Westergaard equation derivation. 016 Hydrodynamics Reading #6 version 3.
Westergaard equation derivation 21a and 11. The vertical stress can be calculated as: Westergaard’s solution is developed in this chapter for uniform biaxial stress \(\sigma \), with a central fracture of length 2a. 2011. For simplicity this can be modeled as In the derivation of these equations, Westergaard employed grossly restrictive assumptions, including infinite slab size, linear temperature distribution through the slab thickness, and Westergaard's assumptions are more close to the field reality, especially for over-consolidated and laminated sedimentary or stratified soils, which exhibit marked anisotropy. This pressure deformation characteristics of rigid pavement lead Westergaard to the define the term radius of relative stiffness l in cm is given by the equation 1. For vertical stress computation, he had proposed a formula in 1938. In this video, we'll cover:An introduction to Boussinesq's Theory and its historical context. . The Westergaard approach is derived for a crack that propagates in Mode I, therefore τ xy =0 along y =0. 2 Wheelload stresses- Westergaard’sstressequation Equations (16)-(20) correspond to Westergaard's equations (4)-(6) and (9)-(10) [1] if A is taken as zero. Stress calculation is based Introduction The use of Airy Stress Functions is a powerful technique for solving 2-D equilibrium problems. 8 m (i. Int. The Fenske’s equation: derivation Multicomponent distillation column design The method is applied to the light key (LK) and to the heavy key (HK) components, under column’s condition of total reflux. Author links open overlay panel G. In cases of relatively simple geometry use can be made of analytical methods, but in view of the complexity of the boundary conditions, numerical solution of the equations is soon necessary. 11 Very Near-tip Displacement Field (Mode I) 43 Generalised Westergaard equations 62 . Fract. Section 6 Superelevation Derivation As said earlier, superelevation is the transverse slope along the width of the road, provided to develop centripetal force to counteract the centrifugal force. Mode I Recall that in 1939, Westergaard found an Airy stress function, \(\phi\), that is the solution for the stress field in an infinite plate containing a crack [1]. Mohr’s circle problem. IRC 58 [16,17] adopted modified Westergaard’s equation’s for the estimation of critical edge stresses. by Pratik. (i) Vertical stress distribution on a Section 8. Q. 2) The formulas take into account load magnitude, [Use Westergaard equation] Q. 1 . Here it is shown that the superposition of the Westergaard functions with a uniform uniaxial stress field provides a complete solution to the equations of two-dimensional linear elasticity for Notation • E = modulus of elasticity • Cx and Cy = Bradbury's coefficients • T = temperature • a = radius of contact • h = thickness of cement concrete slab • k = modulus of subgrade reaction • l = raduis of relative stiffness • t = temperature differential • αt = coefficient of thermal expansion • ε= strain • µ= Poisson's ratio • σ= stress The Westergaard expressions for stresses and displacements in a halfspace of Westergaard material are shown to be invalid because the vertical shear stress does not vanish at the plane boundary of Love, Westergaard, Sokolniko•, Lur e, and Timoshenko and Goodier [2–4,7–9]. Tables show how the stress increase varies with load type, depth below the surface, and distance Boussinesq (1842 – 1929) evolved equations that can be used to determine stresses at any point P at a depth z as a result of a surface point load. 11) H M Westergaard ^ In this study he j analyze s the stres situation resultmg from loads appUed at a comer, at the center of a slab, and at an unsupported edge at some considerable distance from a comer Only the solution of the comer situation will be reviewed here In this solution, Dr. 22. 6: The multi-lateral junction of an oil well is under analysis prior to junction kick-off. (ii) to use the condition of equilibrium of the internal vertical normal stresses and vertical point load to determine the Differential equations govern the bending and deflection of roads under a concentrated load. , species A and B) can be formulated as follows AA R BBRR yx yx a æö æö çççç÷÷÷÷= çççç÷÷÷÷ èø èø (6. Alan Arnold Griffith's energy-based analysis of cracks in 1920 is considered to be the birth of the field of fracture mechanics [1]. 0 234 πρa m ⎛ 2 66 =ρ⎜a −b 2 ⎞ ⎟ ⎠ 2 ⎝ Two dimensional added mass coefficients for a circle, ellipse, and square in 1: Surge, 2: Sway, 6: Yaw Using these coefficients and those tabulated in Newman’s Marine Hydrodynamics on The Lode angle is part of a cylindrical coordinate system (the Haigh–Westergaard coordinates) with height (hydrostatic axis) and radius . 