Is Code-Guidelines for Branching of Penstocks in Hydropower Projects

7/27/2019 Is Code-Guidelines for Branching of Penstocks in Hydropower Projects

1/24

PRELIMINARY

DRAFT CIRCULATION

NOTICE

Our Ref: Date

WRD 14/T-39 31-01-2008

TECHNICAL COMMITTEE: Water Conductor Systems Sectional Committee,WRD 14

------------------------------------------------------------------------------------------------------------

ADDRESSED TO: All Members ofWater Conductor Systems Sectional Committee,WRD 14

Dear Sir(s),

As per the decision taken in the 8th

meeting of the sectional committee, we are posting the

draft standard as mentioned below on the BIS website www.bis.org.in for your ready reference.

Doc.WRD 14(496) Guidelines for Design of Branching in Penstocks for Hydro

Electric Projects

Kindly examine this draft and forward your views stating any difficulties which you are likely toexperience in your business or profession, if this is finally adopted as a national standard andkindly provide your specific suggestions for revising the same in view of latest technology.

Last date for comments :15 March 2008

Comments, if any, may please be made in the format attached to the draft and mailed to

the undersigned at the above address. Comments will be appreciated in electronic form at the

email address [email protected]. In case you have any difficulty in accessing the documentat our website, please write to us for a hard copy.

Thanking you,

Yours faithfully,

[ Bhavana Sharma ]

Asstt Director (WRD)

Note: Please inform your e-mail address for faster communication at the e-mail address above

For official use onlyDoc. WRD 14(496)

February 2008
http://www.bis.org.in/


2/24

BUREAU OF INDIAN STANDARDS

PRELIMINARY DRAFT

Indian Standard

GUIDELINES FOR DESIGN OF BRANCHING IN PENSTOCKS FOR HYDRO

ELECTRIC PROJECTS

__________________________________________________________________________

_

(Not to be reproduced without the Last date for receipt

permission of BIS or used as a of comments is

STANDARD)

FOREWORD

(Formal clauses to be added later)

Different types of branching like bifurcation, trifurcation etc are used in penstocks carrying water from

surge tanks or reservoirs to the power houses.This standards covers guidelines for design of branching in

Penstocks for Hydro Electric Projects.

For the purpose of deciding whether a particular requirement of this standard is complied with, the final

value, observed or calculated. Expressing the result of a test or analysis, shall be rounded off in accordance

with IS 2:1960 Rules for rounding off numerical values (revised). The number of significant places

retained in the rounded off value should be the same as that of the specified value in this standard.

1. SCOPEThis standard covers guidelines for design of branching in Penstocks for Hydro Electric Projects. It covers

Types of Branching, Requirement for Design, Types of Reinforcements, Analysis of Wyes, AnalyticalDesign of Internal Sickle Plate Type Bifurcation and Design of Spherical Branch.

2. TERMINOGY: Definition for some of the key terms used in the Code are givenbelow for ready reference:

2.1 Branch: A penstock is generally called a branch when the flow of water is to bedivided into two or more branches, or when two or more flows are to be gathered to

a main pipe.

2.2 Wye Branch: It is a branch to diverge a main pipe to two branch pipes, attachingstiffening girders on their intersection lines.

2.3 Symmetrical and Unsymmetrical Branch: In a symmetrical branch, as shown inthe Fig.1, below, angles a and b are equal. Otherwise, the branch is unsymmetrical

(see fig. 12).

Branch Pipe

Main PipeAngle a

Angle b

Crotch


3/24

Branch Pipe

2.4 Equibranch: When branches have similar diameter, the branch is known as equibranch.

2.5 Tie Rod: A tie rod is the structural member placed inside the pipe for support, at the branching. The

same is illustrated in Fig. 2 below.

2.6 Spherical Branch: It is a type to connect main pipe and branch pipes to a spherical shell throughreinforcing rings (see fig. 12).

2.7 Sickle Plate: It is a crescent shaped rib inside the branch pipe, to give strength at the joint. TheSickle Plate, shaped as an internal horse-shoe girder, is also called splitter plate. It is provided at

the intersection of the two branches for resistance against the forces being developed there.

2.8 Reinforcement: It is the support provided to counter the unbalanced forces acting on unsupported

areas at the branching junction.

3. TYPES OF BRANCHING

3.1 Geometrically, there are several types of branching possible, such as bifurcation,trifurcation etc. However, in practical applications, generally a bifurcation is

employed.

3.2The Wye Branching is the one in which the main pipe diverges into two branch pipes.In the Wye

branching, the following categories are available:

a) Wyes with sharp transition.

b) Wyes with conical transition.c) Wyes with tie rods.

d) Wyes with sickle.

