Numerical Methods in Soil Mechanics 18.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 "SPECIAL SECTIONS" Structural Mechanics of Buried Pipes Boca Raton: CRC Press LLC,2000 Figure 18-1 Wye showing transition flow from a mainline (inflow) pipe to two branch (outflow) pipes (In fact, flow could be in either direction.) Unit length of cone is x(cos Θ ) ©2000 CRC Press LLC CHAPTER 18 SPECIAL SECTIONS Special sections in pipes are valves, tees, wyes, elbows, caps or plugs, transitions such as cones for changes in diameter (i.e change in flow velocity), and transitions between conduits of differing shapes or sizes such as transitions from a rectangular conduit to a circular pipe Experience and expertise are available from manufacturers of common standard specials However pipeline engineers often need an uncommon section Following are some basic rules and procedures for preliminary design of specials As an example, consider a wye A wye (Y) is a bifurcation of the pipeline from a larger mainline pipe to two smaller branch pipes See Figure 18-1 A wye may require a trifurcation, or branch pipes of different diameters, or at different offset angles, etc Wyes can be either molded (warped surface) or mitered (circular cylinders or cones) The following example is a mitered bifurcation with equal offset angles The basic components are two truncated cones shown dotted with large ends Di and small ends Do to match up with the diameters of the inflow (mainline pipe) and the outflow (branch pipes) The cones are cut and welded together at the crotch to form the wye; and are then welded to the mainline and the branch pipes as shown It is noteworthy that the crotch, the intersection of the two cut cones, is an ellipse in a plane It is like a crotch seam in jeans An ellipse is easy to analyze and to fabricate Because the cut is in a plane, the welded intersection lends itself to reinforcement by internal vane or external stiffener ring or crotch plate For high pressure, fabricators favor welding the crotch cuts to a heavy crotch plate, and welding stiffener rings to the outside of the welded miters Do = t R δ = = = L LT θ = = = Consider the horizontal cross section 0-0 at the crotch of Figure 18-1 A free-body-diagram of half of the cross section shows a rupturing force of pressure times the span of the cut This force is more than twice the rupturing force in each ring of the branch pipes and will cause ballooning of the cross section unless the rings are held together at the center — either by a vane on the inside, or by a crotch plate on the outside A crotch plate is a Cclamp with an elliptical inside cut as shown in Figure 18-2 It is located at the plane of intersection (crotch cut) of the two branch pipes Force On Crotch Plate Due To Internal Pressure See Figure 18-3 From ring analysis (Chapter 2) the force to be resisted by the crotch plate from each of the cone walls at the vertex section A-A-A, is PrA per unit length of the pipe Per unit length of crotch plate (or vane), the force on the crotch plate becomes: w = 2PrA(cos θ ) where w = Notation and Nomenclature D Di = = ID = inside diameter (nominal for steel) inside diameter of inflow pipe (mainline) ©2000 CRC Press LLC inside diameter of outflow pipes (branch) wall thickness radius of bend in the pipe or cone offset angle of contiguous mitered sections length of section of pipe length of truncated cone angle between axes of each branch pipe and the mainline pipe P r = = θ = (18.1) vertical force on crotch plate (vane) per unit length of crotch plate internal pressure radius of the circular cone (or circular cylinder in some cases) offset angle of axes of branch cones (or cylinders) from the axis of the mainline pipe 217 Figure 18-2 Development of the crotch cut and the crotch plate which, together with the stiffener rings, supports the hoop tension at the cuts (all cuts are elliptical) ©2000 CRC Press LLC Figure 18-3 Free-body-diagrams of cone corss sections showing where the crotch is cut, the hoop forces PrA at section A-A-A and PrB at section B-B ©2000 CRC Press LLC Figure 18-4 Free-body-diagram of one limb of the crotch plate showing an approximate procedure for analyzing the forces on the limb assuming it to be a cantilever beam (bottom sketch) loaded at the free end of the statically indeterminate restraint Q of the stiffener rings ©2000 CRC Press LLC α = Li = angle on the cone cross section from the vertical axis to the crotch cut length of longitudinal element on the inside of the mitered cone section But this force w oc curs only on section A-A-A at the vertex of the crotch cut At section B-B of Figure 18-3, the hoop forces are not vertical If the crotch plate (or vane) resists vertical components only, per unit length the vertical force on the crotch