Numerical Methods in Soil Mechanics C.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 "SIMILITUDE" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 APPENDIX C SIMILITUDE Engineering is basically design and analysis; with due regard to cost, risk, safety, etc In the following, that which is designed is a buried pipe The analysis is a model that predicts performance Performance must not exceed some performance limits The major concern, usually, is the model Mathematical models are convenient Physical, small-scale models are used occasionally The most dependable models are full-scale prototypes in service Mathematical models are written to describe prototype performance because it is impractical to perform a full-scale prototype study for every buried pipe to be installed The set of principles upon which a model can be related to the prototype for predicting prototype performance, is called similitude Similitude applies to all models — mathematical, small-scale, and prototype There are three basic steps in achieving similitude Fundamental variables (fv's) — a list of all of the variables that affect the phenomenon All of the fv's must be uniquely interdependent However, no subset of the fv's can be uniquely interdependent For example, force, mass, and acceleration cannot all be used as fundamental variables in a more complex phenomenon, because force equals mass times acceleration Therefore the subset is uniquely interdependent Only two of the three fundamental variables could be used in the phenomenon to be investigated Basic dimensions (bd's) — the dimensions in which the fv's can be written The basic dimensions for buried pipes are usually force (F), distance (L); and sometimes time (T) and temperature Pi-terms (pi's) — combinations of the fv's that meet the following three requirements; a) The number of pi's must be at least the number of fv's minus the number of bd's b) The pi-terms must all be dimensionless c) No subset of pi's can be interdependent This is assured if each pi-term contains a fundamental variable not contained in any other pi-term ©2000 CRC Press LLC The pi-terms can be written by inspection Example Write a set of pi-terms for investigating the maximum wheel load W that can pass over a buried flexible pipe without denting the top of the pipe See Figure C-1 for a graphical model Following the three pi-term requirements; fv's W EI H P D E' g bd's = = = = = = = wheel load wall stiffness height of soil cover all pressures pipe diameter soil modulus soil unit weight F FL L FL-2 L FL-2 FL-3 fv's - bd's = pi's required pi's (W/E'D2) p (EI/D3) p2 (H/D) p3 (P/E') p4 ((D/E') p5 This set of five pi's, by inspection, is not the only possible set If this set is not convenient for investigating the phenomenon, a different set can be written In this case, the maximum wheel load is given by a mathematical model: p = f(p2, p3, p4, p5) (C.1) if we can somehow find the relationship of pi-terms Principles of physics provide one possibility Prototype studies provide the writing of empirical equations of best-fit lines through plots of data If small scale model studies are to be used, Equation C.1 must describe the performance of both model and prototype Therefore, the model must be designed such that corresponding pi-terms on the right side of Figure C-1 Sketch of a physical model for evaluating that wheel load passing over a buried flexible pipe that dents the top of the pipe ©2000 CRC Press LLC Equation C.1 are set equal for model and prototype This can be accomplished, even for small-scale models, because the pi-terms are dimensionless, and, therefore have no feel for size — or any other dimension, for that matter The subscript m designates model In order to design the model, design conditions (dc's) are: (EI/D3)m = (EI/D3) (H/D)m = (H/D) (P/E')m = (P/E') (g D/E')m = (g D/E') Using subscript r to represent the ratio of prototype to model, each of the design conditions can be met according to the following: gravity increases the effective unit weight of the soil in the model Another approximate solution is to draw seepage stresses down through the model (air, or water if the soil is to be saturated) in order to increase the effective unit weight of the model soil For most minimum soil cover studies, it turns out that the effect of the unit weight of the soil is negligible so dc is ignored Once the model is designed and built, from tests, the weight W can be observed when the buried pipe is dented The prediction equation (pe) is the equation of the pi-terms on the left sides of Equation C-1 for model and prototype, i.e., (W/E'D2) = (W/E'D2)m (EI)r = (Dr)3 (Hr) = (Dr) geometrical similarity W = Wm (Dr)2 (Pr) = (E' r) (g r) = (E' r)/(Dr) where Dr is the scale ratio of prototype to model If the scale ratio is (i.e., 1:5 model to prototype), then the load W on the prototype that will dent the buried pipe is 25 times the load Wm on the model that dents the model buried pipe where Dr is scale ratio Because soil is a complex material, it would be convenient if the same soil could be placed and compacted in the same way in both model and prototype The result is that E' r = 1, and g r = But now design conditions and are not met From design condition 3, Pr = Therefore, all pressures P must be the same in the model as at corresponding points in the prototype For example, tire pressures must be the same in model and prototype The soil pressure must be the same at corresponding depths in the model and prototype But this is impossible for a small scale model if the soil has the same unit weight One solution is to test the model in a long-arm centrifuge such that centrifugal force plus ©2000 CRC Press LLC If E' r = 1, then the prediction equation is: In order to write the mathematical model (equation) for the phenomenon, enough tests must be made to provide data plots for p = f(p2) with p3 held constant and for p = f'(p3) with p held constant From the best-fit lines plotted through the data, an equation of combination can be written for p = f[(p 2), (p3)] This becomes the mathematical model In fact, design condition may not be significant Just keep tire pressures the same in model and prototype Then the mathematical model is simply the equation of the best-fit plot of p = f(p2) Of course it can be written in terms of the original fundamental variables ... same in the model as at corresponding points in the prototype For example, tire pressures must be the same in model and prototype The soil pressure must be the same at corresponding depths in the... the soil is to be saturated) in order to increase the effective unit weight of the model soil For most minimum soil cover studies, it turns out that the effect of the unit weight of the soil. .. of pi's can be interdependent This is assured if each pi-term contains a fundamental variable not contained in any other pi-term ©2000 CRC Press LLC The pi-terms can be written by inspection Example