Numerical Methods in Soil Mechanics 09.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 "NON-CIRCULAR CROSS SECTIONS" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 Figure 9-1 Examples of non-circular cross sections of pipes used commonly in the corrugated steel pipe industry, as described by AISI Handbook of Steel Drainage & Highway Construction Products ©2000 CRC Press LLC CHAPTER NON-CIRCULAR CROSS SECTIONS If the pipe cross section is not circular, "ring" analysis must be modified For most buried pipes, a circular cross section is the most effic ient shape But even flexible circular rings deflect out-of-round during installation Morever, a demand exists for non-circular cross sections Some typical examples are shown in Figure 9-1 A standing demand exists in highway departments for culverts with reduced height of cross section Each inch of height of the culvert requires an enormous amount of soil to raise the highway by that amount The pipe arch and low profile arch are examples of efforts to serve highway demands for reduced heights of culverts Multiple culverts serve to reduce heights, but also increase costs, spread stream beds, and trap trash Mean radius is sufficiently accurate for thin-wall pipe analyses Outside radius is more accurate — especially for thick-wall pipes RING COMPRESSION STRESS For non-circular cross section, ring compression stress is simply, s = T/A; or, for plain pipes (smooth cylindrical surfaces, no ribs or corrugations, etc.), s = T/t "Ring compression" is a misnomer in noncircular pipes Nevertheless, the expression ring compression stress is understood to mean circumferential stress in the pipe wall RADIAL SOIL PRESSURE A pertinent variable for ring analysis is radius of curvature, r The basic deflection of a flexible ring is from circle to ellipse, for which radii of curvature are shown in Chapter But non-elliptical deformation could make it necessary to measure radii of curvature Techniques for measuring radii are explained in Chapter Figure 9-2 shows the free-body-diagram of an infinitesimal segment of pipe wall loaded by external radial pressure P The effects of bending moment (ring deformation) can be combined by superposition as discussed in this chapter Reactions are thrust T in the pipe wall From static equilibrium in the vertical direction, and noting that for the small differential angle, sin(dq/2) = (dq/2), the equation of vertical forces is Prdq = 2Tdq/2; and, Another pertinent variable is radial soil pressure P on the pipe From Equation 9.1, if thrust T is constant, P varies inversely as radius r If the ring is flexible, the soil must be able to provide enough pressure P for equilibrium It is conservative to neglect shearing stresses between soil and pipe Shearing stresses reduce radial stresses Moreover, any shearing stresses that develop during installation are easily broken down by earth tremors, variations in temperature, rise and fall of the water table, wetting and drying of the soil, etc Without shearing stress, thrust T is constant around the entire perimeter of the pipe This is evident from Figure 9-3 where, for static equilibrium, T1 = T = T = constant thrust around the entire perimeter From Equation 9.1, P1r1 = P2r2 = Pr = T T = Pr where T = P = r = (9.2) (9.1) tangential (circumferential) thrust in the wall, external radial pressure (plus internal vacuum), mean radius of curvature (assuming that the pipe is thin-walled and cylindrical) ©2000 CRC Press LLC Wherever the radius r is small, the external pressure P is large This introduces the very important concept that for a flexible non-circular cross section, the external soil-bearing capacity must be increased wherever radii are decreased If the corner plates on a pipe arch, Figure 9-4, have a radius equal to one-third the top radius, then the external normal pressure (radial soil support) must Figure 9-2 Free-body-diagram of an infinitesimal segment of pipe from which ring compression thrust in the pipe wall is T = Pr Figure 9-3 Free-body-diagram of sections of a pipe wall of varying radii of curvature from which the ring compression thrust is constant, T = P1r1 = P2r2 = Pr Shearing stress between the soil and the pipe wall is neglected ©2000 CRC Press LLC Figure 9-4 Typical cross section of a corrugated steel structural plate pipe arch