3.3 In case of branching into 3 or more branches, the following types are available:

a) a type in which the main is pipe directly trifurcated

b) manifold type in which pipes are branched in the same direction in successionfrom a straight main pipe, andc) a type to combine bifurcation.

These are illustrated in the Fig.3.

3.4 Another type of branching used in Hydro Electric Projects is the Spherical branching in which the mainpipe is connected to the branch pipe through a spherical shell and having reinforcing rings. In India, so far

the application of Spherical branching has been rather restricted.

Fig.1.

Section of main

pipe at branchTie Rod

Fig. 2 Tie Rod


4/24

4. REQUIREMENT FOR DESIGN OF BRANCHING

For the safe design of a branching, the following considerations are relevant:

a) Hydraulics of Water Flow through the branchb) External Pressurec) Internal Pressure

4.1 Hydraulic Considerations for Flow of Water through the Branch

4.1.1 While designing the branch, the following hydraulic considerations are required to be taken care of,

during pressure flow conditions:

a. Head loss due to branch should be small.b. The total head loss in the penstock before and after the branching should be small.c. Turbulent and secondary flows should not be allowed to be generated.d. For equibranch, head loss in each pipe should be of similar value.e. In case the flow rate in one branch pipe changes, a large vortex or a hydraulic pulsation should not

take place in the flow of the other branch pipes.

4.1.2Design Considerations :

In order to achieve the above conditions, the following design considerations are

recommended:a. In case of equibranch, the angle of branching should be kept symmetrical about the main pipe axis,

i.e., symmetry should be ensured in case of equibranch. In the context of Fig. 1, angle a and angle

b should normally be equal.

b. The head loss coefficient due to branching varies to a large extent due to the distribution ratio ofthe flow. It has been observed that the head loss coefficient is minimal when the angle between the

equi-diverging branches is between 45 to 600, and it rises sharply with increase of angle beyond

this limit. Therefore, in the context of Fig.1, the sum of angle a and angle b should normally be

between 45 to 600.

c. Sharp changes in cross sectional area of a passage should be avoided as far as possible.

d. While deciding on the type of support for a branching, the fact that any hindrance to the flow leadsto significant increase in head loss, should be taken in to view.e. Right angle branches and cylindrical outlets or inlets should be avoided where hydraulic efficiency

is important or cavitation cannot be allowed..

f. While selecting Tie Rod type of support at the branching, the fact that the head loss coefficientincreases largely due to presence of a tie rod, should also be considered. Tie Rod is an obstruction

to the flow in the penstock as shown in Fig.1A .

g. Similarly, the head loss coefficient largely increases due to presence of obstruction to the flow inthe penstock, caused by sickle in Sickle type branching.

h. The velocity of flow in the branches is to be selected so that the Reynolds Number (Re) is greaterthan 104.

i. The use of conical connections with side-wall angles, F , equal to 6-8 degrees, reduces hydrauliclosses to about one third of those resulting from use of cylindrical connections. Therefore, in

practical applications, appropriate conical angles are generally implemented.

4.1.3 Additional Hydraulic Considerations for Spherical Branch:

Apart from the overall hydraulic considerations given above, for the spherical branch the following

additional aspects are also required to be considered suitably:

a. The ratio of sphere diameter to main pipe diameter should not be kept very high in order to limitthe head loss coefficient. When the flow distribution ratio of a branch pipe becomes high, i.e.,

when the % of flow in one branch is much higher than the other branch, the head loss increases

rapidly. However, from construction point of view, it is not desirable to employ a spherical branch


5/24

having too small a diameter compared with the main pipe. Therefore, the ratio of sphere diameter

to a pipe diameter of 1.3 to 1.6 should generally be used.

b. It is normally preferable to install a flow regulating plate inside. However, thismethod is insufficient when the ratio of sphere diameter to a main pipe diameter is

larger than 1.6.

c. While designing the Diameter of the spherical branching, it is to be kept in view

that the head loss in spherical wye increases rapidly with increasing diameter ofwye.

4.1.4 Loss Coefficient for Bifurcation

a) The head loss due to branch, H, can be expressed in the following equation:

H = ag

v

2

20

where v0 is the mean velocity of flow in the main pipe, and a is the head loss

coefficient. Values of a are influenced by the branch angle, change in the sectional area,

distribution ratio of the flow to each branch pipe, and the Reynolds Number. Anestimation of the head loss coefficient for different branch angles can be made from Fig.

4, while influence of Reynolds number of main pipe over the head loss coefficient, incase of conical wye having equal distribution amongst the branches, is given at Fig. 5.

b) The head loss coefficient for conical wyes and manifolds, with various types of

transitions, and with/ without tie rods is given in Fig. 6, wherein Open branch refers to

branch where no gate is provided, while Closed branch refers to branching having gates

for regulating flow through the branch. An estimate of the head loss in spherical wye withincreasing diameter may be made from Fig.7.

