Numerical Methods in Soil Mechanics 22.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 "BURIED TANKS AND SILOS" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 CHAPTER 22 BURIED TANKS AND SILOS A tank is generally either a circular cylinder with end closures, or a sphere See Figure 22-1 Tanks are buried with the axis horizontal Silos (and basins and caissons) are buried with the axis vertical Silos may or may not have closures at the top They are used as access ways, mine shafts, and bins or hoppers for storage of ore, coal, aggregate, etc Bins have an outlet at the bottom for feeding a conveyor in a pipe or tunnel below Tanks are used to store fluids, but are also used as bunkers for hazardous materials ANALYSIS OF BURIED TANKS Tanks are designed either for internal pressure or for external loads For worst-case internal pressure, external soil support is ignored For worst-case external pressure, it is usually assumed that the tank is empty, buried in poor soil with groundwater table above the tank and possibly a vacuum in the tank The performance limit is collapse An accurate model would be complicated Even if a model were devised, it could not be generalized, and it might imply greater precision than can be justified Soil properties are major elements of the interaction, but cannot be quantified precisely under normal installation procedures Boundary conditions are seldom precise Therefore, basic principles of engineering mechanics provide adequate analysis Safety factors are required in any case A freebody-diagram and principles of static equilibrium help in visualizing tank-soil interaction and in dispelling misconceptions, some of which arise from forensic arguments on assessment of responsibility for leaks in buried tanks Leaks are analyzed in Chapter 24 See nomenclature, Figure 22-2 critical on the inside surface of the tank See the infinitesimal cube in Figure 22-3 Because P' is small compared with circumferential and longitudinal stresses, it is usually neglected The maximum principal stress is the circumferential stress For a thick-wall tank (D/t < 10), stress analyses are found in Chapter 19 For a thin-wall tank (D/t > 10), the maximum principal stress is the circumferential stress The following applies to thin-wall tanks Spheres From mechanics of solids, hoop stress due to internal pressure is only half as great in a sphere as in a cylinder of the same diameter and wall thickness Therefore, for a sphere, σ = P'(ID)/4A where σ = P' = ID = t = A = A = (22.1) hoop stress, internal pressure, inside diameter, wall thickness, wall cross sectional area per unit length, t for plain cylinder wall Cylinders (Shells) Cylindrical tanks are short pipes, for which the critical circumferential stress is, σ = P'(ID)/2A (22.2) Maximum Internal Pressure Longitudinal stress is only half as great as circumferential stress End closures are stiffeners which help to resist internal pressure But when inflated, end closures cause stress concentrations in the shell-to-head seam For worst-case analysis of internal pressure, it is customary to neglect support of the embedment Analysis is based on principal stresses which are Equation 22.2 is reasonably accurate for design, but safety factors are essential End closures (heads) are analyzed separately ©2000 CRC Press LLC Figure 22-1 Some typical buried tanks and silos ©2000 CRC Press LLC Figure 22-2 Nomenclature for steel tanks (above) and buried tanks (below) ©2000 CRC Press LLC Maximum External Pressure Spheres The following are worst-case analyses which include burial in saturated soil A major concern is flotation In the following, it is assumed that the soil cover is sufficient to prevent flotation From Chapter 21, soil cover should be at least half the tank diameter The interaction of tank, soil, and groundwater is complex Strength of soil is decreased under water and loads must include external water pressures The tank itself is complex because of the interaction of shell, heads, risers, and welds See Figure 22-2 Codes and standards are available from ASME (boiler code), Underwriters Laboratory, and ASTM (tank standards) Recommendations are available from industries (waterworks, gas, and petroleum) Codes were developed originally for tanks subjected to uniform internal pressure For tanks subjected to external pressure, these codes only cover part of the requirements for performance Collapse of spheres due to uniform external pressure is analyzed by marine engineers for bathyspheres When soil support is included, analysis is complicated Not only are circumferential stresses in a sphere half as great as in a cylinder, but the vertical soil pressure is less because two-way soil arching action (soil dome) is twice as effective as a cylindrical soil arch In the design of buried pipes, the benefit of soil arching action is usually neglected, but is a significant plus for conservative analysis In the design of buried spheres, it may be prudent to test or evaluate the arching action of the soil dome Little information is available on buried spheres Notation Geometry: D = diameter of the circular shell, H = height of soil cover, h = elevation of water table, L = length of the tank (or height of silo), r = D/2 = radius of the circular cylinder, t = wall thickness, A = cross sectional wall area per unit length, R = radii of curvature of the head in a plane that contains the axis of revolution Properties of Materials: E = modulus of elasticity of tank material, υ = Poisson ratio, ϕ = soil friction angle, S = yield stress Forces, Pressures, and Stresses: P = external pressure or internal vacuum, γ = unit weight of soil, σ = stress in the tank (Figure 22-3), Subscripts refer to directions ©2000 CRC Press LLC Cylinders (Shells) A theoretical analysis is available for uniform external pressure at collapse of cylindrical tanks with no soil support In fact, soil provides support Moreover, external water pressure is not uniform, but increases from top to bottom of the tank R Allan Reese (1993) investigated hydrostatic pressures on the bottoms of horizontal steel tanks at collapse as the tanks were lowered in water He concluded that it is sufficiently accurate to design tanks by assuming uniform external pressure according to an equation from Young (1989), P = 0.807E(1-υ 2)-3/4(r/L)(t/r)5/2 (22.3) where, in examples that follow: P = water pressure on the bottom