same force-displacement relationships is also discussed. The results obtained indicate that at low frequencies, the dynamic stiffness coefficients for flexible foundations are lower than those for a rigid foundation of the same area. At higher frequencies the opposite behaviour is observed. The radiation damping coefficients for flexible foundations are significantly lower than those for a rigid foundation of the same area.     

Axisymmetric vibration of an elastic circular plate bonded on a transversely isotropic multilayered half-space

Based on the analytical layer-element method, an analytical solution is proposed to determine the dynamic interaction between the elastic circular plate and transversely isotropic multilayered half-space. The dynamic response of the elastic circular plate is governed by the classical thin-plate theory with the assumption that the contact surface between the plate and soil is frictionless. The total stiffness matrix of the transversely isotropic multilayered half-space is acquired by assembling the analytical layer-element of each soil layer with the aid of the continuity conditions between adjacent layers. According to the displacement condition of coordination between the plate and soil, the dynamic interaction problem is reduced to that of multilayered transversely isotropic half-space subjected to axisymmetric harmonic vertical loading. Some numerical examples are given to study the vertical vibration of the plate, and the results indicate that the dynamic response of elastic circular plate depends strongly on the material properties of the soils, the rigidity of the plate, the frequency of excitation and the external load form.     

Dynamic stresses in an elastic half-space

This paper deals with the problem of time-varying point loads applied onto the surface of an elastic half-space and the stresses that such loads elicit within that medium. The emphasis is on the evaluation of the isobaric contours for all six of the stress components at various frequencies of engineering interest and for a full range of Poisson’s ratios. The extensive set of pressure bulbs presented herein may be of help in predicting the severity of dynamic effects in common practical situations in engineering—or even the lack thereof.     

Torsional vibrations of elastic foundations embedded in an elastic half-space

This paper examines the axisymmetric torsional vibrations of an elastic pile and a hemispherical foundation embedded in a homogeneous elastic half-space. The embedded foundation–half-space system is decomposed into an extended half-space and a fictitious foundation. The deformations of the fictitious system are specified by an admissible function containing a set of generalized coordinates. The Lagrangian equations of motion are used to determine these coordinates associated with the assumed displacement function. Numerical results are presented for torsional impedance of an elastic pile and a hemisphere to illustrate the effects of relative flexibility and geometry. By employing certain simplifications on the pile–half-space system an approximate closed form solution is presented for the torsional impedance of an elastic pile.     

Transient flexural vibrations of an elastic column supported by an elastic half-space

An analysis is presented of the transient flexural vibrations of an elastic column supported by an elastic half-space under the condition that an arbitrarily shaped free-field lateral acceleration and displacement are given as inputs. Applying Laplace transformations with respect to time and numerical inverse Laplace transformations, the time histories of the column acceleration at the interface and free end, and the column and half-space displacement distributions are obtained. After the input free-field acceleration terminates, slightly damped and almost harmonically variable acceleration is observed. The acceleration frequency after the disappearance of the input acceleration nearly coincides with the resonant frequency of the system. The slight damping with the first resonant frequency, even if the half-space is soft compared with the column, is characteristic of the transient flexural vibrations of a column supported by a half-space. Such a phenomenon is not typical of the transient longitudinal vibration problem. Therefore, it may be concluded: when buildings and structures are subjected to an earthquake or an explosive force, their flexural vibrations will continue with their first resonant frequencies, even if their foundations are soft.     

The dynamics of a rigid foundation on the surface of an elastic half-space

The response of a rigid rectangular foundation block resting on an elastic half-space has been determined by considering first the displacement functions for any position on the surface of an unloaded half-space due to a harmonic point force. The influence of the foundation has been taken into account by assuming a relaxed condition at the interface, i.e. a uniform displacement under the foundation and that the sum of the point forces must equal the total applied force. The three motions of vertical, horizontal and rocking have been considered and numerical values for the in-phase and the quadrature components of the displacement functions are presented for a Poisson's ratio of 0.25. The effect of the mass and inertia of the foundation can be allowed for by an impedance matching technique. Response curves and non-dimensional resonant frequency curves are given for a square and a rectangular foundation for different mass and inertia ratios and for several values of Poisson's ratio. These curves are for design purposes and are an addition to similar published curves for circular and infinitely long rectangular foundations. Some of the calculated results have been verified by a laboratory experiment.     

Disturbance produced in an elastic half-space by impulsive normal pressure

Summary The displacement produced in the half-space by uniform impulsive pressure acting over a circular portion of the surface has been obtained in terms of definite integrals. FollowingCagniard andGarvin, the displacement at an interior point on the axis of symmetry has been calculated. The approximate displacement at points far off from the axis of symmetry has been obtained by using saddle point method and Tauberian limit theorem.     