4.1.5 Loss Coefficient for a Trifurcation

A trifurcation is illustrated in figure 12. The loss coefficient for a trifurcation can be

given as 3

2

3

2

2

2 SinQ

QSin

Q

Q

mm

Entry Loss Coefficients

where Qm is the Discharge through main pipe, Q2 and Q3 are the discharges through the branches, and 2and 3 are the horizontal angle of take off (see Fig.8. below). However, the losses would be higher than that

calculated by the above formula, in case when one or more branches are closed.

Fig.8

Main Pipe, Qm

Angle 2Angle 3

Branch Pipe, Q3Branch Pipe, Q2


6/24

5 TYPES OF REINFORCEMENTS

5.1 To compensate for the openings for the branching made in the normal circular section, some

reinforcement is required to be provided to take care of the unbalanced forces acting on unsupported

areas resulted by these junctions. These reinforcements can be either inside the pipes (internal) or

provided externally at the junction (external).

5.2External Reinforcements

Based on the extent of unsupported area, internal water pressure, ratio of main and branch pipe

areas, clearance restriction for fabrication etc., various types of external reinforcements are possible.

Most common practice is to provide one or more exterior girders welded with tie rods or ring girder

or a combination of these. The various types of external reinforcement are:

a) one plateb) two plate reinforcement, and

c) three plate reinforcement.

The placement of reinforcements in case of one plate, two plate and three plate reinforcements is

shown in the figure 9 below.

ONE PLATE

REINFORCEMENT

THREE PLATE REINFORCEMENT

TWO PLATE

REINFORCEMENT


7/24

The exterior girder, also called as yoke, is of horseshoe shape which is welded to the periphery of the

junction of the pipes and finally welded to a tie rod or a ring girder provided at the beginning of the

bifurcation. The section of the girder may be T shaped attached to the penstock surface. Someportion of the penstock steel liner also is assumed to act monolithically as a flange of the yoke girder

as in the case of stiffener rings (see Fig. 10).

5.3Internal Reinforcements

Another type of reinforcement is to provide an internal horse-shoe girder called splitter or sickleplate. This type has been developed by Escher Wyes and generally consists of crescent shaped rib

inside the branch pipe and it is designed in such a way that rib is directly subjected to tension and

has the same magnitude as the stress in shell section of pipes adjacent to it. Apart from being

structurally strong, this type is more economical because of smaller external dimensions taking lesser

space and enables large branch pipes to be fabricated, transported, stress relieved and erected as a

single unit (see Fig.11).

5.4Spherical Branch

It is a type to connect main pipe and branch pipes to a spherical shell through reinforcing rings. This

type is normally treated as an axis symmetrical shell, and it is possible to decrease the local bending

stress of a spherical shell connected to a reinforcing ring if its cross section is properly selected.

Therefore, the plate thickness is comparatively less. The arrangement of a spherical branch, vis--vis

a wye branch is brought out in Figure 12.

5.5 Selection of type of Branch

On account of higher plate thickness in case of wye branch, normally it is preferred for low Design

Head. For comparatively higher discharge values or heads, Spherical branch is normally preferred.However, no strict rules are available and the selection of type of branch is generally left to the

discretion of the designer.

6.0 ANALYSIS OF WYES

6.1 As mentioned above, at the junction where the main pipe diverges into two branch pipes, the attaching

stiffening girders on their intersection lines can be carried out internally or externally.

6.2 The forces on a bifurcation with external reinforcement, are shown in Fig.13. Some typical examples of

reinforcements for non-wye branching are illustrated in Fig. 14, while examples for wye branching are

given in Fig. 15.

6.3 The method of stress analysis used for branch outlets and wyes is approximate, with simplifying

assumptions. The reinforcement is proportioned to carry the entire unbalanced load as indicated by the

loading diagrams in the figures above. The total load carried by the reinforcement is equal to the product of

the internal pressure and the unsupported area projected to the plane of fitting. A portion of the pipe shell is

considered to be acting monolithically with the girders as in the case of stiffener rings.

6.4 Allowable StressesFor branching, it is prudent to use lower allowable stresses as compared to the penstock, on account of

limitation of the designers to determine critical stresses in these complex structures with the same degree of

FIGURE 9 - ONE PLATE, TWO PLATE AND THREE PLATE

REINFORCEMENTS


8/24

accuracy as is possible for the penstock. Consequently, these allowable stresses are based on one half timesthe minimum specified yield stress or one-fourth times the minimum specified tensile strength, whichever

is less.