plate is: w' = 2Pr B(cos θ )sin α ; where α is the angle, in the plane of section B-B, from vertical to the intersection of the two branch pipes The horizontal components of the hoop forces, Pr B, are balanced because of symmetry — i.e because the pressures, diameters, and offset angles are equal in the two branch pipes Figure 18-4 is a plot of w' throughout the length of the crotch cut of Figure 18-3 Clearly, the plot does not deviate significantly from a straight line Therefore, if a straight line is assumed, angle α serves no purpose, and Equation 18.1 provides a value for w for analyzing forces on the crotch plate From the force analysis, the crotch plate can be designed Hydrodynamic Guidelines In pressure lines of high velocity water flow, such as penstocks for hydroelectric power plants, it is prudent to avoid sudden changes in velocity or sudden changes in direction of flow because of turbulence and loss of energy Guidelines used by fluid dynamicists for minimizing energy loss are as follows See Figures 18-5 to Keep the cross-sectional areas of the mainline pipe nearly equal to the areas of the branch pipes For a wye (bifurcation), Do2 = Di2/2 keep δ < 7.5o or even < 6o for very high velocity flows Keep the radius of the bend greater than 2.5 times the pipe diameter (or mean diameter of any mitered cone section); i.e R > 2.5D = 5r It is preferable to keep R > 3D or even > 4D for very high velocity flows See Figure 18-7 Keep the length, Li, on the inside of the bend of each mitered section, greater than half the mean radius of the section (pipe or cone) Keep the cone taper angle minimum The greater the taper angle, the shorter are the length s Li of contiguous cone (or pipe) sections This means a sharper bend (shorter radius R of bend) On the other hand, the smaller the taper angle, the longer the crotch plate must be Consequently, much greater loads must be supported by the cantilever limbs of the crotch plate The crotch plate, a critical structural element of the wye, presents a dilemma — the need for a large taper angle to keep the crotch plate short, and the need for a small taper angle to keep the radius of the bend and the inside lengths, Li, within limits of hydrodynamic guidelines From this point on, design is by trial The relationship of the hydrodynamic guidelines to the structural integrity of the wye are best described by an example Example Consider a penstock for a hydroelectric power plant Suppose that the mainline pipe is 96-inc h steel pipe, bifurcated into two branch pipes to supply water under high pressure and high velocity to two equal sized turbines For steel pipes, diameters are inside Yield strength is 45 ksi For preliminary design, including 100% surge, pressure is P = 225 psi First Trial On mitered bends, keep the inside-of-bend offset angles minimum Inside offset angles are the critical cause of turbulence Inside offset angle should never be greater than δ = 15o It is preferable to keep δ < 10o Most engineers try to ©2000 CRC Press LLC Start with a trial wye — say Figure 18-1 Try a taper angle of 7.5o, for which the truncated length of the cones is LT = 113.94 inches To facilitate fabrication and welding, select the same steel Figure 18-5 Wye with inside offset angles limited to 7.5o Note that branches not clear each other at the lower end Figure 18-6 Same wye, but mitered; i.e., cut near midlength of the cone, rotated 180o, and welded Note that the branches now clear each other ©2000 CRC Press LLC thickness for all of the pipes and cones at the wye The hoop stress is maximum in the mainline pipe where hoop stress is σ = PDi /2t Therefore: t = PDi(sf)/2S (18.2) where t = wall thickness (to be found) σ = hoop stress in the pipe wall P = internal pressure = 225 psi Di = inside diameter of the mainline pipe = 96 inches S = yield strength of steel = 45 ksi sf = safety factor — say 1.5 Solving, t = 0.360 inch This is not a standard, so try standard t = 0.375 inch for analysis From guideline 1, Do2 = Di2/2 So Do = 67.88 inches for the branch pipes Specify diameters of the branch (outflow) pipes to be a standard Do = 66 inches The ratio of areas, inflow to outflow, is 1.058 — not bad Moreover, the slight reduction in outflow areas from perfect gives a slight increase in flow velocities into the turbines This is desirable from the standpoint of turbine efficiency From guideline 2, the inside offset angle, δ , should be less than about 7.5o For the trial wye of Figure 18-1, δ is greater than 7.5o — actually 15o If reduced to 7.5o, the branch offset becomes θ = 15o as shown in Figure 18-5 Obviously, the outflow pipes not clear each other at the lower end In order for the branches to clear each other, two remedies are considered The taper angle could be reduced such that length