Vertical Point Load Research Articles

2-D analysis of arch bridges using an elasto-plastic material model

A general method for the static determination of the continuous distribution of internal forces and displacements in arbitrarily shaped arch bridges under different load conditions is proposed. The basis of the elasto-plastic model are three equations of equilibrium: horizontal equilibrium, vertical equilibrium and equilibrium of moments. Boundary conditions are added to impose restrictions on the horizontal and vertical movement and on rotations at the abutments of the arch. For masonry arches, it is necessary to include material properties that allow the occurrence of cracks and the subsequent formation of hinges. This theory has been implemented in a computer program that was subsequently used to calculate the results of the two examples presented here, an arch loaded with a vertical point load and under the influence of a horizontal displacement of one abutment.

Surface displacements due to batter piles driven in cross‐anisotropic media

AbstractThis article derives the closed‐form solutions for estimating the vertical surface displacements of cross‐anisotropic media due to various loading types of batter piles. The loading types include an embedded point load for an end‐bearing pile, uniform skin friction, and linear variation of skin friction for a friction pile. The planes of cross‐anisotropy are assumed to be parallel to the horizontal ground surface. The proposed solutions are never mentioned in literature and can be developed from Wang and Liao's solutions for a horizontal and vertical point load embedded in the cross‐anisotropic half‐space. The present solutions are identical with Wang's solutions when batter angle equals to 0°. In addition, the solutions indicate that the surface displacements in cross‐anisotropic media are influenced by the type and degree of material anisotropy, angle of inclination, and loading types. An illustrative example is given at the end of this article to investigate the effect of the type and degree of soil anisotropy (E/E′, G′/E′, and ν/ν′), pile inclination (α), and different loading types (a point load, a uniform skin friction, and a linear variation of skin friction) on vertical surface displacements. Results show that the displacements accounted for pile batter are quite different from those estimated from plumb piles, both driven in cross‐anisotropic media. Copyright © 2007 John Wiley & Sons, Ltd.

An analytical model to determine the ultimate load on masonry arch bridges.

This paper proposes an analytical elasto–plastic model to describe the behavior of arches. The modeling is carried out using the equations of (i) horizontal equilibrium, (ii) vertical equilibrium and (iii) equilibrium of moments. The latter equations of equilibrium are ordinary differential equations which can easily be solved by adding boundary conditions, imposing restrictions on the horizontal and vertical movement and on the rotation in the abutments of the arch. For masonry arches, including material properties allowing the occurrence of cracks and the subsequent formation of hinges is required. The latter theory has been implemented in a computer program (Matlab), offering numerical simulations. The software was used to illustrate two case-studies, i.e., the assessment of an arch loaded with a vertical point load and one with a horizontal point load.

地表面の鉛直方向調和振動荷重による地盤の層厚・弾性定数の推定方法とその数値実験による検証

This paper presents a method of estimating the elastic properties of the soils and their variation with depth using the characteristics of P-SV wave fields caused in the surface of an elastic multi-layered half space by the vertical harmonic point load applied also on the soil surface. To demonstrate the capability of the proposed method, the two numerical examples are presented, for the estimation of S wave velocity, P wave velocity and thickness of each soil layer, using a three-layered half space and a five-layered half space. In the numerical examples, the stiffness matrix method is used for an accurate simulation of all P-SV wave fields.

Evaluation of Green’s function for vertical point-load excitation applied to the surface of a layered semi-infinite elastic medium

The topic of this paper is to show that the integrals of infinite extent representing the surface displacements of a layered half-space loaded by a harmonic, vertical point load can be reduced to integrals with finite integration range. The displacements are first expressed through wave potentials and the Hankel integral transform in the radial coordinate is applied to the governing equations and boundary conditions, leading to the solutions in the transformed domain. After the application of the inverse Hankel transform it is shown that the inversion integrands are symmetric/antimetric in the transformation parameter and that this characteristic is preserved for any number of layers. Based on this fact the infinite inversion integrals are reduced to integrals with finite range by choosing the suitable representation of the Bessel function and use of the fundamental rules of contour integration, permitting simpler analytical or numerical evaluation. A numerical example is presented and the results are compared to those obtained by the CLASSI program.