6.5 Analysis of External Reinforcement

6.5.1 In case of simple curved reinforcing plate, it is assumed that the plate is acting as a plane ring and theloads in both directions are uniformly distributed and the plate is circular.

6.5.2 When the external girders are used in combination with tie rods or ring girders, the analysis becomesstatically indeterminate. For analysis, the deflection of the girder at the junctions of the tie rods or ring

girder under the load in both directions are evolved and equated to the deflections of the tie rods or ring

girders.

6.5.3 The loads considered for external reinforcement are shown in the Fig.13. As seenfrom the figure, the yoke is considered as an elliptical cantilevered beam. It is assumed to

be loaded by vertical forces varying linearly from zero at X=0 to P (r1 cos 1 + r2 cos 2) at

X=L and by the forces V1 and V2 due to tie rod at 0 and C and by the moment M (see

Fig. 15).

6.5.4 Analytical Design of External Reinforcement:The method of stress analysis used for branch outlets and wyes is approximate. Simplifying assumptions

are made in the analysis that yield results of efficient accuracy for practical design purposes. The

reinforcement is proportioned to carry the entire unbalanced loads. The total load carried by the

reinforcement is equal to the product of the internal pressure and the unsupported area projected to the

plane of the fitting. A portion of the pipe shell is considered to be acting monolithically with the girders,

similar to the stiffener rings.

The external stiffener(s) (see fig. 9) are analyzed as a C-clamp with a portion of the pipe shell considered as

an equivalent flange. The width (W) of the equivalent flange may be obtained from the formula :

56.1 RtaW

where a is the thickness of girder, R is the radius of the main pipe and t is the shell thickness. The

distribution of the design load in the case of one plate external reinforcement is shown in fig. 15 (b). The

increase in the bending stress, if the radius of curvature at the crotch is small, can be evaluated using a

correction factor in the bending formula for straight beams.

If the external girder is used in conjunction with the ring girder (see fig. 15(c)), the same is statically

indeterminate. To simplify analysis, the ring girders are assumed to be free at the common intersection

point, and loaded with the triangularly distributed design load and the unknown shear load concentrated at

the intersection point. The deflection of each girder is calculated with the direction of the unknown shear

force assumed. The deflection of all intersecting members are equated and the shear forces calculated, and

the direct and bending stresses at any point along the girders and ring may be determined. With the

elongation known, the stress in the tie rod can also be determined.

The sample computation sheet, with figure 13 as reference, illustrates the steps taken in the analysis of a

typical external reinforcement analytically. The same is placed at Annex 2.

6.5.5 The typical design of one plate and two plate external reinforcement usingNomograph:As an alternative to the complex calculations involved in analytical method, a simplifiedgraphical method has been devised, which has simplified the process of design of


9/24

external reinforcement to a great extent. The steps involved in use of Nomograph for

design of external reinforcement are illustrated below. While wye depth dw and basedepth db refer to two plate design, dw and db refer to one plate design:

Step 1. Lay a straight edge across the nomograph through the appropriate points on the

pipe diameter and internal pressure scales. Read off the depth of plate (d) from its scale.This reading is the crotch depth (for understanding crotch, see fig.-1) for 1-inch thick

plate for a two plate 90 degree wye branch pipe.

Step 2a. Based on the deflection angle, use the N factor curve (Fig.17) to get the factorsthat, when multiplied by the depth of the plate found in Step 1, will give the wye depth

dw and base depth db for the new wye branch.

Step 2b. If the wye branch has unequal diameter pipe, the larger diameter pipe willhave been used in steps 1 and 2a, and these results should be multiplied by the Q factors

found on the single plate stiffener curves (Fig.18) to give wye depth dw and base depth

db. While Qw is to be multiplied with dw to get dw, Qb is to be multiplied with db to

get db. These factors vary with the ratio of radius of small pipe to the radius of the largepipe.

Step 3. If the wye depth, dw found so far is greater than 30 times the thickness of the

plate (1 in.) then wye depth dw and base depth db should be converted to conform to agreater thickness t, by use of the general equation:

1dd360

917.01

t

t

in which d1 is the existing depth of plate, t1 is the existing thickness of plate, d is newdepth of plate, t is the new thickness of plate selected, and is the deflection angle of the

wye branch.

Step 4. To find the top depth, dt (for two plate design) or dt(for one plate design), useFig. 19. This dimension gives the top and bottom depths of plate at 90 deg from the

crotch depths (see figure 20).

Step 5. The interior curves follow the cut of the pipe, but the outside crotch radius in

both crotches should equal dt plus the radius of the pipe for two plate design, or, in the

single plate design, dt plus the radius of the smaller pipe. Tangents connected betweenthese curves complete the outer shape.

