Bearing pressures and cracks, asme journal of applied mechanics, vol. 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 2. Westergaard complex stress functions 16 i main topics a. Plane analysis of finite multilayered media with multiple aligned crackspart ii. C westergaards stress function is given, not derived. This problem is also referred to as bearing resistance. Westergaard also introduced the concept of the modulus of subgrade reaction, k, an important value to know when calculating the deflection in soil due to loading on the surface. Westergaard hydrodynamic pressure on bulkhead wall. Harald malcolm westergaard wikipedia, a enciclopedia livre.
To remedy this situation, a reexamination of these solutions using the finite element method is described in this paper. 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. Westergaard solution of stress field for modei youtube. He was professor of theoretical and applied mechanics at the university of illinois in urbana and of civil engineering at harvard. Seismic induced loads on spillway gates phase i literature. Westergaard solutions was conducted, using the tool of finite element analysis tabatabaie and barenberg, 1980. Carthwrigth dj, rooke dp 1976 compendium of stress intensity factors, hmso, london, uk. Westergaard stress functions for several periodic crack. The crack opening displacement approach to fracture mechanics.
We will limit ourselves to mode i, and the summary of the procedure given below is taken from the book by gdoutos 2010. If the wall is stiff such that its natural frequency is high with respect to that of the fill, the soil and ocean water pressure will be out of phase, such that the fill and its pore water think in terms of undrained soil behavior because of cyclic loading with high. Westergaard 1 introduced an airy stress function of complex numbers to describe the stress field in an infinite plate with a crack. Analysis of irwin based on westergaardsanalysis and williams expansions planar symmetric. The westergaard method of plane elastic analysis is used to obtain an exact solution to the problem of an elastic roller crossing a gap, intended to represent a scratch, on a rigid halfspace. Westergaard also introduced the concept of the radius of relative stiffness, l, for ground supported slabs, which is determined in equation 20 chapter 7. Office of naval research contract nonr6lo i06 task order. Barenberg abstract the pioneering analytical work of harold malcom westergaard 18881950 has been at the heart of slabongrade pavement design since the 1920s. Westergaard, bearing pressure and cracks, journal of applied mechanics, vol. Stress solution at crack tip westergaard 1934 general stress solution the procedure for computing the stress state near a crack tip was established by westergaard 1934. Westergaard stress functions for several periodic crack problems. Water pressures on dams during earthquakes, asce 1933 general solution of the problem of elastostatics in an ndimensional homogeneous isotropic solid in an ndimensional space, 1935 bearing pressures and cracks, 1939 theory of elasticity and plasticity, 1952. It also provides an easy means of evaluating stress intensity factors, when particularly applied to fracture mechanics. If the soil class or bearing capacity cannot be determined by test or soil records, but its type can be identified, the soil classification, allowable pressures, and torque values shown in table to 3285.
Fracture mechanics, composite stress intensity factor, fracture angle, unstable rock journal name. Suleiman, shawqi mohammad, stress intensity factors for near surface cracks by boundary collocation 1986. Westergaard, bearing pressures and cracks, 1937, j. For loads applied skew symmetrically with respect to the crack line, say along the x axis, the normal stress o is required to vanish at y0, or re 2 and displacement fields for the three loading modes. They discussed the form, theoretical background, limitations and applicability of westergaards equations, and presented what are considered to be the most accurate versions of his formulae. Basics elements on linear elastic fracture mechanics and crack. Determining finitewidthcorrection factors for fatigue crack. Div f a v v u 37 2d problem, quasistatic, no volume force. Westergaard theory applies under the following assumptions. Culverts are essential parts of the lock and navigation structures. Westergaard developed a solution for the stress field surrounding a crack 1 that has two.
A westergaards 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. Traditionally, design engineers prefer to work with stresses rather than energy, or energy release rates. Westergaard solution of stress field for modei nptelhrd. I main topics a historical perspective b expression of the biharmonic function by harmonic functions. B one of the three most important early papers in fracture mechanics, the other two being by inglis 19 and griffith 1921. A westergaard s 1939 paper presented a simple way to express the. On the general solution of plane crack problems in westergaard. The main features of the theory are briefly presented in the paper, together with several validating examples and some convergence assessments.
Office of naval research contract nonr6lo i06 task order nr. Basics elements on linear elastic fracture mechanics and. The crack opening displacement approach to fracture mechanics in yielding materials show all authors. As such, a shift in emphasis from energy to the stress analysis approach was made in the late 1950s, starting with irwins paper, published in the journal of applied mechanics of asme. Inglis, stresses in a plate due to the presence of cracks and sharp corners, transactions of the institution of naval architects,vol. The stress intensity factor describes the stress state at a crack tip, is related to the rate of crack growth, and is used to establish failure criteria due to fracture.