showing radii of the top plate, corner plates, and bottom plate Figure 9-5 Elliptical cross section of a flexible ring showing the distribution of external pressure required for equilibrium ©2000 CRC Press LLC rx ©2000 CRC Press LLC be three times as great as the pressure on the top of the pipe arch The soil against the short-radius corner plates must have adequate bearing strength It is noteworthy that only a little spreading of the corner plates will allow reversal of curvature of the bottom plate if hydrostatic pressure should act on the bottom Soil support at the corner plates is imperative If a circular cross section is deflected into an ellipse, then Pxrx = Pyry See Figure 9-5 From Chapter 3, the ratio of radii is ry /rx = (b/a) But (b/a) = (1+d)3/(1-d)3, approximately Therefore, negligible if ring deflection is less than about ten percent Equation 9.3 is accurate enough for most ring deflection analyses Example A flexible pipe is deflected into an approximate ellipse shown in Figure 9-5 Initial ring deflection is = 15.9% If pressure on top is P = 1.0 ksf, what is the required horizontal bearing capacity of the sidefill soil at the spring lines? The horizontal pressure Px at spring lines, from Equation 9.3, is, Px = Py (ry /rx) = Py (1+d)3/(1-d)3 = 2.617 Py Px = Py(b/a) = Py (1+d)3/(1-d)3 (9.3) Horizontal bearing capacity of the soil at the spring lines must be greater than 2.62 ksf With a safety factor, specify soil-bearing capacity of ksf Sidefill must be well compacted, otherwise the soil will be at incipient slip, and the ring at incipient collapse Ring compression stress is s = Py ry /A, where A is wall cross-sectional area per unit length where a = r(1-d) = minimum semi-diameter b = r(1+d) = maximum semi-diameter d = /D = ring deflection An accurate solution, from Chapter 3, is, P x = Py + 3d + 4d2 + 4d3 + ) (9.4) - 3d + 4d2 - 4d3 + ) The accuracy of Equation 9.4 is seldom justified The following example illustrates the point Example Assume that ring deflection of the elliptical cross section is d = 10% What is Px in terms of Py? From Equation 9.3, Px = 1.826 Py From Equation 9.4, Px = 1.826 Py This many significant figures of accuracy is not justified — either in practice or theory It must be remembered that: the elliptical cross section is only a theoretical assumption; shearing stresses are ignored; the perimeter is assumed to be constant; the horizontal and vertical ring deflections are assumed to be equal, etc For an elliptical cross section, vertical ring deflection is slightly greater than horizontal ring deflection, but the difference is ©2000 CRC Press LLC MEASURED CHANGE IN RADIUS If the ring deflects into an ellipse, all that is needed to evaluate maximum and minimum radii of curvature is measurement of ring deflection The equations are shown in Figure 9-6 For deformations other than ellipse, the change in radius can be evaluated from changes De in the middle ordinate e of a cord of length L Figure 9-7 shows the analysis, from which the approximate radius of curvature is r = L2/8e for small ratios of e to L The change in radius from r to r' is found from change in the middle ordinate, D e = e - e': 1/r - 1/r' = D e/er (9.5) Procedures for measuring e (either inside or outside the pipe) are described in Chapter Minimum radius is pertinent to soil strength analysis; maximum radius is pertinent to ring stability, Chapter 10 Both are pertinent to circumferential stress analysis CIRCUMFERENTIAL STRESS Assume that within cord length L, a pipe is initially circular and ring compression stress in the pipe wall is Pr/A Now if the ring is deformed, the change in radius of curvature causes a change in Pr/A; and also introduces a flexural stress, (E/m)(r'-r)/2r' See Equation 5.3 The ring compression stress is essentially constant around the ring However, flexural stress is maximum where change in radius is greatest; i.e., where the change in middle ordinate, e, is greatest Knowing the change in middle ordinate, e, the circumferential stress within the cord length is, s = Pr/A + ( e/e)(Ec/r) (9.6) where (See Figure 9-7) e = middle ordinate for the original circle, e = L2/8r, or can be measured before the pipe is deformed, D e = change in middle ordinate due to ring deformation, L = length of cord t = wall thickness m = r/t = ring flexibility