3. Westergaard developed a solution in 1939 for the stress field surrounding a crack that has two advantages over Inglis's earlier solution. If the equation of the elas- method," its derivation, and its use in finding deflections of 2. " These solutions are only available for three particular loading conditions (interior, edge, and corner) and assume a slab of infinite or semi equation by using three-dimensional fmite elements and Shentu et al developed a fmite element model in 1997. ri (10) the load is approximately -0. Westergaard assumed that an elastic soil medium is laterally reinforced by numerous, closely spaced, horizontal sheets of negligible thickness but of infinite rigidity. (115) The equilibtium equations (114) are a copy of Appendix I of a paper by Paris and Sih and a section from Anderson outlining the crack tip stress analysis of Westergaard. This is more than just coincidence, and it leads to the topic of Cauchy-Riemann equations, which are several important relationships between the derivatives of complex functions. Westergaard. 7 %âãÏÓ 306 0 obj > endobj xref 306 39 0000000016 00000 n 0000002397 00000 n 0000002517 00000 n 0000002646 00000 n 0000002973 00000 n 0000005265 00000 n 0000005428 00000 n 0000005588 00000 n 0000005742 00000 n 0000005901 00000 n 0000006057 00000 n 0000006188 00000 n 0000006342 00000 n 0000006472 00000 n . com is for sale. The formula is presented below. moment of the plate. 016 Hydrodynamics Reading #6 version 3. Since their first appearance, beginning in the early 1920s, Westergaard equations have often been misquoted or misapplied in subsequent publications. Then, a generalized worked example of plate rolling was analyzed by the criterion and The Westergaard equations (1938) are thought to provide better results for layered media, however the solution essentially smears out the effect of the different layers so that large errors will still occur for a finite number of layers with high stiffness contrasts. The Download Citation | On Dec 3, 2023, Charles Ike published Solving Westergaard Half-Space Problems Using Potential Theory | Find, read and cite all the research you need on ResearchGate the classical Westergaard added mass formulation and the more advanced analytical FEM-BEM technique. 28) it is important to note that the derivation of both the stress intensity factor and the strain energy release rate is independent of the This pressure deformation characteristics of rigid pavement lead Westergaard to the define the term radius of relative stiffness in cm is given by the equation . A proposed rectangular geometry for natural fractures could derive in a mathematical problem due to rectangle vertex. Based on this criterion of no lateral displacement, Westergaard derived the following equation for a point load, Q, at a depth z from the surface: Westergaard's basic formulation for a vertical wall assumes that the acceleration is oriented normal to the wall. Infinite Layer - Ridgid Rectangle. Westergaard's solution It is shown that the modified Westergaard equation describes the normal stress distribution and the singular stress state ahead of the crack tip in a reasonably accurate manner. 27) and (2. Two columns A and B are situated 5 m apart. Westergard’s formula: Example. These sheets prevent the mass as a whole from experiencing any lateral strains. 3970/CMES. Crit-ical stress calculation is presented in charts (Appendix-I of IRC 58 [16]) for single and tandem axle load with different magnitudes in the edge region. (1) where E is the modulus of elasticity of cement concrete in kg/cm 2 (3. The solutions of the Westergaard correction formula are quite consistent with the hydrodynamic pressures noted in previous reports. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture. 2174 ')(TT] The derivation of equations (4), (6) and (11) is described in the last section of this paper. Solve by Equations . INTRODECTION. Equations for the stresses and strains induced in a homogeneous, isotropic, weightless, linearly elastic half space, with a plane horizontal surface, by a point load perpendicular to the surface and acting at the surface, was first solved in usable form by Boussinesq (1885). 1b depicts the elastic stresses ahead of the crack tip that must be derived in Cartesian coordinates using theory of elasticity. The Westergaard approach permits to easily treat and analyze these kinds of crack problems [70, 71]. [22J and the normal stresses, are both equal to ReZ. Westergaard solution Stress Intensity Factors Energy release rate Mixed mode fracture J and interaction integrals 3 Cohesive zone models Institute of Structural Engineering Method of Finite Elements II 2. Westergaard (1938) proposed vertical stress equation within soil due to a point load by assuming zero Poisson’s ratio value of the soil to prevent any lateral strain and allowing only vertical movement of the soil. Essentially the same result that Westergaard obtained by means of a series solution of a partial differential equation was deduced by Karman using a remarkably simple momentum-balance method. 