Implant Prosthodontics: An in-vitro photoelastic stress analysis

Arguably the greatest controversy in implant dentistry is that of linking a natural tooth and an implant together in a fixed restoration. Some researchers have absolutely contradicted the inclusion of the two, while others have suggested different options such as design considerations in the restoration or in the implant itself. Further more an outcome of a situation in linking a periodontally compromised abutment with an implant in a restoration itself lacks investigation. The study was designed to evaluate stresses transmitted to supporting structures by a fixed partial restoration under different abutment situations using a rigid / movable connector in the prosthesis. Also to evaluate stresses transmitted and the outcome of linking a periodontally compromised abutment with the implant in a fixed restoration. Photoelastic models were fabricated depicting a partially edentulous posterior mandible. The situations depicted were of a completely tooth supported, tooth implant supported and a completely implant supported situations. 20% and 35 % periodontal damages were also incorporated in the anterior abutment in the models. The fixed restorations were fabricated with movable and rigid type connectors. Vertical point loads were applied on fixed points on the restoration and the transmitted stresses under different simulated situations of the study were then recorded and analyzed. The fringe orders were recorded photographically and the fringe orders were tabulated for interpretation. After thorough analysis of the results obtained the study concluded that it is indeed beneficial to connect natural tooth to implants in a fixed partial restoration and that the type of connector advocated is a rigid one. Also conclusive is the fact that periodontally compromised teeth can be integrated in the restoration in combination with an implant as a conservative treatment option. The use of non rigid connectors in any situation may be erroneous.

調和振動荷重によるＰ‐ＳＶ波の地盤振動・波動伝播特性

Elastic waves generated by a controlled dynamic load applied on the surface of soil deposits yield useful tools for non destructive testing of the systems to determine the elastic properties of the soils and their variation with depth. A number of methods for the non destructive testing have been developed and are being used in practice. The objective of this paper is to examine some of the basic concepts related to P-SV wave propagation in layered media and their dynamic response due to the vertical harmonic point load applied on the soil surface. A proper understandings of these concepts is necessary for a correct interpretation of the field data and for improving the accuracy of the estimation of the soil properties at low strain values by using the spectral analysis of surface waves or the Rayleigh wave method.

Three-dimensional steady state Green function for a layered isotropic plate

The elastodynamic response of a layered isotropic plate to a source point load having an arbitrary direction is studied in this paper. A decomposition technique is developed within each homogeneous isotropic lamina to simplify the general three-dimensional plane-wave propagation problem as a separate plane-strain problem and an anti-plane-wave propagation problem. The accuracy of computation is assured by cross-checking the numerical results by different methods. Results are checked numerically for a vertical point load acting on a homogeneous and a layered plate by using a hybrid method. On the other hand, results are checked for a horizontal point load by using dynamic reciprocal identities. Results are presented for both a homogeneous as well as a layered plate.

Displacements and stresses due to a vertical point load in an inhomogeneous transversely isotropic half-space

We present the solutions for displacements and stresses subjected to a vertical point load in a continuously inhomogeneous transversely isotropic half-space with Young’s and shear moduli varying exponentially with depth. Planes of transverse isotropy are assumed to be parallel to the horizontal surface. The solutions for the half-space are obtained by superposing the solutions of two full spaces, one with a point load in its interior and the other with opposite traction of the first full space along the z ¼ 0 plane. The Hankel transform in a cylindrical co-ordinate system is employed for deriving the solutions. However, the resulting integrals for displacements and stresses involve polynomial, exponential function, and Bessel function that cannot be given in closed form; hence, numerical techniques are adopted in this work. In order to check the accuracy of numerical procedures, the comparisons are carried out with the homogeneous solutions of Liao and Wang, and the calculated results agree with those to nine decimal places. Furthermore, two illustrative examples are presented to elucidate the effect of inhomogeneity, and the type and degree of rock anisotropy on the vertical surface displacement and vertical normal stress in the inhomogeneous isotropic/ transversely isotropic rocks subjected to a vertical concentrated force acting on the surface. The calculated results show that the induced displacement and stress are decisively influenced by the inhomogeneity, and the degree and type of material anisotropy. The proposed solutions can more realistically simulate the actual stratum of loading problem in many areas of engineering practice. r 2003 Elsevier Ltd. All rights reserved.