The important depths of the reinforcement plates, dw, db and dt (see Fig. 20) can befound from the nomograph. If a curved exterior is desired, a radius equal to the inside

pipe radius plus dt can be used, both for the outside curve of the wye section and for theside curve of the base section.


10/24

6.6 Three Plate Design

In the case of three plate external reinforcement, as shown in Figure 9, the function of the

third plate is to act like a clamp in holding down the deflection of the two main plates. In

doing so, it accepts part of the stresses of the other plates and permits a more economicaldesign. This decrease in the depths of the two main plates is small enough to make it

practical simply to add a third plate to a two plate design. The two factors that dictate theuse of a third plate are diameter of pipe and internal pressure. When both of these are

above certain limits, a ring plate may be used advantageously. These limits may be

considered as a nominal diameter of 1500 mm and a design pressure of about 2 N/ mm2.

If either of these limits is exceeded, the designer may elect to use a third plate.

The size should be dictated by the top depth (dt). Since the other two plates are flush with

the inside surface of the pipe, the shell plate thickness, plus clearance, should besubtracted from the top depth. This dimension should be constant throughout, and the

plate should be placed at right angles to the axis of the pipe, giving it a half ring shape. Itsthickness should be equal the smaller of the main plates.

7.0 DESIGN OF INTERNAL SICKLE PLATE TYPE BIFURCATION

7.1 The design of internal sickle plate is based on the principle that the stresses which occur in the line of

intersection of two parts of pipes are transmitted to a strengthening collar which lies in the plane of

intersection.

7.2 The design of strengthening collar involves firstly, the determination of resultant of all forces from the

beginning of the collar to the point in question, determined according to size, direction and position for

various points along the line of intersection. The cross-section of the strengthening collar which passes

though this point and is at right angles to the resultant, is made symmetrical to this resultant and

proportional to its size. This results for the strengthening collar in a body of constant strength, which isonly subjected to normal stresses, i.e., with an absolute minimum of material requirements.

7.3 Considering the requirement of equal thickness for ease of fabrication purpose, the collar is obtained inthe shape of sickle, which lies symmetrical to the line of intersection at the crown right inside the branch

piece and at its apexes.

7.4 For designing the strengthening collar, the size, position and direction of the resultant forces which are

transmitted at various points from the pipe walls at the line of intersection to the strengthening collar, must

be known.

7.5 Escher Wyss has developed a purely analytical method for design of strengthening collar. Fig. 21- 24show half of the sickle shaped strengthening rib and also the resultant forces. The coordinates of the

intersection curve are obtained as follows:

7.5.1 The coordinates of the intersection curve (x and y coordinates) are obtained as below. The Notationsare explained below and also in the Fig. 21to Fig.24 :

R1 Radius of the main pipe

- Half angle of bifurcation

f Cone angle

- angle varied from 0 oat vertical to 90 0 at horizontal, in steps of 2.5 0


11/24

x = rsin

sin; y = r cos , where r is given by r =

CotSin

R

tan1

1

Similarly, z is given by z = R1 Sin Cot

CotSinTan .1

Therefore dz = R1 Cos Cot d2

.1 CotSinTan

Therefore x =CotSinSin

SinR

tan1

.1

and y =CotSin

CosR

tan1

.1

Length of the Sickle Plate:

Projection of the intersection curve on the horizontal plane is obtained by putting = 2/ in value of xabove.

x 2/ =CotSin

R

tan1

1which gives the length of the sickle plate.

7.5.2 The pipe walls transmit forces at the point of intersection from both sides on to thestrengthening collar which lies in the plane of intersection AB (see Figure 22-24). Onaccount of their symmetry the resultant of these forces must always fall in the plane of

the intersection. It is assumed that the wall of the pipe are so thin that they can beconsidered as membranes and therefore possess no resistance against bending. Hence,they inflict only tractive and shearing forces on the strengthening structure in the case of

internal pressure.

7.5.3 With an internal pressure p, the forces per unit length in a cylindrical membrane are:

in circumferential direction =cos

pr

in axial direction =

cos2

pr

7.5.4 In an element of the line of intersection of length dl the following forces areinflicted on the strengthening from one side:

(a) As a result of the circumferential stresses :

Pl =cos

prdz {which can be seen in Fig.24}


12/24


13/24

y

x1tan

HV1tan

7.5.7 As seen from the geometry of the triangle RVH in figure 21, when the branches are cylindrical (i.e.,

R1 = R2), and hence the resultant R will be perpendicular to the line OF (see Fig. 21). In the case

of Conical branches, and therefore R has components giving rise to a normal stress fn and shear

stress qt on the plane passing through O.