1 Kinematic equations The kinematic equations for small displacement linear elasticity are six Periodic waves in the Boussinesq approximation, shown in a vertical cross section in the wave propagation direction. Hence, only Westergaard’s Model of Subgrade Reaction Elastic layered theory can’t be applied for jointed rigid pavements due to the fact that one of the assumptions of the layered theory was that layers are infinitely long in horizontal direction ( no effect of discontinuity at joints). Due to various conspicuous features in the results, works of Muskhelishvili [] and Westergaard [] and their transfer to contemporary literature are Introduction The use of Airy Stress Functions is a powerful technique for solving 2-D equilibrium problems. The Westergaard method is applied to a certain class of plane problems in linear elasticity and Derivation of complete crack-tip stress expansions from Westergaard–Sanford solutions. This project solves the underlying differential equation in Stress Distribution in Soils l Boussinesq's Equation for uniformly loaded in circular area | CivilBoussinesq's theory | stress#Boussinesqstheory #Boussinesqs In the Westergaard analysis of stresses in concrete road slabs the radius of load distribution "a" is an important factor Since the computed stress in the slab may have a very appreciable range of variation depending upon the value of "a" it is necessary that this factor for various wheel loads as affected by different types and ar Principal stress: Definition, Equation, Example, Explained. At point P of above figure due to a point load Q, Bousinesq vertical stress and Westergaard vertical stress decreased as the depths moves further away from the point load. There are a number of simplifications and limitations inherent in this method as discussed Westergaard’s 1939 paper presented a simple way to express the stresses and displacements around mode I fractures (later expanded to cover mode II fractures by Sih (1966)). > 1) distillation with the total reflux as shown Figure 6. Displacement field – summary (Westergaard function) 42 . The edge stress at the midspan of the A simple example is presented to illustrate the Westergaard's added mass implementation in PLAXIS to model the effect of hydrodynamic pressure on a dam (with fixed displacements in the vertical direction). The assumptions and limitations of the theory. Learning goals Reviewing basic elasticity equations and solution methods Understanding some basic concepts of Linear Elastic Fracture Mechanics The inertial masses are calculated using Westergaard’s equation of inertial masses at each node on the upstream face of the dam. It is shown that the Westergaard’s values are lower than the experimental ones under the sinusoidal Equation (12) is one of only a few Westergaard functions previously known for displacement- specified crack problems [2]. The topic is central to Westergaard's solution for cracks. 9 vectors defined in equations 3. The equation and its graph are the key results of Westergaard's solution that are shared by most authors by Westergaard’s equation are compared with the experi-mental values under different waves. Identifying critical parameters, such as the maximum deflection and maximum bending moments of a street supported by an elastic subgrade, is key to designing safe and reliable roadways. The scheme for such a column is here below sketched. For this example, the horizontal displacement at the top of the dam at the upstream face was monitored. Recall that Westergaard used complex numbers and Airy stress functions to do so. the difference between crest and trough elevation), and the mean water depth is 5 m, Finally, note that \(dZ \over dz\) is equal to \(\partial Z \over \partial x\). Approximate methods (point load method, two-to-one load distribution method). b = 8 <: p 1:6a2 +h2 0:675 h if a < 1:724 h a otherwise (15. Notice the flat troughs and sharp crests, due to the wave nonlinearity. On the dodecan of the projection area, Tresca locus, twin shear stress yield locus, and Mises circle are the triangle (JB'F, the triangle (m'B. The radius of relative stiffness for the Westergaard equations assuming a liquid Winkler foundation can be calculated using the below equation where E is the Young’s Modulus of the slab, h is the thickness of the slab, v is the Poisson’s On this basis, Westergaard equation was modified taking into account the influence of dam height, elasticity and reservoir bottom condition. Techet Circle Ellipse Square 2 mm d11 22==ρ 2 mb11 =ρπ 2 mm a11 22==. g. WESTERGAARD—AIRPORT RVNWAY STRESSES 199 is found to be approximately equivalent to the condition <r2<0. Rectangle Corner. This The Lode coordinates are most easily computed using the mechanics invariants. It is achieved by raising the outer edge with respect to the inner edge in a transverse direction for the total length of the curved section. Appendix B Westergaard Approach The three crack modes described in Section 4. To remedy this situation, a Westergaard, by deriving closed-form equations for the maximum deflection and bending stress occurring on the unloaded side of an edge capable of load transfer. (12) By substituting equation. Proof – Displacement field derivation 38 - Proof , 1 . The correction factor C x depends on L x /e and the correction factor C y depends on L Equation 9 gives the maximum interior stress at the center of a slab. Westergaard as sumes the subgrade reactions to be verti generalized to an abstract yield criterion in the Haigh-Westergaard coordinates. 016 Hydrodynamics Reading #6 Circle Ellipse Square m 2 2 2 11 =m22 =ρd m11 =ρπb m11 =m22 =151 . believe that the Westergaard equations give a better estimate for the stress. A. In view of the wide Using Westergaard's analysis, Bradbury (1938) developed a simple chart for determining C x and C y, as shown in Figure. It can lead to a more correct and/or simpler solution scheme in comparison with the current technique. Westergaard’s formula for point load: It was proposed by Westergaard in 1938. 1) A cylinder of soil fails under axial vertical stress of 100 kN/m². . πρa m 2 4 66 =0 m 22 =ρπa m66 =. 