The Elastic Sphere Under Concentrated Loads

1952 ◽  
Vol 19 (4) ◽  
pp. 413-421 ◽  
Author(s):  
E. Sternberg ◽  
F. Rosenthal
Keyword(s):  
Three Dimensional ◽  
Plane Boundary ◽  
Elastic Sphere ◽  
Concentrated Loads ◽  
Concentrated Force ◽  
Concentrated Load ◽  
Load Axis ◽  
Present Solution ◽  
Photoelastic Investigation ◽  
Axisymmetric Problems

Abstract This paper contains an exact solution for the stress distribution in an elastic sphere under two equal and opposite concentrated loads, applied at the end points of a diameter. The solution is based on the Boussinesq stress-function approach to axisymmetric problems and is represented as a sum of two solutions: A singular solution in closed form, and a series solution corresponding to surface tractions which are finite and continuous throughout the surface of the sphere. It is shown that the singularity at the points of application of the loads is not identical with that arising at a concentrated load acting normal to a plane boundary. The existence of pseudosolutions to the problem under consideration, as well as to concentrated-force problems in general, is illustrated, and the validity of the present solution is confirmed through a limit process applied to the case of tractions uniformly distributed over a portion of the boundary. The numerical results for the normal stress on the plane of symmetry perpendicular to the load axis, are compared with the corresponding stress values obtained in a recent three-dimensional photoelastic investigation of the same problem by Frocht and Guernsey (1).

Surface Vibration of a Multilayered Elastic Medium due to Harmonic Concentrated Force

2011 ◽  
Author(s):  
Akindeji Ojetola ◽  
Hamid Hamidzadeh
Keyword(s):  
Fourier Transform ◽  
Elastic Medium ◽  
Three Dimensional ◽  
Concentrated Force ◽  
Double Complex ◽  
Concentrated Load ◽  
Propagator Matrix ◽  
Displacement Vectors ◽  
Bottom Interface ◽  
Similar Information

Dynamic response of a multi-layer elastic medium subjected to harmonic surface concentrated load is considered. In development of the analytical solution, the three-dimensional theory of elasto-dynamic is utilized for derivation of the governing partial differential equations for each layer. These equations are solved in the Fourier domain by employing the Double Complex Fourier Transform technique. In the analysis, each layer of the medium is assumed to be extended infinitely in the horizontal x and z directions and has uniform depth in the y direction and is considered to be linearly elastic, homogeneous, and isotropic. Utilizing the Integral Fourier Transform, displacements and stresses at any point in each layer can be determined in terms of boundary stresses for each layer. Also, the presented solution provides the relation between stress and displacement vectors for the top and bottom of each layer in matrix notation. By satisfying the compatibility of displacements and stresses for each interface, a propagator matrix relating displacements and stresses at the top of the medium to the bottom interface will be obtained. This relates the displacement and stress vector on the top surface to the bottom interface by eliminating similar information for the other interfaces. In this study, the displacements on the surface of the layered medium are computed for the two cases where the surface of the medium is subjected to a concentrated harmonic vertical or horizontal harmonic force.

Design of face-shell bedded hollow masonry subject to concentrated loads

1996 ◽  
Vol 23 (1) ◽  
pp. 98-106 ◽  
Author(s):  
Ezzeldin Y. Sayed-Ahmed ◽  
Nigel G. Shrive
Keyword(s):  
Finite Element ◽  
Load Capacity ◽  
Three Dimensional ◽  
Element Model ◽  
Concentrated Loads ◽  
Concentrated Load ◽  
Strength Enhancement ◽  
Design Detail ◽  
Current Design ◽  
The Face

Many parameters affect the behaviour and failure of face-shell bedded hollow masonry subject to concentrated load. Detailed study of these parameters is needed to develop realistic design rules for this situation. The effects of loaded length and wall dimensions on capacity of the face-shell bedded hollow masonry subject to concentrated load are studied; the effect of mortar joint strength is also evaluated. The current design detail of filling some of the blocks under the concentrated load with grout is reviewed. The study was performed with a nonlinear elastoplastic finite element model that takes into account geometric and material nonlinearities as well as damage due to progressive cracking. The methodology, when combined with substructuring, allows analysis of substantially larger walls than would more typical three-dimensional analyses. The results indicate that the length of the loading plate is the significant parameter for load capacity. A possible design equation for plain hollow masonry subject to concentrated loads, concentric across the width of the wall, is provided. Adjustments could be made given the precise loading detail specified. Improvement details are explained. Key words: masonry, hollow concrete masonry, finite element modelling, cracking, failure, strength-enhancement factor, concentrated loads.

