Numerical Methods in Soil Mechanics B.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 "RECONCILIATION OF FORMULAS FOR RING DEFLECTION" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 APPENDIX B RECONCILIATION OF FORMULAS FOR RING DEFLECTION M G Spangler was the first to predic t the ring deflection of buried flexible circular pipes His Iowa Formula is: Dx = Df KsWcr3/(EI+0.061E'r3) where (See Figure B-1.) Dx = horizontal increase in diameter due to vertical soil pressure P, P = vertical soil pressure on top of pipe, D = vertical decrease in diameter which Spangler assumed to be equal to D x, d = ring deflection = D/D , D = mean circular diameter of the pipe, r = D/2 = mean radius of the pipe, Df = deflection lag factor which can be incorporated into a time-dependent soil modulus and can be ignored (Spangler assumed Df = 1.5.), Ks = bedding factor Spangler found it can vary from 0.083 to 0.110 depending upon the bedding angle, a (A reasonable assumption is Ks = 0.1.) Wc = Marston load per unit length, (Wc PD for most flexible pipes.) e = vertical soil strain, (compression) in sidefill soil due to pressure P, EI = stiffness of the pipe wall per unit length (See Figure B-2.) E' = soil modulus (Spangler defines this as horizontal modulus of soil reaction, E", at the spring lines It is more relevant to relate ring deflection to vertical soil modulus from confined compression tests.) Rs = E'/(EI/D3) = stiffness ratio = ratio of soil stiffness E' to ring stiffness EI/D3 Rs = 53.77R' s R' s = E'/(F/D) = stiffness ratio based on pipe stiffness, F/D The Iowa Formula can be written in the following form: d/e = R s/(80+0.061Rs) IOWA FORMULA This is the relationship between two dimensionless variables: d/e = ring deflection term, and Rs = stiffness ratio This relationship is shown on Figure B-3 The graph approaches a horizontal asymptote at d/e = 1.64 But empirical ring deflections not exceed vertical compression of the sidefill soil Therefore, d/ e does not exceed unity More reasonable is the empirical graph of Figure B-3 which is expressed by the formula: d/e = Rs/(30+Rs) EMPIRICAL FORMULA The empirical formula is an upper 90 percentile of ring deflections from tests at USU in the 1960s Figure B-1 Assumed loads on a flexible ring in Spangler's Iowa Formula ©2000 CRC Press LLC Figure B-2 Pertinent notation for calculating the ring deflection of buried flexible pipes Figure B-3 Comparison of the Iowa Formula and an Upper Limit Empirical graph for predicting ring deflection of buried flexible pipes ©2000 CRC Press LLC Iowa Formula The Iowa Formula was derived to predict horizontal ring deflection of buried flexible pipes Figure B-1 is the basis for the derivation Bedding angle, a , could vary from to 180 o Horizontal passive soil resistance acted over a parabola subtended by an arc of 100o If an arc of 90o seemed reasonable, 100o must be conservative The passive resistance was better represented by a sine curve throughout the height of the ring, but a sine curve appeared to be more difficult to integrate Integrations were performed using virtual work to solve the two deformation equations required for analysis Three equations were available from static equilibrium With load, Wc, and soil resistance, h, known, five unknowns can be solved The mathematical analysis by Spangler was elegant and correct after a modification of definition of h A critical variable was the soil resistance, h, assumed to be a function of horizontal soil modulus of elasticity, E-prime (E') E' was assumed to be constant for any soil and density (compaction) For details see text by Spangler, M.G and Handy, R.L., Soil Engineering, 3rd Ed., IEP, New York 1975 E-prime (E') E-prime (E') is the horizontal soil modulus in the Iowa Formula In the derivation of the formula it was assumed that all materials are elastic and that E' is constant for any given soil embedment In fact, E' is not constant Pipe-soil interaction is not elastic Flexible pipes are best analyzed by mechanics of plastics Soil can vary from viscous through plastic to granular (best analyzed by the mechanics of particulates) Values of E' were investigated at USU in 1996 and 1997 E' was found to be a function of soil depth (confinement), ring stiffness, and soil type and density Figure B-4 is a graph of E' in silty sand (Unified Classification SM) Figure B-5 is a proposal for conservative values of E' for silty sand Other soils require similar tests to evaluate E' EXTERNAL PRESSURE DESIGN BASED ON RING STIFFNESS (BUCKLING) ©2000 CRC Press LLC Like the Iowa Formula, some ring design equations are based on elastic stiffnesses of the ring and the soil These equations are subject to the same cautions as any equation based on elasticity Two such design equations are considered in the following One equation is simplified AWWA C950: (B.1) P = \/R wEs(EI)/(0.149r3) where P = uniform collapse pressure, Rw = buoyancy factor = 1- 0.33(H'/H), where H'