4. Here it is shown that the superposition of the Westergaard functions with a uniform uniaxiat stress field Westergaard found an Airy stress function of complex numbers that is the solution for the stress field in an infinite plate containing a crack. 151πρ m66 =0 2 ma22 =ρπ 4 ma66 =. Parametric analyses of gravity dams with extra features: (a) exact and approximate Westergaard versus FE (reprinted from Salamon and Manie 2017 with permission); (b) USSD 2018 “blind prediction” of hydrodynamic pressure (reprinted from USSD 2018 with permission); (c) impact of the numerical procedure (reprinted from Calayir et al. Buy it today! One of such method is due to Westergaard, who introduced the following stress function, $$ \Phi = \text{Re} \overline{\overline{Z}} + y ~\text{Im} \overline{Z} $$ Solution Equations (2. The failure plane makes an angle of 50° with horizontal. The modified Westergaard formula solutions are quite #CivilEngineering #CivilEngg #Engineers Westergaards Equation | Soil Mechanics Fracture 1977, Volume 3, ICF4, Waterloo, Canada, June 19 - 24, 1977 THE MODIFIED WESTERGAARD EQUATIONS* R. This case (drawn on scale) shows a wave with the wavelength equal to 39. and the Again in 1933, Westergaard [12] published a paper on the stress analysis at the crack tip in reinforced concrete members in bending. Among these different stress functions the Westergaard stress function has become most popular because it can the point-load equations will be discussed subsequently. " The Boussinesq equations are named after the French scientist J. First, Westergaard's solution directly applies to cracks rather than an ellipse approximating a crack. 3, the shear and moment results are substantially different. 15. Added mass is a common issue because the object and surrounding fluid cannot occupy the same physical space simultaneously. 2) where a is the radius of the wheel load distribution in cm and h is the slab thickness in cm. Using this condition and employing In 1939 Westergaard[1] introduced a complex variable formulation for the plane elastic crack problem which, although limited in scope, provided a convenient method for solving certain types of crack problems that had previously not been amenable to solution. The ratio of vertical stress at a point 10 m below the ground surface at points vertically below B to that of A is ? (Use Boussinesq equations) The effect of the Westergaard assumption is to reduce the stresses substantially below those obtained by the Boussinesq equations. It's just different strokes for different folks. In the preceding example, we discussed the case where the position of the point of separation of the crack surface and the wedge is definite and known beforehand. de Wit** INTRODUCTION Sih [1] and Eftis and Liebowitz [2] have pointed out that the Westergaard method, which applies to a certain class of plane problems in linear elasticity and is most frequently used in fracture mechanics [3], 1 Introduction. (1) where E is the modulus of elasticity of cement concrete in kg/cm (3. You would need to look into the derivation of the equations and usually the equations are simply derived differently. function (i. , 8 (1972), pp. England, 1971. In that contribution, which appears to be one of the first attempts to apply fracture mechanics concepts to quasi-brittle materials, Westergaard modeled the cracked beam as two completely disconnected half-beams the derivation of stress intensity factors, and a variety of methods have been developed to approach the problem. 1 have symmetric and antisymmetric properties. (356 mm) on centers, as shown below, instead of over a 6-in. It presents the Boussinesq, Westergaard, and 2V:1H methods for calculating stress increases due to point loads, line loads, strip loads, and circular loads. Crossref View in Scopus Google Scholar. 16. 2 Framework of The Theory of Elastic Half-Space Problems Elastic half-space problems simultaneously satisfy the kinematic equations, generalized Hooke’s stress-strain relations, and the differential equations of equilibrium [4 -6], [57-59]. The method presented, which requires no more than the evaluation of integrals, significantly simplifies the analysis. Finally, we apply Irwin's near crack tip approximation to the displacement equations to obtain expressions in terms of the stress intensity factor, \(K same approach as Skarlatos and Westergaard, closed-form equations are derived for the maximum deflection and maximum bending stress occurring on the unloaded side of an edge capable of load transfer. The reader should denote the meaning of each subscript assigned to each elastic stress intensity factor. 0 10 ), is the Poisson's ratio of concrete (0. On the modified westergaard equations for certain plane crack problems. This chapter discusses Westergaard equations and some modifications relating to it. One of the In the context of linear fracture mechanics, a specific complex formulation has been initially proposed by Westergaard (Westergaard, 1939) and subsequently improved by different The Westergaard method yields a succinct solution to linear elastic crack problems. 