r = initial radius of curvature at some location before the pipe is deformed, r' = radius of curvature at the same location after the pipe is deformed, A = cross-sectional area of the pipe wall per unit length of pipe, P = radial pressure on the pipe, E = modulus of elasticity of the pipe, I/c = section modulus per unit length, e = change in middle ordinate due to ring deformation, c = distance from the neutral surface of the pipe wall to the most remote surface For a plain pipe, c = t/2, and (Ec/r) = (E/2m) to be used in Equation 9.6 From Equation 9.6, for any allowable stress, s , or strain, e , and external pressure, P; the allowable change in middle ordinate, De/e, can be found Ring deformation must then be controlled so that the measured De/e does not exceed the allowable ©2000 CRC Press LLC Internal pressure P' (no external constraint) causes hoop stress s = P'r/A A non-circular ring tries to re-round and change its radii Equation 9.6 still applies Changes in radii would have to be measured and then used to calculate flexural stresses Or knowing allowable stress, maximum allowable changes in radii, or changes in middle ordinate from a cord can be used by inspectors for control of the pipe shape during installation Internal pressure in a pipe with external constraint is sometimes analyzed by neglecting any changes in radii of curvature The presumption is that the soil is rigid But, if the ring is constrained by compressible soil, or by concentrated point loads and reactions, further analysis may be necessary A finite element analysis may be a good option For flexible pipes — even non-circular rings — flexural stresses due to change in radius, are not performance limits in general Brittle linings may pose an exception Many common pipe materials can yield without fracture Consequently, the flexural stress term can be neglected The ring simply sustains permanent deformation without fracture or inversion In summary, the circumferential stress analysis of non-circular pipes is based on Equation 9.6 which always includes ring compression stress and, possibly, flexural stress Pr/A = ring compression stress (or hoop tension), (D e/e)(Ec/r) = (D e/e)(E/2m) = (Mc/I) = flexural stress where (E/2m) = (Ec/r) = arc modulus — used if either change in radius of curvature or D e is known (I/c) = section modulus — used if the moment M is known The moment M can be evaluated from circumferential strains measured by electrical resistance strain gages positioned both inside and outside the pipe at locations where critical moment is anticipated Below yield, both thrust and moment can be found from these strains Circumferential stress due to ring deformation is dependent upon either the section modulus or the arc modulus 9-3 Figure 9-8 shows the cross section of a corrugated steel culvert comprising circular panels with radii as follows: For brittle pipes — the ring compres sion stress and ring deformation stress must be combined for analysis Steel pipes with brittle linings or coatings are not brittle pipes Small cracks are not serious ry = 82.50 inches for top and bottom panels, rx = 26.25 inches = radius of side panels For flexible pipes — deformation is caused by the soil Changes in wall thickness make little difference If soil is placed such that ring deflection is constant, the performance limit is wall crushing at yield stress due to ring compression stress, Pr/A This is an important basis for design See Chapter On the other hand, if ring deformation stress exceeds yield, the deformation is permanent But deformation is not a performance limit until it becomes excessive For analysis of flexible buried pipes, the stress due to ring deformation is not an appropriate basis for design Design by ring compression stress and design by ring deflection are the two basic design procedures For corrugated and profile wall pipes, the combination of ring compression stress and ring deformation stress may result in dimpling of corrugations, or plastic hinging But for buried pipes, dimpling and incipient hinging are not collapse This culvert is to be installed under a highway with the major diameter horizontal Soil cover is ft including an asphalt concrete pavement which is assumed to be flexible enough that the load is not spread by the pavement The road is designed for HS-20 