The numerical solutions of Green's functions for transversely isotropic elastic strata

In this paper, a model of transversely isotropic elastic strata is used to simulate the soil layers situated on a half space. Instead of the half space, an artificial transmitting boundary is used to absorb the vibration energy. The displacement formulas at any soil layer interface under vertical or horizontal harmonic ring loads are obtained by using the thin layer element method. From these formulas, the explicit solutions of Green's functions—the displacement responses at any interface of these strata under vertical and horizon harmonic point loads—are derived. The examples show that the method presented in this paper is close to the theoretical method and the transversely isotropic property has evident influence on the Green's functions.

The Boussinesq problem for soils with bounded non-homogeneity

The response of a compressible continuously non-homogeneous elastic soil to a static vertical point load on its surface is analytically investigated by using classical integral transform techniques and the extended power series method for obtaining the solution in the transform domain. The non-homogeneity is described by means of a depth-function which is non-zero at the surface and bounded at infinity and is capable in modelling both increasing and decreasing soil stiffness with depth. The influence of non-homogeneity on the displacements and stresses at the surface and in the interior is examined over a wide range on the governing parameters. © 1998 John Wiley & Sons, Ltd.

Elastic solutions for a transversely isotropic half-space subjected to a point load

We rederive and present the complete closed-form solutions of the displacements and stresses subjected to a point load in a transversely isotropic elastic half-space. The half-space is bounded by a horizontal surface, and the plane of transverse isotropy of the medium is parallel to the horizontal surface. The solutions are obtained by superposing the solutions of two infinite spaces, one acting a point load in its interior and the other being free loading. The Fourier and Hankel transforms in a cylindrical co-ordinate system are employed for deriving the analytical solutions. These solutions are identical with the Mindlin and Boussinesq solutions if the half-space is homogeneous, linear elastic, and isotropic. Also, the Lekhnitskii solution for a transversely isotropic half-space subjected to a vertical point load on its horizontal surface is one of these solutions. Furthermore, an illustrative example is given to show the effect of degree of rock anisotropy on the vertical surface displacement and vertical stress that are induced by a single vertical concentrated force acting on the surface. The results indicate that the displacement and stress accounted for rock anisotropy are quite different for the displacement and stress calculated from isotropic solutions. © 1998 John Wiley & Sons, Ltd.

Green functions of the earth at borehole sensor installation depths for surface point load.

Green functions of the earth at borehole sensor installation depths (0 to 962.5m, at each 68.75m) for the surface vertical point load were calculated using spherical earth model 1066A. Correct calculation requires both eigen functions and Legendre functions whose order n exceeds 1 million. Green function behavior at depth=0 and depth≠0 are significantly different, especially at a short distance from the loading point. In the θ (angular distance) <0.01 degree range, spherical green functions and their Boussinesq approximations coincide well except for those including radial displacement, both at depth=0 and ≠0, but diverge at large θ due to the earth's nonuniformity and surface curvature. Atmospheric coefficients for areal and volume strain show scale dependence reflecting the earth's inner structure and the relative magnitude of installation and skin depths.

Numerical modelling of elastic interaction between circular rafts and cross-anisotropic media

A technique of numerical modelling of elastic interaction between a soil represented by a half-space and an axisymmetric circular raft is outlined. Also discussed are two numerical methods of solution of the related problem of a circular plate under static, axisymmetric loading conditions. The method allows for homogeneous, cross-anisotropic and isotropic material properties of the half-space to be employed and assumes a frictionless plate-half-space interface. The soil-structure interaction derives from the Koning solution to the problem of a vertical point load applied at the surface of a homogeneous cross-anisotropic half-space, whilst the plate behaviour is assumed to follow the thin plate theory and is modelled using either a finite element technique or a finite difference procedure. The effectiveness and usefulness of the proposed numerical approaches and the developed computer programs are tested with the included examples.

On the Collapse of Masonry Arches

In this paper a constitutive equation for masonry arches is defined and its main properties are proven; in this equation to each pair of generalized strains (e, κ), with e the extensional strain and κ the curvature change of the centre line, is assigned the pair of generalized internal forces (N,M), where N is the normal force and M the bending moment. Subsequently, the collapse of masonry arches is characterized and the static and kinematic theorems proven. Finally, a method for determining the collapse load in the case of circular arches subjected to their own weight and a vertical point load applied at a point of the extrados is presented. The results obtained, of interest in some applications, are summarized in a series of graphs.