2 Theoretical analysis approaches . These equations, valid for general fluid motion, turn into the more manageable linear wave equation, namely D’Alembert’s equation 𝛻2𝑝− 1 𝑐2 ⋅ 𝜕2𝑝 𝜕 2 =0 (1) where p is the pressure and c is the sound speed in the medium equal to √𝑘⁄𝜌, where k Westergaard stress functions for displacement-prescribed crack problems - I H. 0 234πρ 222 mab66 ρ =− Two dimensional added mass coefficients for a circle, ellipse, and square in 1: Surge, 2: Sway, 6: Yaw Using these coefficients and those tabulated in Newman’s On the basis of the analysis of the hexagons inscribed and circumscribed of the Mises yield circle on the π-plane in Haigh Westergaard stress space, a new equation called equal area linear yield criterion or EA criterion was proposed, and its plastic specific work rate was derived. Case 2: Dowel bars are present at contraction joint. Westergaard’s gives a relation for equivalent radius of the resisting section in cm in the equation 15. Polar coordinates will be used for this purpose in a later chapter. The following statements provide the basis for The equation derivation that follows assumes that the load is distributed over a Westergaard is a British scientist. However, several concrete dams have been constructed with inclined faces upstream (or downstream). 383-392. At y = 0 the shearing stress o, according to Equations [4] and , It is noted, furthermore, that the Airy function defining the stresses by the equations Accordingly Equations [23] PLAXIS 2D/3D latest CONNECT Edition (V20 Update 1) introduced the newly implemented Westergaard's added-mass feature to deal with the hydrodynamic effects induced by the impounded water on the response of a dam subjected to earthquake excitation. This feature is implemented as fully dynamic behaviour, not as a pseudo-static calculation enhancement. In parallel with various methods proposed by researchers, ranging from finite element modelling to boundary integral equations, some simplified formulations were derived for the need of practicing designers. 16 William’s Eigen Function Approach 65 Irwin arrived at the definition of \(K\) as a near-crack-tip approximation to Westergaard's complete solution for the stress field surrounding a crack . The ratio of load transferred by column A to column B is 2. The complete set of equations for the stress field is \[ \begin{eqnarray} \sigma_{xx} & = & Westergaard formulation and the Goursatt-Kolosoff formulation are equivalent and that one can be obtained from the other by suitable re-definition of the corresponding complex functions[8]. While these approaches represent remarkably insightful procedures for obtaining a solution to Boussinesq’s problem, there is the question as to whether there is a more direct approach via which Bous-sinesq’s solution can be obtained. In particular, within the plastic correction, we formulate a unique system of nonlinear equations which is common for the both type of the return. Research was made for concrete arch dam with structural height of H=130m The elastic stress field around a crack tip is fully defined through multiparameter equations such as the general relations of Westergaard proposed by Sanford or the most recent of Atluri and Comparing the USACE / Westergaard ("Army Corp") equations for seismic hydrodynamic loading on walls vs ACI 350. Show more. 1 Solid Mechanics Part II Kelly 242 contrast to classical Newtonian fluids for example, where the stress levels are governed by the rate of deformation through the viscosity of the fluid. This was introduced to account for the effect of shear stresses in the vicinity of the load, which was neglected in the "ordinary theory" of medium-thick plates. A yield surface is a surface in the 3D space of principal stresses Determination of Vertical stress using Westargaard formula The basic understanding of the derivation of stress intensity factor is very important if meaningful results are to be achieved. Theoretically, the stress ahead of a sharp crack tip becomes infinite and cannot be On the other hand, Fig. 6 and 3. The results show an agreement Following a brief presentation of essential fundamentals in Sect. Every code of practice published since then makes reference to the "Westergaard solutions. Vertical Stress Caused by a Vertical Strip Load (Finite width and infinite length) Such conditions are found for structures extended very much in one direction, such as strip and wall foundations, foundations of retaining walls, embankments, dams and the like. The Westergaard formulation was modified in 1966 by Sih[3] and later by Eftis and Liebowitz[4] to correct an obvious shortcoming of Westergaard's original formulation. It involves calculating the stresses in the pavement slab under wheel loads and comparing them to the slab's strength to determine if it is sufficiently strong. 21b and the Mohr-Coulomb failure model, follow the equations step by step and derive the critical collapse pressure equation a given by Equation 11. TADA, 1 H. Westergaard's assumptions are more close to the field reality, especially for over In fluid mechanics, added mass or virtual mass is the inertia added to a system because an accelerating or decelerating body must move (or deflect) some volume of surrounding fluid as it moves through it. We also provide a detailed derivation of consistent tangent operators, in order to simplify the implementation of the scheme, and an illustrative numerical example. 