of the cone is increased But then, the length of the crotch plate would have to be increased That's bad An alternative remedy might be to miter the cones as shown in Figure 18-6 The length of the crotch plate is increased only slightly That's not so bad The total offset angle from the axis of the mainline to branches is θ low = 22.5o It is noteworthy that the inside offset angle from ©2000 CRC Press LLC tapered cone to pipe is less critical than the inside offset angle for bends in pipes because the taper-tocylinder transition is a symmetrical squeeze-down of flow Bends are not symmetrical If one or the other has to be mitigated, the taper (rather than the inside offset angle in a bend) is allowed to exceed the recommended maximum Figure 18-7 shows how mitered bends are formed Because a planar cut across any circular cone (or pipe) is a perfect ellipse, mitered bends can be achieved by cutting any cone (or pipe) at an angle of δ /2 with the diameter, rotating one section 180o, and then welding the cut The ellipses match The resulting offset angle is δ It is not always necessary to miter the mainline pipe See the mainline-to-cone cut in Figure 18-7 When this particular cone tilts to angle θ = 15o, its horizontal radius is approximately the same as the radius of the mainline pipe Of course, the cut of the cone is an ellipse, but the ellipse is so nearly circular, that the cone and pipe can be pulled together for welding If the ring cut were mitered, the stiffener rings would come in at some angle such as the ring cut angle of 6o shown on Figure 18-1 For the transition, upper stiffener ring A is a circle — an easy cut See Figure 18-8 The lower stiffener rings B (at the miter cuts in the cones) intersect at the angles shown Second Trial Figure 18-8 is the second trial wye for analysis and design Of primary concern is the crotch plate In this case, the length of the crotch plate limbs is 109.5 inches This compares not too badly with a crotch limb length of 94.8 inches for the first trial shown in Figure 18-1 A free-body-diagram of the force w on the 109.5-inch limb can be calculated by means of Equation 18.1 However, there are two values of θ , upper section and lower section For the upper section, cos θ = cos 15o = 0.966 For the lower section, cos θ = cos 22.5o = 0.924 Not justified is any attempt to interrelate the two by applying Equation 18.1 over the upper and lower sections separately Conservatively, we use the larger value, cos θ = 0.966, and we analyze the full ©2000 CRC Press LLC Figure 18-8 Second trial configuration of mitered wye showing the crotch plate and stiffener rings; and a full circle stiffener ring at A ©2000 CRC Press LLC Figure 18-9 Mitered wye showing cross-hatched areas which, when multiplied by pressure P, are the loads at the mitered cuts where stiffener rings and crotch plate are required ©2000 CRC Press LLC 109.5-inc h length of the limb as a single free-bodydiagram with a straight line distribution of the wforce on the cantilever The radius of the cones at the vertex of the crotch cut is about 33.5 inches Consequently w = 2Pr(cos 15o) = 14.6 kips per inch With this information, forces on the crotch plate can be found The above simplifications are justified by noting that Equation 18.1, for finding w, applies not only to the crotch plate, but to the stiffener rings as well In fact, each stiffener ring is simply two crotch plates with the ends of the limbs welded together All mitered cuts result in a w-force in the plane of intersection of the two contiguous sections Any part of the mitered section that is not part of a full ring (tension hoop), when pressure P is applied, must be supported by a crotch plate or stiffener ring This is shown in Figure 18-9 Areas shown crosshatched, when multiplied by pressure P, represent the w-force distribution diagrams on each of the cuts where crotch plate or stiffener rings are located The areas are shown in the plane of the page, but represent the vertical w-force The proof is evident in the column of values at the right margin, all of which, when multiplied by constant pressure P, are simply Equation 18.1 for w Clearly, rings at A and B must resist w-forces from the mitered joints as well as interaction from the crotch plate However, the areas at the A-cut and B-cut are small compared with the areas at the crotch cut and are usually ignored Moreover, almost any reasonable stiffener ring at the A-cut can resist the w-force acting on it The w-force at B is insignificant The B ring only needs to help support the crotch plate Moreover, the reduced radius rB at the B-cut, where the wall thickness is still 0.375 inch, results in a much stronger cone at the B-cut than at the A-cut Crotch Plate Design Figure 18-10 is a free-body-diagram of a cantilever