Discretely Supported Rails Subjected to Transient Loads

SUMMARY In this paper, the vertical displacement of a railway rail due to a travelling vertical point load of variable intensity is calculated. The model of the track is largely due to Grassie, albeit that the sleepers are modelled more realistically. The rail is modelled as an Euler beam. The most salient features of the paper are that the travelling of the variable load and the irregular discrete support of the sleepers are taken fully into account. The complex Fourier transform is used throughout this paper.

Dynamic plate-soil interaction — finite and infinite, flexible and rigid plates on homogeneous, layered or Winkler soil

An integral method to calculate the solution of a homogeneous or layered soil due to a harmonic point load is described. An infinite plate at the surface of the soil can be introduced in this integration in wavenumber domain, too. Finite structures on the soil are calculated by a combined finite element and boundary element method, which makes use of the point load solution of the soil. The compliance functions for a vertical point load and some vibration modes are calculated for realistic parameters of the plate and the soil and for a wide range of frequencies. The influence of the stiffness of the soil and the foundation is investigated, showing that the soil mainly affects the low-frequent response whereas the structural properties are more important at higher frequencies. A rigid approximation of flexible plates is only found at low frequencies, if the elastic length 1 = B G 3 is used as the radius of a rigid disk. At higher frequencies, a characteristic behaviour of the flexible plate of approximately u p ∼ (iω) 1 2 is observed, what is in clear contrast to the compliance of rigid foundations. A plate on a visco-elastic support (Winkler soil) shows similar displacements as a plate on a homogeneous half-space, but the maximal stresses between the plate and the soil are considerably smaller which is found to be more realistic for a plate on a layered soil. For practical applications, a normalized diagram and some explicit formulas of the exact and the approximate solutions of an infinite plate on a homogeneous half-space are given, which is a useful model to represent the soil-structure interaction of flexible foundations.

Wave Propagation in Layered Soils: Theoretical Solution in Wavenumber Domain and Experimental Results of Hammer and Railway Traffic Excitation

A method of calculating the wave field of horizontally layered soils is presented. It combines numerical integration in the wavenumber domain with the matrix method for layered soils in the most simple way. The three-dimensional problem of a harmonic horizontal or vertical point load acting on the surface of the soil is investigated. The method yields the radial, transverse and vertical displacements as functions of frequency and distance, which describe the complete wave field due to a point load of any direction. Results are presented for the vertical displacement due to vertical excitation, as that is the situation which is usually of interest in soil prospecting and vibration analysis. The amplitudes and partly the phases (which yield the dispersion curves of the soil profile) of the complex transfer functions are given for some elementary layered soils, such as a soft layer on a stiffer half-space, a stiff layer on a softer half-space, and continuous variation of stiffness and damping. Comparison of the results with those for homogeneous half-spaces of different wave speeds leads to a general conclusion that usually only the soil material down to half a wavelength of the Rayleigh wave has an influence on the response of the surface, but some significant deviations from that rule are also shown. Experimental results for three different sites due to impulse, harmonic and railway traffic excitation are presented and compared with the theoretical results, and these demonstrate the strong influence of the soil profile on the distribution of wave amplitudes in distance and frequency.

Time‐harmonic boussinesq problem for a continuously non‐homogeneous soil

AbstractAn analytical solution is presented for the response of a non‐homogeneous, compressible, elastic half‐space to a time‐harmonic vertical point load on its surface. The shear modulus is assumed to increase continuously with depth. The model is chosen so as to describe uniformly deposited cohesionless soils. Expressions for displacements and stresses in the interior of the half‐space medium are derived by means of Hankel transforms and contour integration. Selected numerical results are presented to demonstrate the influence of non‐homogeneity. Finally, some effects are pointed out to be used in connection with vibration tests for subsoil investigation.

Basic Problem on Optimal Spatial Cable Layout

This paper deals with the determination of the optimal profile and orientation of a cable carrying a vertical point load located in close proximity to a right angle corner. The cable volume is to take on a minimum value. In China, it has been the traditional practice to either design the cable such that the load is found at the mid-span or such that the cable span is shortest. In view of checking the correctness of these intuitive solutions, the necessary optimality conditions are derived for determining the optimal solution.