0 updated 8/30/2005-4- ©2005 A. The Westergaard method is an early analytical approach for designing rigid pavements that is still commonly used today. The functions described in Equation 3-6 and Equation 3-7 enter into expressions that define various kind of yield and failure surfaces. For the safe design of the component, it is necessary to know about the maximum stresses acting inside the component and its location. Boussinesq considered the soil medium as homogenous, but in Westergaard's theory, fine sheets of hard material are used, which are stuffed in a homogeneous soil medium. Herein the influence of the applied techniques on the distribution and intensities of the manifested stresses is equation of dynamic motion of dam, ADAD-IZIIS (2008). It is possible to calculate the following pressure distributions by equation (1) of Boussinesq and present them graphically. Finite Layer - Ridgid Circle "Good engineers don't need to remember every formula; they just need to know where they can find them. M. Westergaard (1933) was the first to derive an expression for the hydrodynamic pressure that an Closed-Form Equations of Hydrodynamic Pressure of Resevoir Water According to Italian Ministerial Decree of 1982 In this paper, the impacts of boundary Thus, the Westergaard formula was revised with consideration to the influence of the height of dam, the elasticity of dam, and the absorption characteristic of the reservoir bottom on the hydrodynamic pressure. When it is laterally unconfined. [121 i2zy) + iz—y . NUMERICAL EXAMPLE To illustrate the use of the formulas the possibility of using a 6-in Westergaard’s Theory Assumptions: Elastic medium of semi-infinite extent but containing numerous, closely spaced horizontal sheets of a negligible thickness of an infinite rigid material which permits only downward deformation as a whole without allowing it to undergo any lateral strain. Share. Vertical pressure within an earth mass Boussinesq's Formula Equation. 5. J. A copy of his paper can be found here. Infinite Layer - Ridgid Circle. 1927) its principles might be explained. Hello. A generalized method to determine the stress intensity factor equations for cracks in finite-width specimens of functionally graded materials (FGMs), based on force balance in regions ahead of the crack tip is provided. 5-kN) load is applied on a set of duals spaced at 14 in. The method analyzes three loading conditions - interior, edge, and corner - with corner loading usually Hypothesis of Westergaard In order to understand the hypothesis of Westergaard (Westergaard, H. PARIS 1 Equations (11) and (1 la) are well-known functions for the Volterra [-16] dislocation or for an edge dislocation with the Burger's vector in the y-direction A. For the case where the Stokes equation. C. Figure 1: Fenske’s distillation column with total reflux Step 1 The pioneering analytical work of Harold Malcom Westergaard (1888-1950) has been at the heart of slab-on-grade pavement design since the 1920s. Second, it uses rectangular coordinates rather than elliptical coordinates, making the expressions easier to interpret. 11 and the expressions for f n and d n in equation 12, we obtain the following maximum n-th pressure at the fixed end of the pipe: (13) 6. 078. 1. Stress Distribution of Soils | Stress Distribution in Soils l Boussinesq's Equation for point load l Boussinesq's theory | stress#Boussinesqstheory #Boussine 2. Engineers and hydrologists working on groundwater dynamics often rely on hydraulic groundwater theory that simplifies the description of water flow through unconfined aquifers based on assumptions attributed to Dupuit and Forchheimer (Dupuit, 1863; Forchheimer, 1930). An Introductory Discussion of the Standard Methods of Finding Defec- method the equation of the elastic curve is determined; namely, by integration of an "equation of Rexure" of the form d2y/dx2 ,II/EI. a) Go through the Westergaard‟s solution. See Fig. Central Circle. Boussinesq who derived a version of the equations to find solutions for solitary waves on a water surface []. 1 Total Reflux: Fenske Equation For a multi-component (i. e. 2. Hence, only Equation (1) defines the relationship between the principal stresses at failure. 6. For instance, the stress components in a two-dimensional elasticity problem can be expressed as: The derivation of these expressions is based on closed-form expansions of complex exact solutions (either Kolosov-Muskelishvili or Westergaard-Sanford formulations) into complex power series and then polar power series expansions on the r variable. Westergaard's Equation for Point Loads. He was motivated by Inglis's linear elastic solution for stresses around an elliptical hole [2], which predicted that the stress level approached infinity as the ellipse flattened to form a crack. I) displacement of a dam water surface profile equations for dam height and elasticity in the Westergaard correc- tion formula, respectively ; and e = correlation coef fi cient of the elasticity modulus of the dam . The stress field caused by the constant A does not contain Irwin[2] used Westergaard's formulation to obtain the near-field equations, the tangent formula and other results of interest to early researchers. But before getting to the function, a little more notation is needed. 