representing the crotch plate with the w-force and reactions at A, B, and O as shown The reactions at A and B are the restraints by stiffener rings which can deform under Q-loads Therefore the analysis ©2000 CRC Press LLC is statically indeterminate, depending upon the spring constants of the stiffener rings at A and B For a true ring, such as A, the spring constant is, Q/∆ = 6.72 EI/r3 (18.3) where Q = diametral load on the ring ∆ = deflection of the diametral load E = modulus of elasticity of the steel ring I = moment of inertia of the cross-sectional area of the ring wall r = radius to the neutral surface of the ring cross section The spring constants of ring B and the crotch plate are more difficult to evaluate because of their shapes A reasonable simplification of the crotch plate for preliminary design is to assume that ring A will, at least, prevent rotation of the crotch plate cantilever limb at section B See Figure 18-11 To the left of section B, the crotch plate is approximately a half ring, wherein section B does not rotate under load Consequently, the spring constant for the crotch plate can be analyzed by an equivalent circular ring To analyze the equivalent ring, it is only necessary to neglect QA and to double the load QB on the equivalent ring The spring constant can be calculated from Equation 18.3 It is noteworthy that simulating the crotch plate by a ring with twice the Q-load on it, we are assuming that the limb of the crotch plate does not rotate at B A more accurate analysis would prove that rotation at B is small Figure 18-12 is proposed as a reasonable trial cross section of the crotch plate at section C-C Try a 1.5-inch by 30-inch plate for a web with a x 12inch flange on the outside On the inside of the web are the steel cone walls double welded to the crotch plate and splitter plates securely welded to provide the equivalent of a flange and to provide abrasion resistance to head-on flow of water containing sediment The crotch cross section shown is not a standard I-beam section Nevertheless, assuming that the walls of the cones and the splitter plates combine to provide an area Figure 18-10 Forces on a cantilever beam which is used as an approximation of the upper limb of the crotch plate and showing (below) the general shape of the stiffener rings at A and B ©2000 CRC Press LLC Figure 18-11 Upper limb of crotch plate showing how the spring constant for the load Q can be analyzed approximately by means of an equivalent ring with a load of 2Q ©2000 CRC Press LLC Figure 18-12 Trial cross section of crotch plate at Section C-C ©2000 CRC Press LLC of steel equivalent to the 1x12 flange, an approximate moment of inertia, I, can be easily calculated The spring constant for stiffener B may requ i r e a computer analysis However, at this point, in a sequence of many simplifications, to simulate ring B by an equivalent circular ring may be sufficient for preliminary design The basic function of ring B is to help support the crotch plate A layout of the crotch cut can be accomplished by any of a number of methods for scribing an ellipse Shown on Figure 18-13 is one method based on evaluation of the major and minor semi-diameters A cutting plane is passed through the cone on the right at θ = 18.26o θ can be found graphically, or by the law of sines for the small cross-hatched triangle shown The major diameter is the length of the cutting plane The minor diameter is the length of a line segment piercing the cone perpendicular to the page at the geometrical center G of the ellipse This can be found graphically as shown by the projection below the cone Or it can be found by trigonometry With the semi-diameters known, the elliptical crotch cut can be developed as shown on the left This procedure is based on two circles centered at G with radii equal to the major and minor semi-diameters Any radial line from G intersects the two circles at the latitude and departure of a point on the ellipse as shown A few numerical values of latitudes and departures are shown at the lower left This is the crotch cut No allowance is made yet for the thickness of the cone wall or the thickness of the crotch plate If the crotch cut is laid out on the inside of the cone, the cone wall thic kness is not an issue Depending on the method used to lay out the cut on the cone, half the thickness of the crotch plate must be allowed in the layout Alternatively, a new crotch cut can be analyzed by graphical techniques allowing for half the thickness of the crotch plate in the elliptical cuts in the cones With the crotch cut drawn to scale, a trial crotch plate can be laid out See Figure 18-14 Note that the crotch plate is cut well inside the crotch cut line This allows ample surface for welding the cones and ©2000 