ANCHORAGES AND SUPPORTS

CHAPTER 93ANCHORAGES AND SUPPORTS

Both fixed and semirigid types of penstock are secured in place at points by a n c h o r b l o c k s . Anchorages represent the fixed supports of the penstock and are located at either vertical or horizontal bends in the line. All three possible types of anchor blocks are used for the penstock shown in Figs 1/93 and 2/93.

1. Forces transmitted by the adjacent penstock sections to an anchor block located at a point where the change in slope of the terrain, and thus the bend in the penstock, is concave when viewed from above, tend to displace the anchor block over the terrain. These forces, however, have a component normal to the terrain — the magnitude of which depends upon the angle of

CHAPTER 93Hand Hf(are the design heads at the points indicated

Fig. 1 ¡93. Location of anchorges and types of anchor blocks

27 Mosonyi

Conduit length L Length of one section lf -¿/H} Saddle spacing bfDead veight of bf tong section .* 60

Dead weight of section 1} -ISgof section 1**16%

402 CHAPTER 93

403 CHAPTER 93

the bend — and this favourably influences the stability conditions of the anchor block.

2. At bends, where the penstock deflects downwards, i.e. which are convex when viewed from above, the resultant of forces acting upon the anchor has a component directed away from the terrain and at such places large concrete volumes are usually required to ensure the stability of the block.

3. As already mentioned, intermediate anchor blocks spaced at least at 100 to 150 m are necessary at long straight penstock sections as well. These blocks are subject to a force caused by pipe thrust and approximately parallel to the terrain, and constitute thus an intermediate case between cases 1 and 2, as far as stability is concerned.

Let us consider in the following the forces acting upon an anchor block located as described under point 2, i.e. where the bend is convex and where stability presents the greatest problem. The block fixing the penstock section no. 2 in Fig. 1 ¡93, together with the forces acting upon it, is redrawn to a larger scale in Fig.3/93.

Axial forces transmitted to the anchor from the upstream part of length I2 may be listed as follows:

1. The dead weight of the pipe, according to Eq. (24/92),

Po = + £ Go sin p2 [kg]. (1/93)

ANCHORAGES AND SUPPORTS 404

(Forces acting in a direction towards the anchor will be denoted hereafter as positive, and opposite ones as negative.)

2. The friction force over the supports, according to (Eq. 25/92),

E P'f « db (i E (Go + G'w) cos [kg] (2/93)

where the “ + ” sign applies to forces due to temperature increase, while the “ — ” sign indicataes forces due to a temperature drop. These forces, as pointed out earlier, act along a line below the centreline of the penstock, yet, in order to simplify construction, may be transferred to the centreline since the error introduced thereby will be on the side of safety.

Fig. 2/93. Location of penstock on rugged terrain. Various kinds of anchor blocks are exemplified in the picture, a — simple intermediate anchorage, b — anchorage at concave vertical bend, c — anchorage at convex vertical

bend. (After a catalogue of Christiani & Nielson, Copenhagen)

405 CHAPTER 93

3. The friction force at the expansion joint packing, according to Eq. (29/92),

Pp~ ±pind1eyHt [kg]. (3/93)

The sign convention under point 2 applies.4. The direct water pressure at the expansion joint, according to Eq. (30/92)

P'=+ndidiyH' [kg]. (4/93)

Fig. 3/93, Forces acting upon the anchor block


5. The axial component of hydrostatic pressure acting on the section and due to the change in direction, is one of the most significant forces. The magnitude thereof due to water pressure prevailing in the section of diameter d\ is

K'= + yH fcd • (5/93)

6. The drag of flowing water caused by conduit friction is according to Eq. (32/92),

P'i= + y I'z /= + y AH' [kg], (6/93)

where AH' is the differential head in metres over the section of length l2.

7. An impulse force is exerted by the flowing water on the pipe wall. This force acts along the mitre line of the angle included by sections 2 and 3 and is directed outward, i.e. away from the terrain. The impulse force may be resolved into two components, both of which act along the centreline of the penstock. With J( denoting the mass of water flowing per second at design velocity i?i, the component acting along the centreline of section 2 is

P\=+Jtvi=y- vl [kg]. (7/93)g 4

Axial forces transmitted from the downstream part of length l2 may be listed in a similar manner. Forces directed towards the anchor will again be denoted as positive, while opposite ones — directed downstream — will be negative.

1. The dead weight of the pipe

Po=-£GZsmfo [kg]. (8/93)

2. The friction force over the support (or occasionally supports)

L Pf~ ±pE(GoJr Gv) cos [kg]. (9/93)

An increase in temperature gives rise, as agreed upon previously, to positive, whereas

407 CHAPTER 93

a decrease to negative forces.

3. The friction force at the expansion joint packing:

Pp - ± px 7i dt e y H" Peg] (10/93)

in keeping with the sign convention adopted.4. The direct water pressure at the expansion joint:

P" = + d2 S2 y H" [kg]. (11/93)

5. The axial component of hydrostatic pressure due to the change in direction is computed from the head prevailing in the section of diameter d2

K {di)ndl

4yH.