The principal stresses, to be kept within permissible limits, are

ft = fnqt 2

1

4fn2

1 22

and fn = fnqt2

14fn

2

1 22

To determine the position of the resultant R, moments of the forces with respect to O are calculated as

below:

Moment of Vertical Forces Mv =

0.dVx and MH =

0.dHy

The integration of V, H, Mv and MH are done numerically using Simpsons Rule or other such numerical

techniques.

Total Moment = Mv + MH

The distance of the resultant from O is given by l =R

M.

For ,2/ l 2/ =2/

2/

R

M

7.5.8 Theoretical width of sickle plate at its crown, i.e., 2/ , is given by the following expression:

2

Bax - l 2/

butax = x 2/ =)tan1(

1

CotSin

R


14/24

However for practical consideration, since additional width is required to be provided to project outside the

intersection, the actual width is enhanced by a factor Ax where Ax ranges from 30 to 50mm.

7.5.9 Width of the plate at any other section is then calculated in proportion to the value of R , i.e., b = B

2/R

R

The thickness of sickle is such that the principal stresses are within acceptable limits. Theoretical estimate

of the sickle thickness S may be taken as

S =31

2/2

BpR

tR

7.6A procedure for design of internal sickle plate using analytical method given above, is enclosed at

Annex 1 in a tabular form.

8 Design of Spherical Branch

8.1 When a model consisting of a sphere and a cylinder having cave cover as Fig. 25 is considered :

s HC =2

pr

where p is the internal pressure and r is the radius of the connecting main branch.

The horizontal component of a spheres membrane tensile force pr/2 is :

s HS =2

pacos =

2

pr, where a is the radius of the sphere and angle is the angle from vertical to

the point where the cylindrical main branch meets the sphere.

Thus, the horizontal direction force is kept balanced with sHC = sHS

8.2 On the other hand, a reinforcing ring is attached so as to resist the vertical component s v of aspheres membrane tensile force as illustrated in Fig.26. The tensile force T1 generated in a reinforcing ring

by the internal pressure acting on the reinforcing rings breadth b is T1 = prb, where b is the width of the

reinforcing ring . The tensile force T2 generated in a reinforcing ring by the vertical component of a

spheres membrane tensile force is T2 =2

1 pro a cos , where r0 is the radius of the C.G of the reinforcing

ring, and is the angle shown in Figure 26.

8.3 Supposing only the membrane tensile force acting on a sphere, i.e. a sphere under membrane stress

condition, the cross sectional area Sof reinforcing ring is :


15/24

S =

00

21

rr

PT(br + 0.5 ro a cos )

where0r

: Tensile stress of reinforcing ring.

This formula includes the stress of a reinforcing ring 0r but if is possible to determine

the formula which does not include 0r from a displacement boundary condition as

follows:

The radial force Vacting on a reinforcing ring is :

V = pb +2

pa cos =p( b +

2

acos

Then, the displacement dr , of a reinforcing ring in radial direction is :

cos2

sin 222 abp

SE

aV

S

rr

, where E is the modulus of elasticity and S is

the cross area of the reinforcing ring.

The displacement dS, of a sphere in radial direction at connecting point with reinforcing

ring is :

sin12

2

Eh

pas

where h is the spheres plate thickness.

dr = dS is to be essential in order that a sphere is under the membrane stress condition, which gives

sinhcos2

ab

1

2S

The spheres plate thickness h is, saying the spheres membrane tensile stress s so

h =

so2

pa

While, a cylinders plate thickness t is to be determined only by the tangential stress s po with noconsideration given to the reinforcing rings effect :

t =

po2

pr

If thickness of cylinder t is to be determined so that the radial displacement dc of cylinder may be equal

to the radial displacement dr of a reinforcing ring, it is not required to consider the effect of a cylinder of

a reinforcing ring and the sphere, and the sphere can be kept under a membrane stress.

11

'

2

Et

pr

c


16/24

sin12

2

EL

pa

sr

8.4 With dc = dS, rSina. , = 0.3, and the corrosion allowance is expressedwith e, the cylinders plate thickness required is expressed with the following equation :

ha

rt 43.2'

The range to increase t to t is determined from the following equation :

3.1

2 '' trb where b is the new width of the reinforcing ring.

8.5 Practically, it is seldom to satisfy the above equations in computation of S, t and h, and so it isnecessary to calculate a bending moment and a shearing force at each point by a statically indeterminate

calculation method etc., to attempt a strict solution.

8.6 When considering an actual use of steel penstocks, concave covers do not exist, and thus the axial

forces HS and HC in Fig.25 do not act. As a practical solution in this case, there is a concept to

solve a structure model shown in Fig.27 with an assumption that a pipe is embedded in concrete and

the pipes axial displacement is restrained and fixed at a certain distance point from a reinforcing

ring. Internal pressures acting on the branch are explained above, but it is also necessary to examine

the external pressure.