of the tank at collapse (sudden inversion), E = mod/elast = 30(106) psi for steel, t = wall thickness of plain wall, L = length of the tank, D = diameter of the tank, r = radius of of the shell, υ = Poisson ratio = 1/4 for steel It is better to use the pi-term (r/t) than the common (D/t) because radius includes out-of-roundness At Figure 22-3 Principal stresses on the inside of a tank wall sujected to internal pressure P' The same stress analysis applies for negative internal pressure (vacuum), with reversed signs Figure 22-4 Uniform external pressure on steel tanks at collapse — graphs of Equation 22.4 ©2000 CRC Press LLC some location on the shell, the radius may be greater than D/2 Designers use Equation 22.3 in the following form for steel tanks buried in saturated soil: P = 72(106)psi(D/L)(t/D)5/2 (22.4) Poisson ratio for steel is usually about ν = 0.27 Some designers use ν = 0.3 in Equation 22.4 The differences are not significant; i.e., If ν = 0.25, the coefficient is 71.87, If ν = 0.27, the coefficient is 72.52, If ν = 0.30, the coefficient is 73.54 If Poisson ratio is increased from 0.25 to 0.30, P increases by only 2.3% Poisson ratio is often neglected Without heads to support the shell, from pipe theory, at D/t = 575, P = 2E/(1-ν2)(D/t)3 P = 0.34 psi Example What is the external pressure on the bottom of an empty 12,000-gallon steel tank at collapse if the tank is lowered in water until it collapses? Diameter D = 96 inches, wall thickness t = 0.167, length L = 32 ft, D/t = 575, and L/D = Substituting into Equation 22.4, collapse pressure is P = 2.27 psi, which is equal to a depth of water of 5.25 ft above the bottom of the tank This is less than the diameter of the tank Equations 22.3 and 22.4 are not applicable If wall thickness is increased to 0.2391 inch, P = 5.56 psi At collapse, the water surface is 4.83 ft above the tank Figure 22-4 shows graphs of Equation 22.4 Noteworthy are the effects of wall thickness and length of tank on collapse pressure For comparison, the bottom graph is pressure at collapse of pipes (no heads or ring stiffeners) Even though Equations 22.3 and 22.4 are conservative, safety factors are recommended because tanks are never perfectly circular ©2000 CRC Press LLC Heads (End Closures) Heads are usually analyzed separately — not compound head-shell analysis The load is external hydrostatic pressure plus any internal vacuum in the tank If the water table is above the tank, analysis is sufficiently accurate if the pressure is assumed to be the average pressure distributed uniformly over the head The shapes of heads vary from hemisphere to dish (concave in or concave out) to flat heads (with or without stiffeners) Hemispherical Heads Hemispherical heads are easily analyzed by classical methods (Timoshenko, 1956) Circum-ferential stress is half as great in a hemisphere as in a cylinder of equal radius and wall thickness When pressurized, the change in radius is not the same for head and shell For a cylinder, change in radius is (P/E)(r/t)(1-υ /2) For a sphere, change in radius is (P/2E)(r/t)(1-υ ) For equal values of P, E, υ , and r/t, the increase in radius is greater for the shell than for the hemispherical head If the head fits inside the shell, internal pressure tends to open a gap between shell and head To avoid this, for many buried tanks, the head is a cap that fits on the outside of the shell But, then, the possibility that the head might shear down past the shell should be investigated Consider the case of uniform external pressure on a sphere and cylinder of equal radii Poisson ratio is 1/4 If the hemisphere has the same thickness as the cylinder, the tendency to decrease in radius is 3/7ths as great as the cylinder Stress between shell and head is less if the head is half as thick as the shell See Chapter 24 This is not a priority for most tank fabricators Discontinuities in stress and strain at the head-to-shell joint are accommodated by a good fit and a good weld Some of the incompatibilities of shell and hemispherical head can be reduced by ellipsoidal heads or composite surfaces of revolution (dishes) For analysis, an infinitesimal element is isolated by two meridian cuts and two parallel cuts as shown in Figure 22-5 This element is a free-body-diagram on which stresses are related as follows: σt /rt + σ m /rm = P/t from which σm = Pr/2t = 18 ksi compression Substituting into Equation 22.6, the tangential stress in the head is σt = -18 ksi tension Stress at A in the head is of concern because of the opposite sign Shearing stress becomes critical See Chapter 19 (22.5) Flat Heads where: σt = σm = rt = rm = P = t = S = tangential stress in direction of the parallel, stress in direction of the meridian, radius of curvature of the parallel, radius of curvature of the meridian, external pressure on the head, thickness of the head, yield stress If the head is a hemisphere, from Equation 22.5, rt = rm, and P = 2St/r Example Figure 22-6 shows a steel tank with the head attached to the shell by a reduced meridional radius of curvature, rm All of the steel is the same thickness When ring compression stress in the shell reaches yield, what are the meridional and tangential stresses at point A in the head? Given: t = 0.1875 inch, rt = 36 inches = D/2 = radius of the shell, rm = 12 inches, S = 36 ksi = yield stress At yield stress in the shell, external pressure is P = St/rt = 187.5 psi Substituting values into Equation 22.5, P/t = ksi/inch, and σt /36 + σ m /12 = 1000lb/in (22.6) Equation 22.6 contains two unknowns A second equation comes from a free-body-diagram of the head isolated by a cutting plane of a parallel through point A, Figure 22-6 Equating horizontal forces on the head, Pπr2 = σ m2πrt ©2000 CRC Press LLC (22.7) If the performance limit of a pressurized head is yield stress, analysis can be based either on plate theory or membrane theory, whichever gives the higher stress However, if the performance limit is deformation (rupture), the membrane theory is more responsive At tensile yield stress in a membrane the entire thickess is at yield stress Rupture is incipient On the other hand, at flexural yield stress in a plate, only one surface reaches yield stress — not the entire thickness Rupture is not incipient Surface yielding allows the disk to deform, tending toward a membrane with more uniform stress throughout its thickness Very thick disks (D/t