Bending of an Infinite Beam on an Elastic Foundation

1937 ◽  
Vol 4 (1) ◽  
pp. A1-A7 ◽  
Author(s):  
M. A. Biot
Keyword(s):  
Elastic Foundation ◽  
Poisson Ratio ◽  
Elementary Theory ◽  
Three Dimensional ◽  
Bending Moment ◽  
Two Dimensional ◽  
Concentrated Load ◽  
Elastic Continuum ◽  
Infinite Beam ◽  
A Value

Abstract The elementary theory of the bending of a beam on an elastic foundation is based on the assumption that the beam is resting on a continuously distributed set of springs the stiffness of which is defined by a “modulus of the foundation” k. Very seldom, however, does it happen that the foundation is actually constituted this way. Generally, the foundation is an elastic continuum characterized by two elastic constants, a modulus of elasticity E, and a Poisson ratio ν. The problem of the bending of a beam resting on such a foundation has been approached already by various authors. The author attempts to give in this paper a more exact solution of one aspect of this problem, i.e., the case of an infinite beam under a concentrated load. A notable difference exists between the results obtained from the assumptions of a two-dimensional foundation and of a three-dimensional foundation. Bending-moment and deflection curves for the two-dimensional case are shown in Figs. 4 and 5. A value of the modulus k is given for both cases by which the elementary theory can be used and leads to results which are fairly acceptable. These values depend on the stiffness of the beam and on the elasticity of the foundation.

scholarly journals ANALISIS PENGARUH BENTUK DAN LOKASI BUKAAN PADA BALOK TINGGI MENGGUNAKAN METODE ELEMEN HINGGA

2020 ◽  
Vol 3 (4) ◽  
pp. 1209
Author(s):  
Anthony Fariman ◽  
Leo S. Tedianto
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Wall Structure ◽  
Deep Beam ◽  
Stress Distributions ◽  
Stress Concentrations ◽  
Offshore Structure ◽  
Deep Beams ◽  
Concentrated Load ◽  
Cable Networks

ABSTRAKBalok tinggi beton bertulang merupakan salah satu struktur khusus yang dapat memikul beban cukup besar dan umumnya digunakan sebagai transfer girder, struktur lepas pantai, struktur dinding, dan pondasi. Kehadiran bukaan pada balok tinggi dapat memfasilitasi jalur saluran AC, saluran pipa, jaringan kabel dan lain-lain. Dengan adanya bukaan pada balok tinggi dapat memberikan beberapa efek samping yaitu terjadinya diskontinuitas geometri, tegangan terdistribusi non-linier pada balok tinggi, berkurangnya kekuatan dari balok, dan timbulnya konsentrasi tegangan di sekitar bukaan. Penelitian ini bertujuan untuk menganalisis efek dari kehadiran bukaan pada balok tinggi di atas dua perletakan (sendi-rol) dan dibebani beban terpusat di tengah bentang balok lalu memvariasikan bentuk bukaan (persegi, persegi panjang, dan lingkaran) dan lokasi bukaan. Tegangan lentur pada balok tinggi dan konsentrasi tegangan yang terjadi di sekitar bukaan merupakan hal yang akan dibahas dalam penelitian. Analisis akan dibantu dengan Midas FEA yang merupakan program berbasis elemen hingga dan  pemodelan dilakukan dengan elemen solid tiga dimensi. Hasil dari analisis ini menunjukkan bahwa kehadiran bukaan pada balok tinggi menyebabkan kenaikan tegangan secara signifikan. Lokasi dari bukaan yang mendekati daerah tengah bentang balok juga sangat mempengaruhi besarnya tegangan yang terjadi.ABSTRACTReinforced concrete deep beam is one of the special structures that can carry quite a big load and generally used as a transfer girder, offshore structure, wall structure, and foundation. The appearance of openings in deep beams can facilitate AC pipelines, plumbing pipes, cable networks, etc. The existence of openings in deep beams can provide a few side effects such as geometric discontinuity, non-linear stress distributions over the deep beams, reduced strength of the deep beams, and stresses concentration will emerged around the openings. The purpose of this research is to analyze the effects from the existence of openings in deep beams on two supports (hinge and roller) and loaded by concentrated load in mid-span then variate the shape of openings (square, rectangle, and circle) and location of the openings. Flexural stresses in deep beams and the stress concentrations that occur around the openings are discussed in this research. The analysis will be assisted by Midas FEA which is a finite element based program and modelling will be executed in three dimensional solid elements. The result of this analysis showed that the existence of the openings in deep beams can cause stresses to increase significantly high. The location of the openings close to the mid-span of the deep beams also affect the amount of the stresses that occurs.