CRC Press LLC the splitter plates to the crotch plate These welds must be of excellent quality, for they must resist the entire hoop tension in the walls of the cones Stress analysis of the crotch plate and stiffener rings starts with assumed dimensions See Figures 18-12, 18-14, and 18-15 A free-body-diagram of the upper limb of the crotch plate is represented as a cantilever in Figure 18-14 With four unknowns, Mo, Qo, QB, and QA, it is statically indeterminate Moreover, QA and QB are reactions from rings which deflect under the Q-loads A reasonable simplification is to assume that section BA is a cantilever as shown at the bottom of Figure 18-14, and that the vertical deflections of points A and B are equal Solving, QA = 59.3 kips The remainder of the load is QB = 800 - 59 = 741 kips to be divided between crotch plate and "ring" B This is a conservative simplification because the 800-kip total force is actually distributed — not concentrated — and to assume it to be concentrated at A and B results in higher stresses in the crotch plate and ring B than would the distributed w-force The division of QB between crotch plate and ring B is proportional to the spring constants of the crotch plate and ring B The data for the trial cross sections of crotch plate and ring B are shown in Figures 18-12 and 18-15 To assume that ring B is circular is an approximation The ring is not circular — neither inside cut nor outside cut The two halves of the ring are not even in the same plane Nevertheless, using an average radius, r, an approximate spring constant can be evaluated From Equation 18.3, the ratio of spring constants for crotch plate and ring B is, (Q/∆) r = [9140/2(2293)](59/57.8)3 = 2.12 From this ratio, the crotch plate must take 68% of the 741 kip load (506 kips), and ring B must take 32% (235 kips) Stresses in the trial sections of crotch plate and ring B can be estimated because the moment in a ring due to a concentrated Q-load is: M = 0.318Qr (18.4) Figure 18-13 Graphical construction of the crotch cut based on the fact that it is an ellipse (constructed here by one of various methods) ©2000 CRC Press LLC Figure 18-14 Dimensions of the proposed crotch plate and loads on the upper limb shown (above) for the entire limb, and (below) for the section BA between stiffener rings ©2000 CRC Press LLC Figure 18-15 Trial cross sections for stiffener rings A and B ©2000 CRC Press LLC where M = Q r = = moments at locations of Q and at 90o to Q-locations on the ring concentrated diametral load radius to neutral surface of the ring Crotch Plate I/c Ao Qo r M σ σ = 571 in = 75 in = 506 kips = 57.8 mean = 20,333 kip inches = 2Q o /2Ao + M/(I/c) = 6.75 + 35.61 = 42.36 ksi Ring B 229 in 38 in 235 kips 59 mean 70 inches 5231 kip inches I/c AB QB r rmax M = = = = = = σ σ = Q B /2AB + M/(I/c) = 3.09 + 22.84 = 25.94 ksi Stress in the crotch plate is high; i.e., yield stress is 45 ksi However, considering that flexural yield stress is not failure for pipes (failure is a plastic phenomenon, not an elastic phenomenon), and considering that surge pressures of 100% can be reduced by slow-closing valves and controlledresponse turbines, this preliminary crotch plate design seems acceptable A final analysis may be advisable The stress in ring B is greater than that typical allowable, 45/2 = 22.5 ksi, with a safety factor of Nevertheless, the same mitigating arguments apply The preliminary design of ring B is acceptable The stress in ring A is σ = QA /2A + M/(I/c) = 1.24 + 10.13 = 11.37 ksi ©2000 CRC Press LLC where I/c = A = QA = r = M = 104.24 in 24 in 59 kips 56 mean 0.318QA /(I/c) = 1056 kip inches Stress is low enough to justify elimination of the flange Without the flange, neglecting contribution of the pipe wall, I/c > (16)2/6 = 42.67 in 3, A > 16 in 2, and σ < 1.85 + 24.75 = 26.60 ksi Based on the same arguments used for the crotch plate and ring B, the flange is not needed However, there may be some question about web buckling in compression at 9:00 and 3:00 o'clock On the other hand, buckling could occur only when there is pressure in the pipeline, and pressure would reduce greatly the compressive buckling stress in ring A For preliminary design, the flange is to be eliminated Example Figure 18-16 is a preliminary layout of a trifurcation in a water supply pipeline (penstock) for a hydroelectric power plant Flow in the penstock is to be divided equally into three turbines Rather than a fork with outflow pipe axes all in the same plane, a cluster of outflow pipes is proposed From the standpoint of hydrodynamics, it is an efficient transition The crotch plates are three heavy vanes at 120o all welded to a keel as shown 120° sections of the transition cones are then welded to the crotch plates The inflow is a 126-inch diameter