Numerical Methods in Soil Mechanics 18.PDF

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).

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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

Notation and Nomenclature

D = ID = inside diameter (nominal for steel)

Di = inside diameter of inflow pipe

(mainline)

Do = inside diameter of outflow pipes (branch)

t = wall thickness

R = radius of bend in the pipe or cone

δ = offset angle of contiguous mitered sections

L = length of section of pipe

LT = length of truncated cone

θ = angle between axes of each branch pipe and the mainline pipe

Consider the horizontal cross section 0-0 at thecrotch of Figure 18-1 A free-body-diagram of half

of the cross section shows a rupturing force ofpressure times the span of the cut This force ismore than twice the rupturing force in each ring ofthe branch pipes and will cause ballooning of thecross section unless the rings are held together atthe center — either by a vane on the inside, or by acrotch plate on the outside A crotch plate is a C-clamp with an elliptical inside cut as shown in Figure18-2 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) theforce to be resisted by the crotch plate from each ofthe cone walls at the vertex section A-A-A, is PrAper unit length of the pipe Per unit length of crotchplate (or vane), the force on the crotch platebecomes:

w = 2PrA(cosθ) (18.1)where

w = vertical force on crotch plate (vane) per unit length of crotch plate

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 cutsare elliptical)

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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

Figure 18-4 Free-body-diagram of one limb of the crotch plate showing an approximate procedure foranalyzing 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

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α = angle on the cone cross section from

the vertical axis to the crotch cut

Li = 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' = 2PrB(cosθ)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, P rB, 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 7

1 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

2 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

keep δ < 7.5o or even < 6o for very high velocityflows

3 Keep the radius of the bend greater than 2.5times the pipe diameter (or mean diameter of anymitered cone section); i.e R > 2.5D = 5r It ispreferable to keep R > 3D or even > 4D for veryhigh velocity flows See Figure 18-7

4 Keep the length, Li, on the inside of the bend ofeach mitered section, greater than half the meanradius of the section (pipe or cone)

5 Keep the cone taper angle minimum The greaterthe taper angle, the shorter are the length s Li ofcontiguous cone (or pipe) sections This means asharper bend (shorter radius R of bend) On theother hand, the smaller the taper angle, the longerthe crotch plate must be Consequently, muchgreater loads must be supported by the cantileverlimbs of the crotch plate The crotch plate, a criticalstructural element of the wye, presents a dilemma

— the need for a large taper angle to keep thecrotch plate short, and the need for a small taperangle to keep the radius of the bend and the insidelengths, Li, within limits of hydrodynamic guidelines.From this point on, design is by trial Therelationship of the hydrodynamic guidelines to thestructural integrity of the wye are best described by

an example

ExampleConsider 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 waterunder high pressure and high velocity to two equalsized 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

Start with a trial wye — say Figure 18-1 Try ataper angle of 7.5o, for which the truncated length ofthe cones is LT = 113.94 inches To facilitatefabrication and welding, select the same steel

Figure 18-5 Wye with inside offset angles limited to 7.5 Note that branches do not clear each other at thelower end.

Figure 18-6 Same wye, but mitered; i.e., cut near midlength of the cone, rotated 180o, and welded Note thatthe branches now clear each other

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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 do not clear each other at the lower end In

order for the branches to clear each other, two

remedies are considered

1 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

2 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 θlow = 22.5o

It is noteworthy that the inside offset angle from

tapered cone to pipe is less critical than the insideoffset angle for bends in pipes because the taper-to-cylinder transition is a symmetrical squeeze-down offlow Bends are not symmetrical If one or theother has to be mitigated, the taper (rather than theinside offset angle in a bend) is allowed to exceedthe recommended maximum

Figure 18-7 shows how mitered bends are formed.Because a planar cut across any circular cone (orpipe) is a perfect ellipse, mitered bends can beachieved by cutting any cone (or pipe) at an angle of

δ/2 with the diameter, rotating one section 180o, andthen welding the cut The ellipses match Theresulting offset angle is δ

It is not always necessary to miter the mainline pipe.See the mainline-to-cone cut in Figure 18-7 Whenthis particular cone tilts to angle θ = 15o, itshorizontal radius is approximately the same as theradius of the mainline pipe Of course, the cut of thecone is an ellipse, but the ellipse is so nearly circular,that the cone and pipe can be pulled together forwelding If the ring cut were mitered, the stiffenerrings would come in at some angle such as the ringcut angle of 6o shown on Figure 18-1 For thetransition, upper stiffener ring A is a circle — aneasy cut See Figure 18-8 The lower stiffenerrings B (at the miter cuts in the cones) intersect atthe angles shown

Second TrialFigure 18-8 is the second trial wye for analysis anddesign Of primary concern is the crotch plate Inthis case, the length of the crotch plate limbs is 109.5inches This compares not too badly with a crotchlimb length of 94.8 inches for the first trial shown inFigure 18-1 A free-body-diagram of the force w onthe 109.5-inch limb can be calculated by means ofEquation 18.1 However, there are two values of θ,upper section and lower section For the uppersection, cos θ = cos 15o = 0.966 For the lowersection, cos θ = cos 22.5o = 0.924 Not justified isany attempt to interrelate the two by applyingEquation 18.1 over the upper and lower sectionsseparately Conservatively, we use the larger value,cos θ = 0.966, and we analyze the full

Figure 18-8 Second trial configuration of mitered wye showing the crotch plate and stiffener rings; and a fullcircle stiffener ring at A.

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-body-diagram with a straight line distribution of the

w-force 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

cross-hatched, 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

is statically indeterminate, depending upon the springconstants of the stiffener rings at A and B For atrue 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 plateare more difficult to evaluate because of theirshapes A reasonable simplification of the crotchplate for preliminary design is to assume that ring Awill, at least, prevent rotation of the crotch platecantilever limb at section B See Figure 18-11 Tothe left of section B, the crotch plate isapproximately a half ring, wherein section B doesnot rotate under load Consequently, the springconstant for the crotch plate can be analyzed by anequivalent circular ring To analyze the equivalentring, it is only necessary to neglect QA and to doublethe load QB on the equivalent ring The springconstant can be calculated from Equation 18.3 It isnoteworthy that simulating the crotch plate by a ringwith twice the Q-load on it, we are assuming thatthe limb of the crotch plate does not rotate at B Amore accurate analysis would prove that rotation at

B is small

Figure 18-12 is proposed as a reasonable trial crosssection of the crotch plate at section C-C Try a1.5-inch by 30-inch plate for a web with a 1 x 12-inch flange on the outside On the inside of the webare the steel cone walls double welded to the crotchplate and splitter plates securely welded to providethe equivalent of a flange and to provide abrasionresistance to head-on flow of water containingsediment The crotch cross section shown is not astandard I-beam section Nevertheless, assumingthat the walls of the cones and the splitter platescombine to provide an area