9.0 DESIGN OF BRANCHES BY NUMERICAL TECHNIQUES

The Wye piece, designed based on the above mentioned criterion is an indeterminatestructure. It is necessary that stress concentration occurring at the various intersection

points are ascertained and suitable strengthening measures carried out, if found

necessary. For this either physical model studies / photo elastic studies or mathematicalmodeling deploying methods as Finite Element etc. with appropriate boundary and

loading conditions, are deployed.


17/24

ANNEX -1

PRACTICAL PROCEDURE FOR DESIGN OF INTERNAL SICKLE PLATE

Data to be provided :R1 Radius of the main pipe

F - angle of cone half angle of intersectiont thickness of sickle plate (assumed to be checked through stresses)

Table 1

1 2 3 4 5 6 7 8

Alpha

(deg)

Alpha

(Radia

ns)

r =

CotSin

R

tan1

1

x =

rsin

siny =

r cos y

x1tan

Radians

t1=

tansintan1

Radians

xR

y

cot1tan1

1

Degree

R

(x

0

2.5

5.0

..

..90

Table 2

OUTER & INNER PROFILE OF THE SICKEL PLATE

1 2 3 4 5 6 7

Alpha (deg) xo yo x y xi yi

= x + h Sin = y + h Cos = x b Sin = y - b Cos

0 (b to be determined

subsequently)

2.5

5.0

..

..90

Table 3

CALCULATIONS OF VERTICAL FORCE

ALPHA= X dV/dX dV avg dV V K

cos

1sin

cos

cossincot2[

23

KK =[ dV/dX (R1)2

] x[2(a

susequent a

current)] = 0.5(dV

prev.+dV

current) = S avg dV

=

1+tan sin

0

2.5

5.0..

90

Table 4

CALCULATIONS OF HORIZONTAL FORCE

ALPHA dH/dX dH avg dH H Resu


18/24

= X

cos

)1cos(1cos

cos

sincoscot2

23

2

KK =[ dH/dX (R1)2

] x[2(a

subequenta

current)]

=0.5(dH

prev.+dH

current) = S avg dH

R =

2V

0

2.55.0

7.5

..

90

Table 5

CALCULATION OF VERTICAL AND HORIZONTAL MOMENT

1 2 3 4 5

ALPHA= X dM V /dX dMV avg dMV M V

sincos

1sin

sincos

cossincot2

[ 34

2

K

Sin

K =[ dMv/dX ] x[2(a

susequent a

current)] = 0.5(dMv

prev.+dMv

current) = S avg dMv

0

2.5

5.0

7.5

..

90

6 7 8 9

dM H /dX dMH avg dMH M H

cos

)(1

cos

cot2[

34

3

K

TCosCosCos

K

CosSin

=[ dMH/dX ] x[2(a

susequenta

current)] = 0.5(dMH

prev.+dMH

current) = S avg dMH

10 11 12 13 14

M=Mh+mVResultant =

pR/36Resultant Moment =

(R1)3pM/36 M/R b

Col.12 x 1000

Col.11

B*(Resultant)1000*(Resultant

at alpha 90)

B = 2 (x at alpha 90 * 1000 M/R at alpha 90)

TABLE 6


19/24

CALCULATION OF THE VALUES OF PRINCIPAL TENSILE AND COMPRESSIVE STRESS

1 2 3 4 5 6

Alpha Gamma Theta Theta - Gamma Normal Stress Shear Stress

Tan-1

(V/H) From Table 1 Col.3-Col2Resultant (from col.11 oftable 5 above) *Cos (col.4)

Resultant (from col.1table 5 above)* Sin (c

02.5

5.0

7.5

..

90

7 8 9 10

Fn Qt Pt Fc(Col. 5)*100000/(thickness in cm * b fromtable 5 * 1000)

(Col. 6)*100000/(thicknessin cm * b from table 5 *1000)

(Col. 7) + sqrt(col.72

+ col.82)

2(Col. 7) - sqrt(col.7

2+ 4*co

2


20/24

ANNEX 2

SAMPLE COMPUTATION SHEET FOR ANALYTICAL DESIGN OF EXTERNAL

REINFORCEMENT (SEE FIG. 13 AND FIG. 13-a)

NOTATIONS:

1. AR Cross sectional area of circular ring

2. AU Cross sectional area of U beam of section considered3. E Modulus of elasticity of steel4. IR Moment of inertia of circular ring5. IU Moment of inertia of U-beam at section considered

6. IUC Moment of inertia of U-beam at centre of element s

7. MU Bending Moment of U beam at section considered8. IUC Bending Moment of U beam at at centre of element s due to reaction between circular

ring and U beam, and unbalanced pressure

9. R Radius of main pipe

10. R1 Radius of branch pipes at point M

11. R2 Radius of centroidal axis of circular rings

12. R3 Radius of curvature of inside of U beam at point considered13. T Total vertical shear in U-beam at section considered