truck loads Unit weight of soil is 135 pcf Because the pipe is a drainage culvert, the water table is never more than a few inches above the invert of the pipe What is the minimum sidefill soilbearing capacity required at the spring lines if safety factor is to be 2.0? (Px = 12.2 ksf) 9-4 In Problem 9-3, if the sidefill soil is cohesionless with a soil friction angle of f = 35o, what is the safety factor against soil shear failure? (sf = 0.4 — incipient collapse) 9-5 The elliptical culvert of Problem 9-3 is rotated 90o to serve as a livestock underpass See Figure 99 At what soil friction angle of sidefill at the spring lines would the culvert collapse by reversal of curvature of the 82.5-inch-radius side panels? Neglect H-20 surface live load Why? PROBLEMS 9-1 Derive Equation 9.6 Remember that flexural stress is Mc/I and that M/EI = 1/r - 1/r' for a circle w here r is the original radius and r' is the deformed radius 9-2 In order to check the assumption that a flexible pipe, ID = 42, with ring deflection of d = 10%, is an ellipse, what should be the middle ordinate inside the pipe to the spring line from a vertical cord (straight edge) that is 10 inches long? (e = 0.83) ©2000 CRC Press LLC 9-6 In cohesionless soil what soil friction angle is required to assure that shearing failure does not occur in the embedment of a pipe arch for which: rc = 18 inches at corner plates, rt = 60 inches at top plate, rb = 180 inches at bottom plate, Corrugations are 2/3 x ½ Assume high fill Neglect shearing stresses between the pipe and the embedment soil (f = 33o) 9-7 In Problem 9-6, what is the minimum rc in terms of rt if the soil friction angle is 30o? (rc = rt /3) Figure 9-8 Horizontal "ellipse" of corrugated steel plate to be used as a culvert and for which rx = 26.25 inches and ry = 82.5 inches and showing the radial soil pressure acting on it Figure 9-9 Corrugated steel plate underpass for livestock with H=2 feet of soil cover ©2000 CRC Press LLC Figure 9-10 Cross section of a flexible circular ring that has been deformed into an ellipse during installation such that d=D /D=16% and for which the required horizontal soil pressure is greater than the vertical soil pressure at the crown 9-8 In Problem 9-7 discuss the performance limits based on soil-bearing capacity and soil compression What about deformation of the ring due to soil compression? 9-9 A corrugated steel culvert has the following properties: Corrugations 2/3 x 1/2 D = ft t = 0.1046 inch A = 1.356 in 2/ft S = 36 ksi = yield strength sf = d = D/D Assume elliptical cross section What is the horizontal soil-bearing capacity required at the spring lines for this culvert if it is deflected by d = 16% into an ellipse under a fill height of 20 ft at soil unit weight of 125 pcf? See Figure 9-10 9-10 What is the approximate ring deflection of Problem 9-9 if the embedment is silty sand compacted to 80% density AASHTO T-180? (d = 2.3%) 9-11 What is the maximum ring compression stress in Problem 9-9? (s = 6.4 ksi) 9-12 In Problem 9-9, what is the middle ordinate from a cord 12.65 inches long at yield point stress of 36 ksi at the crown of the deflected pipe? 9-13 If ring deflection, d = 16%, in Problem 9-9 is increased by an additional Dd = 2%, what is the required horizontal soil-bearing capacity? 9-14 See Figure 9-11 What vacuum in the soil is required at invert level of a small-scale model buried pipe if the internal vacuum at collapse is to be the same for the model and a large prototype pipe? The water table is at ground surface Saturated unit weight of soil is 125 lb/ft3 t = 0.281 tm = 0.0359 ID = 84 IDm = H = 48 Hm = Figure 9-11 Hydrostatic pressures (and vacuum) in soil required such that P’m = P’ ©2000 CRC Press LLC ... pipes, the combination of ring compression stress and ring deformation stress may result in dimpling of corrugations, or plastic hinging But for buried pipes, dimpling and incipient hinging are not... Or knowing allowable stress, maximum allowable changes in radii, or changes in middle ordinate from a cord can be used by inspectors for control of the pipe shape during installation Internal... outside the pipe) are described in Chapter Minimum radius is pertinent to soil strength analysis; maximum radius is pertinent to ring stability, Chapter 10 Both are pertinent to circumferential stress