15), h is the slab thickness in cm and K is the The Westergaard method yields a succinct solution to linear elastic crack problems. Calculate the value of cohesion and the angle of internal friction of soil Using equation 3 the hydrodynamic pressure at the fixed end (x = L) of the n-th mode of vibration can be obtained in an analogous way. BOUSSINESQ THEORY Boussinesq (1885) developed a theory to calculate stresses and deformation due to point load applied at the surface of semi infinite soil mass. We start by expressing the stresses in terms of the biharmonic function F σ ∂ ∂ xx F y = 2 2 σ ∂ ∂ yy F x = 2 2 σ ∂ ∂∂ yx F xy = − 2 (16. Plane strain 39 . Derivation of Bous Civilengineering-softstudies. 109 Corpus ID: 56416695; Generalized Westergaard Stress Functions as Fundamental Solutions @article{Dumont2011GeneralizedWS, title={Generalized Westergaard Stress Functions as Fundamental Solutions}, author={Ney Augusto Dumont and Elvis Yuri Mamani}, journal={Cmes-computer Modeling in Engineering \& Sciences}, 3. Westergaard found an Airy stress function of complex numbers that is the solution for the stress field in an infinite plate containing a crack. Materials commonly known as “plastics” are not plastic in the sense described here. 5m. The method uses the Westergaard's stress distribution ahead of the crack in an infinite plate and is based on the requirement of isostrain deformation of layers of 1) The documents discuss formulas for calculating vertical stress due to point loads and line loads on the soil surface using Boussinesq's formula and Westergaard's formula. Contact pressure distribution due to loaded areas. These invariants are a mixture of the invariants of the Cauchy stress tensor, , and the stress deviator, , and are given by [3] = = [() ()] = = ‖ ‖ = = which can be written equivalently in Einstein notation = = [() ()] = = = = where is the Levi-Civita symbol (or permutation symbol) and the last two forms for are (Westergaard [4], von Karman, Bakhmeteff, Zanger [6], Housner [6], Kulmaci [7][8], Chopra. The general solution for The Westergaard analysis equations can be used to calculate the stresses produced in a rigid pavement under loading from a wheeled vehicle. Concept of active zone. 9 jth-mode generalized coordinate of dam boundary of arbitrarily shaped reservoir interior angle of pointed boundary I k2 _ (~}2 m c o definition defined in equation 2. 15), is the slab thickness in cm and is the modulus of sub-grade Hydrodynamic Pressure on Gravity Dams with Different Heights and the Westergaard Correction Formula The single stress function approach of Westergaard has been effective for a certain class of stress-prescribed crack problems. The equitibrium equations are: ∂x∂vy+∂y∂ruy=0∂s∂rnr+∂y∂vy=0rvp=rv⎭⎬⎫ equation: vr(σx+ϵ2)−(∂x2∂2+∂y2∂2)(cx+ϵ2)=0. 2. Occurs when a load is applied on Using Westergaard's formulas, determine the maximum stress in the above Examples if the 10,000-lb (44. The function helps in simplifying the complex equations of elasticity, converting them into more manageable forms. Soil mass is elastic continuum having Westergaard coordinates. ERNST 2 and P. Examples are gravity dams with inclined upstream faces, or arch dams with double curvature, or spillway pier noses. 1) i = ′ ( ) ( ) = • ′ ′ ′ (i) to use the Hankel transformation method to obtain bounded solutions to the Cauchy - Navier displacement equations of equilibrium for the Westergaard problem for the case of point load Q 0 acting at the origin of the semi-infinite elastic space. 0 × 10 5 ), μ is the Poisson’s ratio of concrete (0. Rotation of Rigid Footings. , its real and imaginary parts solve the Laplace equation) instead of partial derivatives of a biharmonic function. If Q is the point load and σ z is the vertical stress due to the point load, Equation 1 b employs Westergaard's "special theory" in which radius b replaces the true radius, a, of the loaded area. Since A is constant the Westergaard method for symmetric loading is complete with the exception of a constant uniaxial stress axx = 2A which acts parallel to the line of the crack. (152-mm) circular Using the Westergaard Stress Function, engineers and scientists can derive stress components in a material. 3 radius of small circle (Fig. The critical load locations are (after Bradbury, 1938 [1] and Westergaard, 1926 [2]): Interior loading. The influence factor for Boussinesq's theory is larger, whereas the influence factor for Westergaard's theory is lower. 7) Considering a dry cohesionless backfill, we have c = 0, hence – σ 1 = σ z = γh (in active case) and σ 3 = p a By H. 1 m, the wave height is 1. 11) This is accomplished by first inserting Westergaard's complex-valued expressions for stress into Hooke's Law to obtain strains. Solved example to calculate effective stresses and water pressure is given below: Faq near-tip fields, Westergaard, Williams and Generalised Westergaard solutions, Influence of the T-stress and higher order terms, Role of photoelasticity on the development of stress field equations in fracture mechanics, Equivalence between SIF and G, Various methods for evaluating Stress Intensity Factors, Modeling plastic zone at the crack-tip, Westergaard's assumptions are more close to the field reality, especially for over-consolidated and laminated sedimentary or stratified soils, which exhibit marked anisotropy. In the present study, the Westergaard approach was successfully extended to displacement-prescribed crack problems. Boussinesq’s Equation Derivation For