scholarly journals Strengthening of Reinforced Concrete Beams Subjected to Concentrated Loads

2022 ◽  
Author(s):  
Paolo Foraboschi
Keyword(s):  
Reinforced Concrete ◽  
Carrying Capacity ◽  
Concrete Beams ◽  
Shear Capacity ◽  
Reinforced Concrete Beams ◽  
Load Carrying Capacity ◽  
Reinforced Concrete Structure ◽  
Concentrated Loads ◽  
Concentrated Load ◽  
Load Carrying

Renovation, restoration, remodeling, refurbishment, and retrofitting of build-ings often imply modifying the behavior of the structural system. Modification sometimes includes applying forces (i.e., concentrated loads) to beams that before were subjected to distributed loads only. For a reinforced concrete structure, the new condition causes a beam to bear a concentrated load with the crack pattern that was produced by the distributed loads that acted in the past. If the concentrated load is applied at or near the beam’s midspan, the new shear demand reaches the maximum around the midspan. But around the midspan, the cracks are vertical or quasi-vertical, and no inclined bar is present. So, the actual shear capacity around the midspan not only is low, but also can be substantially lower than the new demand. In order to bring the beam capacity up to the demand, fiber-reinforced-polymer composites can be used. This paper presents a design method to increase the concentrated load-carrying capacity of reinforced concrete beams whose load distribution has to be changed from distributed to concentrated, and an analytical model to pre-dict the concentrated load-carrying capacity of a beam in the strengthened state.

Three-dimensional direct numerical simulation of flow induced by an oscillating sphere close to a plane boundary

2021 ◽  
Vol 33 (9) ◽  
pp. 097106
Author(s):  
Samayam Satish ◽  
Justin S. Leontini ◽  
Richard Manasseh ◽  
S. A. Sannasiraj ◽  
V. Sundar
Keyword(s):  
Numerical Simulation ◽  
Direct Numerical Simulation ◽  
Three Dimensional ◽  
Plane Boundary ◽  
Oscillating Sphere

Design of Laterally Loaded Plating—Concentrated Loads

1983 ◽  
Vol 27 (04) ◽  
pp. 252-264
Author(s):  
Owen Hughes
Keyword(s):  
Experimental Data ◽  
Load Parameter ◽  
Uniform Pressure ◽  
Concentrated Loads ◽  
Rigid Plastic ◽  
Concentrated Load ◽  
Load Effect ◽  
Permanent Set ◽  
Multiple Location ◽  
Single Location

In the design of plating subject to lateral loading, the principal load effect to be considered is the amount of permanent set, that is, the maximum permanent deflection in the center of each panel of plating bounded by the stiffeners and the crossbeams. The present paper is complementary to a previous paper [1]2 which dealt with uniform pressure loads. It first shows that for design purposes there are two types of concentrated loads, depending on the number of different locations in which they can occur; single location or multiple location. The hypothesis is then made that for multiple-location loads the eventual and stationary pattern of plasticity which is developed in the plating is very similar to that for uniform pressure loads, and hence the value of permanent set may be obtained by using the same formula as for uniform pressure loads, with a load parameter Q that is some multiple r of the load parameter for the concentrated load: 0 = rQP. The value of r is a function of the degree of concentration of the load and is almost independent of plate slenderness and aspect ratio. The general mathematical character of this function is established from first principles and from an analysis of the permanent set caused by a multiple-location point load acting on a long plate. The results of this theoretical analysis provide good support for the hypothesis, as do also the relatively limited experimental data which are available. The theory and the experimental data are combined to obtain a simple mathematical expression for r. A more precise expression can be obtained after further experiments have been performed with more highly concentrated loads. Single-location loads produce a different pattern of plasticity and require a different approach. A suitable design formula is developed herein by performing regression analysis on the data from a set of experiments performed with such loads. Both methods presented herein, one for multiple-location loads and the other for single-location loads, are valid for small and moderate values of permanent set and can be used for all static and quasistatic loads. Dynamic loads and applications involving large amounts of permanent set require formulas based on rigid-plastic theory. Such formulas are available for uniform pressure loads and were quoted in reference [1]. A formula for single-location loads has recently been derived by Kling [4] and is quoted herein.

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