steel pipe The three outflows are all 72-inch diameter Static pressure is 106 psi, but with surge, it is designed for 212 psi For preliminary design, the proposed wall thickness is 0.375 inch Yield strength of the pipe wall is at least 42 ksi It may not be necessary to miter short cone sections if the trifurcation is encased in reinforced concrete Even though the cones are long, crotch plates will not be required if the cones are supported from outside by the reinforced concrete casing See Figure 18-17 All hydrodynamic guidelines are met for high-pressure, high-velocity flow Figure 18-16 Layout of a cluster trifurcation from a 126-inch diameter intake to three 72-inch diameter outlets showing the top crotch plate on the left and the bottom transition cone on the right ©2000 CRC Press LLC Figure 18-17 Cross section of a hexagonal reinforced concrete encasement for the trifurcation of 126-D to three 72-D pipes ©2000 CRC Press LLC Figure 18-18 Branch in a steel penstock, showing a pair of crotch plates in pipes of equal diameter (left) and a stiffener ring that is a curved plate (right) For high pressure, it is possible to add a flat plate as shown dotted It may be advisable to contruct a small scale model — to check dimensions, to note any fabrication or assembly problems, and to test the special section to rupture ©2000 CRC Press LLC BRANCH SECTIONS Branch sections also require reinforcing by crotch plates or stiffener rings Figure 18-18 is a sketch of two common branches which are explained in "Buried Steel Penstocks; Steel Plate Engineering Data — Volume 4," published by the American Iron and Steel Institute Design procedures are explained, and graphs are presented for detailing this text However, once costs are known, the safety factors (zones of safety) can be identified In legal action, comparative damage is based on the percent encroachment of each of the adversaries into the zone of safety For example, a buried pressure pipe joint ruptures To what percent did the manufacturer, designer, welder, and installer each encroach into the zone of safety? PROBLEMS OTHER SPECIAL SECTIONS Wyes and branches are not the only special sections However, most special sections are variations of the wye For example, a tee (T) is a junction of three pipes just as a bifurcation (wye) is a junction of three pipes The junction can be reinforced with ring stiffeners and crotch plates in the same way as the wye The development of special sections has been mostly empirical Because specials are costly, the attitude of manufacturers has been, "Make it stout Our product must not fail." For example, instead of stiffening mitered sections with rings, the thickness of the steel is increased Competition is forcing reconsideration of this policy For costly specials, or specials for which failure is costly, physical tests are performed Small scale model tests are often adequate The conditions for similitude discussed in Appendix C are not sufficient for hydrodynamic model studies However, hydrodynamic similitude is described in most texts on hydrodynamics Hydrodynamicists have led the world in physical model studies SAFETY FACTORS Safety factors are based on monetary equivalents Analysis of the cost of risk is outside the scope of ©2000 CRC Press LLC 18-1 What are the major and minor diameters of the crotch cut of Figure 18-1? 18-2 What should be the diameter of the three steel outflow pipes of a trifurcation from a meter steel inflow pipe if all outflow rates are equal and the flow velocity is to remain constant? (1.732 m) Given: mitered wye in a welded steel water pipeline for which, Di = 51 inches = inside diameter, inflow pipe Do = 36 inches = inside diameter, outflow pipes P = 196 psi = design pressure, includes surge Taper = 9o = taper angle of the transition cones 18-3 What should the wall thickness be if the allowable hoop stress in the steel is 20 ksi? (t = 0.250 inches) 18-4 What is the length of the two transition cones? (LT = 47.35 inches) 18-5 What is the w-force on the crotch plate? (w = 6.6 k/in) 18-6 What should be the miter cut angle in the transition cones if the outflow ends are to be separated by inches to allow for welding? ... reduction in outflow areas from perfect gives a slight increase in flow velocities into the turbines This is desirable from the standpoint of turbine efficiency From guideline 2, the inside offset... Di = = ID = inside diameter (nominal for steel) inside diameter of inflow pipe (mainline) ©2000 CRC Press LLC inside diameter of outflow pipes (branch) wall thickness radius of bend in the pipe... the ring Crotch Plate I/c Ao Qo r M σ σ = 571 in = 75 in = 506 kips = 57.8 mean = 20,333 kip inches = 2Q o /2Ao + M/(I/c) = 6.75 + 35.61 = 42.36 ksi Ring B 229 in 38 in 235 kips 59 mean 70 inches