Circular Area. • The equations for two cylinders in contact are also valid for: – Cylinder on a flat plate (a flat plate is a cylinder with an infinitely large radius) – Cylinder in a cylindrical groove (a cylindrical groove is a cylinder with a negative radius) x y z F F 2 b E 2, ν 2 E 1, ν 1 R 1 p max L Rectangular contact area with semi-elliptical %PDF-1. A useful observation from Equation (101 is that the value of q at y = 0 is ImŽ — 1mŽ. Keywords: elastoplasticity, small-strain, nonsmooth plastic potential, implicit return-mapping scheme, Westergaard Type Materials (Layered Soils) Point Load. In other words, as the values of radial and vertical distances increased, both values of Bousinesq vertical stress and Westergaard vertical stress decreased. In addition, a sound, engineering approach to numerical, experimental, and field data interprelation is proposed, founded on the principles of dimensional analysis. An approximate stress distribution assumes that the total applied load on the surface of the soil is distributed over an area of the same shape as the loaded area on the surface, but with dimensions that increase by an The purpose of this work is to review the derivation of the Boussinesq equation for vertical and lateral stresses in a soil body and to present several new, closed-form solutions for various surface load cases, including finite line and finite area loads. A. Assumptions: 1. 1996 with Concept of Westergaard’s Analysis. They are covered here because the approach was used by several researchers in the mid 1900's to develop analytical solutions to linear elastic problems involving cracks. The formulations are presented as Cartesian coordinate functions such that the stress at Fig. 1 Westergaard (1926) Westergaard formula assumes that slab acts as a homogenous, isotropic, elastic solid in equilibrium and that the reactions of the foundations are vertical and believe that the Westergaard equations give a better estimate for the stress. Westergaard stress functions 43 The original equations developed by Westergaard (1926 [2]) for three critical load locations will be presented. To obtain the interior stress in the case ofa square loaded area matrices defined in equation 3. The principal stress gives the The single stress function approach of Westergaard has been effective for a certain class of stress-prescribed crack problems. Figure 2: Graph of values of σ B and σ 1 Derivation and Representation of EA Criterion The projection of the principal stress components (Jj , (J2' and (JCj on the rr-pla ne in Haigh Westergaard stress space is shown in Fig. Later, Boussinesq equations have been presented in many different versions, and there is no strict consensus concerning the use of “Boussinesq equations” in relation to a single set of differenlial) to be applied to the Westergaard equation to deter mine the maximum combined tensile slre s in the lab under edge loading. 11. For all mode-I configurations here considered, only odd indexed coefficients are non zero. Lc = (2* σst Superelevation Concept and Formula Derivation with Practice The document discusses methods for analyzing vertical stress increases in soil due to different types of surface loading. Then the strain equations are integrated to obtain displacements. DOI: 10. Following these assumptions, Boussinesq posed a general equation for For more details about the derivation steps, see these pages: (i) Westergaard's Solution for Cracks, (ii) Stress Intensity Factor, and (iii) Crack Tip Displacements. 2, the complex potentials and stress functions provided by established literature as well as resulting stress and displacement fields are discussed in Sect. There is much confusion in the literature on fracture mechanics about the value of A, e. Add to Mendeley. Based on this criterion of no lateral displacement, Westergaard derived the following equation for a point load, Q, at a depth z from the surface: For solving crack problems analytically, several complex forms of Airy stress function have been proposed (Westergaard 1939; Muskhelishvili 1953; Paris and Sih 1965; Sih 1966, 1973; Rice 1969; Goodier 1969; Eftis and Liebowitz 1972). To solve this problem Substituting all the values in the above equation, we get, the length of the contraction joint as 4. K=a-y\/(2irr) (Irwin[19]) (10) K=o-^(2r) (Paris [31]). 1, the equation for vapour-liquid equilibrium (VLE) at the re-boiler for any 2 components (e. Alien [24] considers the Westergaard equation for the stress Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. The inertial masses are the partial representation of water during seismic motion and follow parabolic variation with depth for the whole time period of earthquake motion [ 10 ] and are calculated by the equation given: Since the classical work by Westergaard [1], the question has attracted a considerable amount of research attention. 5: Using Equations 11. In theory, accurate stresses can be calculated for any The Westergaard function Z for the concentrated forces P and Q applied at the point x = b (b < a) of a crack AB of length 2a in an infinite plate (Figure la) is 1 % It is known that the principal stresses are related to the shear parameters of the backfill material by the Bell’s equation as follows – σ 1 = σ 3 tan 2 α + 2c tan α (15.
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