14. Y Reaction between inner circular ring and external U-beam girder.

15. b1 Effective width of inner ring (equivalent flange)

16. b2 Effective width of U-beam (equivalent flange)

17. c Distance from centroidal axis to extreme inner fiber of U beam

18. c Distance from centroidal axis to extreme outer fiber of U beam

19. e Distance from centroidal axis of extreme inner fiber of inner ring

20. e' Distance from centroidal axis of extreme outer fiber of inner ring

21. ki, ko Ratio of actual stress in inner or outer fiber, respectively, to stress computed by flexureformula for straight beam

22. mc Bending moment of U-beam at center of element s due to unit load at C.

23. p Internal pressure

24. q Constant for steel25. Poissons ratio for steel

26. x Horizontal distance from centre line of circular ring

27. a Angle of cross section of U beam with vertical

28. dc Deflection of U beam at point C

Note: positive moments produce tension of inner fiber

+ sign of tension

- sign of compression


21/24

FORMULAE:

A. Calculation of Y

1. Deflection dc of U beam at point C

a. Moments and Vertical Shear (for a and L see Fig. 13) :

MUC = Yx for x from C to K T = -Y

MUC = YxCosaxpRL

2

1)(

3

1 for x from K to M T = YCosaxpRL

2

1)(

1

MUC = YxCosLaxpR )3

2(

1

for x from M to N T = YLCospR1

mc = x For x from C to N

b. Deflection dc =

UC

cUC

EI

smM

1 2 3 4 5 6

x MUC S MUC x S IUC Col.4/ Col.5

Edc = S

2. Deflection dc of circular ring:

a. Hoop Stress S1 due to internal pressure =q

dApR

R

2

12

RTq

1285. Find the value of q with this formula.

b. Radial deflection dc at point C:

Due to S1 Edc1 = (+) S1R . Find Edc1 based on this formula.

Due to Triangular load Edc2 = +0.0274R

I

pbRR22

Due to Y Edc3 = +0.0745

RI

YR2

3. Solve for Y : Edc = Edc = Edc1 + Edc2 + Edc3

b. Stress in U Beam:


22/24

1 2 3 4 5 6

x MU c c k1 IU

7 8 9 10 11 12

Bending Stress Combined stress

uI

ckMU 1

uI

kocMU ' T

UA

Tsin Inside Outside

c. Stress in Circular Ring:

Point MR e e IR Bending Stress

Si =

R

R

I

eM So =

R

R

I

eM'

C -0.07074pbR22 0.3183YR2 - +

E -0.065pbR22 + 0.0353YR2

F +0.6959pbR22 + 0.1817YR2 + -

Point Direct Stress Combined Stress

Inside Outside

C S1

ES1 = YpbR

AR

354.01768.01

F

S1 = YpbRA

R

54.05.01

4. Determination of Values for ki and ko:Case I ( See Fig. 13-a(a)-considering tie rod and inner circular ring)

A= b1t + t1h; h1 = r-R3 ; h2 = R6 r

r =

4

5

1

3

4

1R

RLogt

R

RLogb

A

ee

ki =c

I

AeR

hu

3

1; ko = '

5

2

c

I

AeR

hu

Case II( See Fig. 13-a(b)-considering tie rod, inner circular ring and external girder))

A= b1t + t1h + b2t2; h1 = r-R3 ; h2 = R6 r


23/24

r =

7

6

2

4

7

1

3

4

1R

RLogb

R

RLogt

R

RLogb

A

eee

ki =c

IAeR

h u

3

1; ko = '

5

2

cI

AeRh u

5. Determination of C, Rmax, Rmin and R3 pertaining to elliptical intersection of U-beam:

See Fig. 13-a (c). F and F1 are the foci of the ellipse. R is the Radius of curvature at any point P.

r1 = a +cx; r2 = a -cx; r1 + r2 = 2a;

c =a

ba 22; Rmax =

b

a2

; Rmin =a

b2

; R3 =

23

21

ab

rr


24/24

This document was created with Win2PDF available at http://www.win2pdf.com.The unregistered version of Win2PDF is for evaluation or non-commercial use only.This page will not be added after purchasing Win2PDF.
http://www.win2pdf.com/http://www.win2pdf.com/http://www.win2pdf.com/http://www.win2pdf.com/http://www.win2pdf.com/

Date post:	14-Apr-2018
Category:	Documents
Upload:	anonymous-mchqifbnv1
View:	225 times
Download:	0 times

Is Code-Guidelines for Branching of Penstocks in Hydropower Projects

Documents