ASH-source in an elastic half-space with a non-homogeneous surface layer

Summary The response of an elastic half-space with a non-homogeneous surface layer due to aSH-source operating in the non-homogeneous layer of an elastic half-space have been studied. The elastic and physical constants vary exponentially in the surface layer. The surface displacement consists of several integrals. On approximate evaluation of these integrals, they are identified as direct or reflected pulses.     

Transient flexural vibrations of a timoshenko column supported by an elastic half-space

Transient flexural vibrations of an elastic column supported by an elastic half-space are investigated analytically under the condition that an arbitrarily shaped free-field lateral acceleration is given as an input. Applying the Timoshenko theory to the column and making use of Laplace transformations with respect to time and numerical inverse Laplace transformations, the time histories of the column free end acceleration are presented. Numerical results obtained from the Timoshenko theory are compared with those of a previous paper¹ (applying the Bernoulli-Euler theory to the column), and the effects of column slenderness and foundation stiffness on the transient flexural vibrations of the column are clarified.     

On the thermal bending of a thin elastic simply supported rectangular plate resting on an elastic foundation

Pure and Applied Geophysics - Deflection and bending moments of a thin isotropic rectangular plate, having its edges simply supported and maintained at zero temperature, have been investigated in...     

Seismic response of reduced micropolar elastic half-space

This paper explores reduced micropolar theory to simulate ground motion during an earthquake. In this theory, rotational motions are kinematically independent of translational motions. Analytical expressions for ground displacement and rotational motions due to a buried seismic source are presented in this paper. This theory requires two additional material constants which characterise the microstructure of the medium compared with linear elastic theory. Ground motions are simulated for an earthquake of magnitude (M _w) 5.0. The sensitivity of ground motion to these new material constants is reported. It is observed that rotations are sensitive to microstructure of the medium. A comparison with recorded rotations of the M _w 5.2 Izu peninsula, Japan event is also presented in this article.     

Modal analysis of circular flexible foundations under vertical vibration

A methodology using modal analysis is presented to evaluate dynamic displacements of a circular flexible foundation on soil media subjected to vertical vibration. The interaction effects between the foundation and the underlying soil are represented using modal soil impedance functions determined by an efficient procedure developed. The displacements of the foundation can then be easily solved by modal superposition. Comparing with existing solutions, the presented method is found to provide accurate results with less computational effort using only a few vibration modes. In addition, parametric studies for modal responses of the flexible foundation indicate that the response of the foundation are significantly influenced by relative stiffness between the foundation and the soil medium, load distributions, vibration frequency range, and the foundation mass. Besides, justification for flexible foundations to be considered as rigid are investigated.     

Vibration of beams on partial elastic foundations

The natural response of a beam or pile having only a portion of its span supported by an elastic foundation is investigated for the two cases when both ends are either simply supported or free. The derivation of the shape functions and the computed natural frequencies are compared with the extreme cases where the element is either completely supported by, or fully detached from, the elastic foundation.     

Vibration of beams on non-homogeneous elastic foundations

The natural response of a beam or pile supported by an elastic foundation is investigated for the case when the coefficient of subgrade reaction varies linearly along the span of the member.     

Stability and dynamics of beams on winkler elastic foundations

The dynamical properties of a foundation beam on a Winkler soil have been recently discussed by using the power series method¹⁴ and the finite element method.^{15, 17 – 19} The first method can be applied only if the Winkler soil is linearly distributed along the whole span: on the contrary, the second method is more general, but it requires considerable computational efforts. In this paper a recently developed discretization method has been shown to be particularly well suited for dynamic analysis of foundation beams, in the presence of second order destabilizing effects. The procedure is based on the Hamilton principle, and the equation of motion is deduced from the corresponding Lagrange equations. According to the proposed method, the structure is reduced to a set of rigid bars, connected together by elastic cells. The convergence of the method is shown to be quite rapid, so that a small number of Lagrangian coordinates has to be introduced, and the resulting eigenvalue problem can be easily solved. Numerical comparisons are also performed with the available data of the literature, and the agreement is shown to be excellent. Two particularly unusual cases are illustrated, in order to emphasize the method's peculiarities. Finally, the influence of the axial forces on the fundamental vibration frequency is discussed, both for conservative and non-conservative forces.     

Response of a plate and elastic half-space to harmonic waves

Analytical results are presented for the dynamic interaction of an elastic flexural plate and an elastic half-space subjected to harmonic seismic waves. Displacements and contact stresses are presented for square, massless plates having a practical range of flexural stiffness and subjected to incident waves oriented parallel to either an edge or a diagonal of the plate. The behaviour of massive plates is also briefly discussed; such plates exhibit additional resonances higher than those normally associated with the motion of a rigid mass.     

Displacement due to impulsive loadings on the surface of elastic half-space

Summary The displacement due to vertical impulsive load has been obtained by function theoretic method. Graphs have been drawn for horizontal as well as for vertical displacement. It has been shown thatPekeris's [12]²) method is easier to apply than that ofCagniard [2] for the same purpose.