Numerical Methods in Soil Mechanics F.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 "STRAIN ENERGY ANALYSIS" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 437 APPENDIX F STRAIN ENERGY ANALYSIS An infinitesimal cube is subjected to triaxial principal stresses The general case would include shearing stresses, but the cube can be oriented such that all stresses are principal stresses Moreover, for pipes, principal stresses can usually be identified directly Shearing stresses are zero on plains of principal stresses terms can be added The sum is, 2EU = σ12+σ22+σ32-2ν (σ1σ 2+σ 1σ 3+σ 2σ 3) If yield stress is the performance limit, the total EU must be equated to the strain energy term at tensile yield stress: Uf = σ f εf /2 = σ f2/2E Rewriting, 2EUf = σf2 = 2EU σ12 + σ22 + σ32 - 2ν(σ1σ2 + σ 2σ + σ 3σ 1) = σ f2 (F.2) For most pipe materials, the strain energy of volume change should not be included in the total strain energy at failure (yield stress) Subtracted out of Equation A.2, the result is the Huber-Hencky-von Mises equation: U = strain energy (scalar), s = normal stress, sf = yield strength, e = normal strain, E = mudulus of elasticity, n = Poisson ratio (0.27 for steel) Subscripts, 1, 2, and refer to maximum, intermediate, and minimum values σ12+σ22+σ32 - (σ1σ 2+σ 2σ 3+σ 3σ 1) = σ f (F.3) For pipes, the square of the smallest stress, σ3, is often ignored with results: σ12 + σ22 - σ1σ = σ f2 (F.4) On coordinate axes a plot of σ1 vs σ 2, at elastic limit, is the von Mises ellipse as shown below Strain energy is U = S se de As stress increases from to s within the elastic range, the strain energy is average stress times strain, i.e.: f U = Sse /2 (F.1) From mechanics of materials, e = s 1/E-n(s 2+s 3)/E Strain energy for each stress can be written as follows Strain Ee = σ1-ν(e 2+e 3) Ee = σ2-ν(e 1+e 3) Ee = σ3-ν(e 1+e 2) Strain Energy Term 2EU1 = σ12-νσ1(σ2+σ3) 2EU2 = σ22- νσ2(σ1+σ3) 2EU3 = σ32- νσ3(σ1+σ2) Because energy is scalar, the three strain energy ©2000 CRC Press LLC Figure F-1 Infinitesimal cube showing principal stresses acting on it ...437 APPENDIX F STRAIN ENERGY ANALYSIS An infinitesimal cube is subjected to triaxial principal stresses The general case would include shearing stresses, but the cube can be oriented... such that all stresses are principal stresses Moreover, for pipes, principal stresses can usually be identified directly Shearing stresses are zero on plains of principal stresses terms can be... coordinate axes a plot of σ1 vs σ 2, at elastic limit, is the von Mises ellipse as shown below Strain energy is U = S se de As stress increases from to s within the elastic range, the strain energy