Numerical Methods in Soil Mechanics 03.PDF Numerical Methods in Geotechnical Engineering contains the proceedings of the 8th European Conference on Numerical Methods in Geotechnical Engineering (NUMGE 2014, Delft, The Netherlands, 18-20 June 2014). It is the eighth in a series of conferences organised by the European Regional Technical Committee ERTC7 under the auspices of the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE). The first conference was held in 1986 in Stuttgart, Germany and the series has continued every four years (Santander, Spain 1990; Manchester, United Kingdom 1994; Udine, Italy 1998; Paris, France 2002; Graz, Austria 2006; Trondheim, Norway 2010). Numerical Methods in Geotechnical Engineering presents the latest developments relating to the use of numerical methods in geotechnical engineering, including scientific achievements, innovations and engineering applications related to, or employing, numerical methods. Topics include: constitutive modelling, parameter determination in field and laboratory tests, finite element related numerical methods, other numerical methods, probabilistic methods and neural networks, ground improvement and reinforcement, dams, embankments and slopes, shallow and deep foundations, excavations and retaining walls, tunnels, infrastructure, groundwater flow, thermal and coupled analysis, dynamic applications, offshore applications and cyclic loading models. The book is aimed at academics, researchers and practitioners in geotechnical engineering and geomechanics.
Anderson, Loren Runar et al "RING DEFORMATION" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 Figure 3-1 (top) Vertical compression (strain) in a medium transforms an imaginary circle into an ellipse with decreases in circumference and area (bottom) Now if a flexible ring is inserted in place of the imaginary ellipse and then is allowed to expand such that its circumference remains the same as the original imaginary circle, the medium in contact with the ring is compressed as shown by infinitesimal cubes at the spring lines, crown and invert ©2000 CRC Press LLC CHAPTER RING DEFORMATION Deformation of the pipe ring occurs under any load For most buried pipe analyses, this deformation is small enough that it can be neglected For a few analyses, however, deformation of the ring must be considered This is particularly true in the case of instability of the ring, as, for example, the hydrostatic collapse of a pipe due to internal vacuum or external pressure Collapse may occur even though stress has not reached yield strength But collapse can occur only if the ring deforms Analysis of failures requires a knowledge of the shape of the deformed ring For small ring deflection of a buried circular pipe, the basic deflected cross section is an ellipse Consider the infinite medium with an imaginary circle shown in Figure 3-1 (top) If the medium is compressed (strained) uniformly in one direction, the circle becomes an ellipse This is easily demonstrated mathematic ally Now suppose the imaginary circle is a flexible ring When the medium is compressed, the ring deflects into an approximate ellipse with slight deviations If the circumference of the ring remains constant, the ellipse must expand out into the medium, increasing compressive stresses between ring and medium See Figure 3-1 (bottom) The ring becomes a hard spot in the medium On the other hand, if circumference of the ring is reduced, the ring becomes a soft spot and pressure is relieved between ring and medium In either case, the basic deformation of a buried ring is an ellipse — slightly modified by the relative decreases in areas within the ring and without the ring The shape is also affected by non-uniformity of the medium For example, if a concentrated reaction develops on the bottom of the ring, the ellipse is modified by a flat spot Nevertheless, for small soil strains, the basic ring deflection of a flexible buried pipe is an ellipse Following are some pertinent approximate geometrical properties of the ellipse that are sufficiently accurate for most buried pipe analyses Greater accuracy would require solutions of infinite series ©2000 CRC Press LLC Geometry of the Ellipse The equation of an ellipse in cartesian coordinates, x and y, is: a2x2 + b2y2 = a2b2 where (See Figure 3-2): a = minor semi-diameter (altitude) b = major semi-diameter (base) r = radius of a circle of equal circumference The circumference of an ellipse is p(a+b) which reduces to 2pr for a circle of equal circumference In this text a and b are not used because the pipe industry is more familiar with ring deflection, d Ring deflection can be written in terms of semidiameters a and b as follows: d = D/D = RING DEFLECTION (3.1) where: D = decrease in vertical diameter of ellipse from a circle of equal circumference, = 2r = mean diameter of the circle — diameter to the centroid of wall crosssectional areas, a = r(1-d) for small ring deflections (