Stresses from Loads over Rectangular Areas

Vitone, D. M. A.; Valsangkar, A. J.

doi:10.1061/(asce)0733-9410(1986)112:10(961)

Cited by 17 publications

(17 citation statements)

References 3 publications

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“…Therefore, one equation will be sufficient for both of these loading regimes. The vertical stress solution of the loading case shown in Figure 7(a) was developed by Gray [15] and the vertical stress under the specific linear load configuration illustrated in Figure 7(c) developed by Vitone and Valsangkar [14] by the superposition of Gray's equation [15]. Same superposition technique can now be adopted by summing Equations (12) and (13) for the settlement under corner a of the specific linear load configuration demonstrated in Figure 7(c).…”

Section: Derivationmentioning

confidence: 99%

“…If it is required, the stiffness and depth corrections can also be introduced to the presented equations in the usual way [20]. The presented general solutions can be used for the settlement under the corners of all common types of linearly distributed loading regimes including the loading cases adopted to obtain the vertical stresses by Steinbrenner [19], Newmark [10], Vitone and Valsangkar [14], Gray [15] and Algin [17,18]. Furthermore, the presented solutions can also be used for estimating the rotation and the deferential elastic settlement of the eccentrically loaded foundations.…”

Section: Introductionmentioning

confidence: 99%

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Elastic settlements under linear surface pressures on rectangular areas

Alğin¹,

Alğın²

2008

Num Anal Meth Geomechanics

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SUMMARYThis paper presents the practical closed-form solutions for elastic settlement under the linear full-contact pressures on rectangular areas resting upon an elastic mass. The linear pressure distribution is mostly assumed in practice for the rectangular footing having biaxial bending. The equations presented here are determined by evaluating the integration of Steinbrenner's strain approximation based on the stress solutions from the Boussinesq equations. The presented formulae are validated to be used for the elastic settlement under any point of linear full-contact loading, triangular and embankment types of pressure cases. In this respect, it represents a significant step forward in the understanding of elastic settlement and rotation under the practical loading cases. The presented solutions are concise and easy to use. The numerical examples are demonstrated and the resulting influence factors in graphical forms are presented for their practical use especially in the elastic settlement calculations of rectangular footing subjected to the biaxial bending.

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Section: Derivationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Elastic settlements under linear surface pressures on rectangular areas

Alğin¹,

Alğın²

2008

Num Anal Meth Geomechanics

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“…This has been conducted on a limited number of load cases [1,2,5,7,8]. For example, the stress distribution under a uniform circular load was presented by Jurgenson [8] and an arithmetical equation of total vertical stress at a depth below a corner of a uniformly loaded rectangular area has been described by Newmark [7] (see Figure 2).…”

Section: Introductionmentioning

confidence: 99%

“…Figure 3. Vitone and Valsangkar's special load con"guration [1] which was developed by superimposing the two triangular pressure regimes formulated by Gray [5]. Occasionally, the foundation may be loaded with triangular or other types of non-uniform surface load distributions.…”

Section: Introductionmentioning

confidence: 99%

“…Occasionally, the foundation may be loaded with triangular or other types of non-uniform surface load distributions. Vitone and Valsangkar [1] presented an arithmetical solution for a linear loading case by applying the method of superposition to the speci"c triangular load con"guration which was formulated by Gray [5]. The solution of the linear pressure con"guration illustrated in Figure 3 is currently used for the case of an optimally designed rectangular footing subjected to biaxial bending.…”

Section: Introductionmentioning

confidence: 99%

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Stresses from linearly distributed pressures over rectangular areas

Alğin

2000

Int. J. Numer. Anal. Meth. Geomech.

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This paper presents a general algebraical formula to determine the vertical stresses resulting from a linearly distributed surface pressure resting on an elastic medium. This problem and its various derivations has received considerable attention because of its applications in foundation engineering [1}5]. The equation presented in this paper has been determined by the integration of the Boussinesq [6] equation with respect to the general surface function of a linearly distributed loading regime. The equation can be applied to the case of an optimally designed rectangular footing subjected to biaxial bending. Unlike the current design formulae [1}7], it provides the #exibility of having various contact pressures at each of the four corners of the rectangular area. This may enable more accurate design procedures for pad foundations in the future. This paper explains the validity of the new formula by comparing its results with those produced by classical methods. It shows how it can be applied to determine the vertical stress distributions resulting from all types of linearly distributed surface pressures which occupy a rectangular area on an elastic mass.

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Vertical stress distribution in isotropic half-spaces due to surface vertical loadings acting over polygonal domains

D’Urso

Marmo

2013

Z. Angew. Math. Mech.

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By integrating the classical Boussinesq expression we derive analytically the vertical stress distribution induced by pressures distributed with arbitrary laws, up to the third order, over polygonal domains. Thus, one can evaluate in closed form either the vertical stress produced by shell elements, modelling raft foundations by finite elements, acting over a Winkler soil or those induced by a linear pressure distribution simulating axial force and biaxial bending moments over a pad foundation. To this end we include charts and tables, both for rectangular and circular domains, which allow the designer to evaluate the vertical stresses induced by linear load distributions by hand calculations. The effectiveness of the proposed approach is witnessed by the comparison between the analytical results obtained with the proposed formulas and the numerical ones of a FEM discretization of the soil associated with the loading distribution induced by a foundation modeled by plate elements resting on a Winkler soil.The classical approach to finding stress and displacements in an elastic, homogeneous and isotropic half-space due to surface tractions has been first developed by Boussinesq [4] who provided the solution for a point load by making use of the potential theory [3].The case of uniform pressure applied within a circular domain was addressed by Lamb [19] while Love [20] considered the case of isotropic uniform pressure applied within a rectangular region.However, even in many natural soils deposited through a geologic process of sedimentation over a period of time stiffness becomes greater with depth. Analytical continuum models reflecting this property were first studied by Gibson [16] by assuming incompressibility and a linear increase of soil stiffness with depth. In later papers Gibson further refined his model and presented on account of his results in a Rankine lecture [17].Subsequently Vrettos [30] solved the Boussinesq problem for an half-space exhibiting an exponential behavior of stiffness yet with a bounded value at infinite depth. Half-spaces of this kind were later studied by Selvadurai in the framework of contact static problems and a survey account of his result were presented in [24].Since the pioneering work by Gibson an abundant literature has been published on the evaluation of stresses and displacements in inhomogeneous and/or anisotropic half-spaces, see e.g. [31] for a fairly complete account.Nevertheless, Boussinesq solution is still widely used mainly for its elegance and conciseness, for instance to evaluate the order of magnitude of the depth of influence of a foundation load [7] and the related settlement. Actually, it is usually assumed that the variation of vertical stress with depth below a loaded region on the surface can be predicted with sufficient accuracy using linear, homogeneous, isotropic elastic theory.Further applications of this theory range, to quote only a few, from space geodesy to contact mechanics, rock mechanics and geomechanics.For instance, in the former case position...

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Stresses from Loads over Rectangular Areas

Cited by 17 publications

References 3 publications

Elastic settlements under linear surface pressures on rectangular areas

Elastic settlements under linear surface pressures on rectangular areas

Stresses from linearly distributed pressures over rectangular areas

Vertical stress distribution in isotropic half-spaces due to surface vertical loadings acting over polygonal domains

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