Elasticity solutions are developed for finite multilayered domains weakened by aligned cracks that are in a state of generalized plane deformation under two types of end constraints. Excessive contact pressure can lead to a typical bearing failure such as a plastic deformation similar to peening. Hydrodynamic pressure on culvert gates during an earthquake ali rasekh phd, senior structural engineer with tetra tech abstract. Stress analysis of cracks chapter 3 fracture mechanics. Plane analysis of finite multilayered media with multiple aligned crackspart i. Harold malcolm westergaard 9 october 1888 in copenhagen, denmark 22 june 1950 in cambridge, massachusetts, united states was a danish structural engineer. The unsoaked california bearing ratio cbr value at 2. Water pressures on dams during earthquakes, asce, 1933. He served as a commander in the civil engineer corps of the us navy during world war 2 and was a member of a. The excess pore pressures are positive in front of the wall and negative behind it. Determining finitewidthcorrection factors for fatigue.
Eftis j, leibowitz h 1972 on the modified westergaard equation for certain plane crack problems. Williams, fracture mechanics of polymers, ellis horwood limited, halsted press. Analysis on fracture mechanics of unstable rock authors. Selected papers on foundations of linear elastic fracture. The full equations for stresses and displacements around a central crack, in an infinite plate subjected to uniaxial and biaxial tension, are determined using the westergaard stress function. Stresses in a plate due to the presence of cracks and sharp corners c. The crack opening displacement approach to fracture mechanics in yielding materials. The formulation is naturally applicable to notches and generally curved cracks. Contact pressures as an elastic roller crosses a scratch. The crack opening displacement approach to fracture mechanics in yielding materials f. General solution of the problem of elastostatics in an ndimensional homogeneous isotropic solid in in an ndimensional space, 1935. Westergaard equations have often been misquoted or misapplied in subsequent publications. Bearing pressure is a particular case of contact mechanics often occurring in cases where a convex surface male cylinder or sphere contacts a concave surface female cylinder or sphere. Westergaards solution for cracks fracture mechanics.
Aem 644 westergaard solution part i basic mathematics duration. Stress distribution at predetermined depths based on. Calculation method of hydrodynamic pressure in seismic response analysis of gates by takashi sasaki1, tomoya iwashita 2 and yoshikazu yamaguchi 3 abstract in order to maintain dam storage functions, it is essential to guarantee the safety of the dam body and appurtenant structures which are use to control the flow of water from the dam reservoir. Calculation method of hydrodynamic pressure in seismic. Westergaards method, that the stress field in the area of the crack tip is completely determined by the quantity k after kies, a colleague of irwin, 19521954, called stress intensity factor. Plane analysis of finite multilayered media with multiple. First, westergaards solution applies directly to cracks, not to an ellipse that approaches a crack in the limit. A more natural procedure should make use of fundamental solutions that represent the square root singularity of the gradient field around the crack tip a greens. Hydrodynamic pressure on culvert gates during an earthquake. Thats not a simple question, as it depends on the relative stiffness of wall and fill, freq content of the eq, etc. Inglis transactions of the institution of naval architects 19 18. Biography from karleugen kurrer, the history of the theory of structures from.
Printed in great britain compendium westergaard stress functions for several periodic crack problems the crack problems with the geometry and loading configurations which are shown in figs. Stress intensity factors for near surface cracks by boundary collocation shawqi mohammad suleiman. A boundary element implementation for fracture mechanics. Bearing pressures and cracks, 1939 theory of elasticity and plasticity, 1952. First, westergaard s solution applies directly to cracks, not to an ellipse that approaches a crack in the limit. Other possible modes of deformation at a crack tip are. The crack opening displacement approach to fracture. Application of bousinesqs and westergaards formulae in. Westergaard journal of applied mechanics 1939 23 the distribution of stress in the neighbourhood of a crack in an elastic solid i. Stress distribution at predetermined depths based on the bousinesqs and westergaards formulae. Second, the solution is expressed in rectangular coordinates rather than elliptical coordinates.
Limitations of the westergaard equations for experimental. B one of the three most important early papers in fracture mechanics, the other two being by. He served as a commander in the civil engineer corps of the us navy during world war 2 and was a. W westergaard stress coefficient i w westergaard vertical stress n point load from the end bearing pile z vertical distance from the end point of pile r radial distance from z murthy, 1992. In the parameter k subscript i refers to mode i loading, i. Growth behavior of surface cracks in the circumferential plane of solid and hollow cylinders. Stress intensity factors for near surface cracks by. In the traditional boundary element methods, the numerical modelling of cracks is usually carried out by means of a hypersingular fundamental solution, which involves a 1r2 kernel for twodimensional problems. They transfer water between upper and lower chambers, lakes, rivers andor ocean side of the navigation system. Westergaard developed a solution for the stress field surrounding a crack that has two advantages over ingliss solution 2. Thus the total hydrodynamic pressure acting on a submerged wall is twice that given by. Siqi chen, hongkai chen, ming yang, tao chen, kexuan guo keywords.