Pi^ ^2 j T Ttt

y------AH4

[kg] (13/93)

Pf- + y ndl .2 g 4 vi [kg], (14/93)


K-2£yH [kg]. (12/93)4

It should be noted that the axial water pressure due to the reduction in cross-section and considered during the structural analysis of the pipe, should not be allowed for separately in this instance. This force is already included in forces P'w

and P„ transmitted from section ¡2 and ({, Namely, P'w may be resolved into two components, one of which is that acting on the taper section (confuser)

Pc —AFyH— - yH,4

while the other is the water pressure acting on the projection of the reduced section

Obviously

6. The drag of flowing water includes a force

pointing downstream.

7. The component of the impulse force acting along the centreline of section 3

points upstream, i.e. towards the anchor block.

To check the stability of the anchor the parallelogram of forces should be drawn as in Fig. 3(93, assuming an increase in temperature, which is obviously a less favourable condition for stability than that of temperature drop.

409 CHAPTER 93

Starting from the point where the centrelines of the two penstock sections intersect, the forces pointing downstream

p^+rp}+p;+p;+p;+p;-fpi

(15/93)

are plotted on the extended centreline of the section preceding the anchor, while the forces pointing upstream

- Po + E P;+ Pf; + Pi+ Pi - P2 4- P" (16/93)

are plotted on the extended centreline of the section following the anchor. The vectorial sum of these forces is force Z, the resultant of all forces transmitted to the anchor from the adjacent penstock sections.


The resultant acting upon the anchor is influenced primarily by the effects listed subsequently in the order of their significance:

From upstream: the hydrostatic pressure due to the change in direction (P*),the direct water pressure transmitted at the expansion joint (Pelthe component due to dead weight (Pi>), and the frictional force over the supports (EP*ef}).The rest is usually too small to be of significance.

From downstream: the hydrostatic pressure due to change in direction (Pwlthe direct water pressure transmitted at the expansion joint

(#)■

Components due to dead weight and friction over the supports are usually negligible from among those acting upon the anchor from downstream, inasmuch as expansion joints are often located immediately downstream of anchor blocks. An arrangement frequently applied is to locate the expansion joint between the anchor and the first downstream support so that this component does not develop at all.

Both the shape and dimensions of the anchor block of width B should be adjusted until the resultant R of the force Z and of the dead weight G of the anchor does not pass within the middle third of the base of the block. With notations asin Fig. 3/93

x>A/ 3, and xo — A/2 — x and

the maximum pressure transmitted to the rock is

= (1 + [kg/cm2] • ( 1 7/93)

The downstream face of the anchor block is inclined for reasons of stability and the base is frequently stepped for economical reasons. Flat and stepped bases may be horizontal, yet may be constructed at a counter slope as shown in Fig. 3/93. This latter

411 CHAPTER 93

arrangement involves a larger excavation volume and requires more material, but offers increased safety against sliding. For the case illustrated, the safety against sliding may be expressed as

jiRsin 9~nRcos 9 (18/93)

where, neglecting the bond between the rock and the concrete and assuming good-quality rock, a sliding coefficient /¿=0.6 to 0.7 may be assumed. In rocks of power quality, showing a tendency towards saturation, sliding coefficients significantly lower, amounting even to but fractions of the above value, may be encountered. Safe resistance against sliding should be ensured very carefully whenever there is some doubt as to the magnitude of the sliding coefficient. Values of thelatter have been given in Chapter 69 (Volume I). The coefficient of safety should be at least n~ 1.5.

Simple petrographical investigations of the underlying rock are, however, un-satisfactory in themselves and should be accompanied already in the preliminary stages of designing by the careful exploration of the hill slopes selected for locating the penstock. In some instances the rock lying close to the surface and consequently to be taken into account for foundation may prove satisfactory as far as load-bearing properties are concerned, whereas the dip of rock strata towards the valley may be excessive, so that the anchor blocks may slide together with the underlying rock. At greater depths deposits may be encountered between the rock strata that may become unstable when overloaded. Hillsides that show evidences of rock slides or avalanches should be avoided.

Sliding of the top layers covering the hillside presented in 1946 a source of danger for the lower portion of the penstock and for one of the anchor blocks at the Cubatdo power station in Brazil. This was one of the reasons why engineers decided in favour of the underground arrangement for the proposed extension.

A lean mix containing from 150 to 200 kg cement per m3 concrete is generally used for the anchor blocks. For their foundation even a mixture containing 120 kg/m3 may be used.

When there are heavy loads to be supported, the simple embedding of the penstock into the anchor block may not prove adequate, and hold-down straps may be necessary to fix the pipe to the anchor as illustrated in Fig. 4/93. In order to prevent the covering concrete


layer from cracking, reinforcement is placed also into the latter. Hold-down straps, as well as the reinforcement around the pipe, should be dimensioned to resist the force Z determined previously. Flanges should be welded at close intervals to the pipe pieces to prevent it from sliding. An interesting solution was applied at the Rjukan power station in Norway, where the

Fig. 4/93. Anchor detail showing hold-down straps for penstock, Arnstein power plant. (After A.Schoklitsch)

Fig. 5¡93. Anchor block loaded by rock fill, Rjukan power plant, Norway. (After A. Ludin)

413 CHAPTER 93

upper part of the anchor block was designed to form a box filled with stone (Fig. 5/93).It may be seen in Fig. 6/93 that the resultant of forces transmitted to the anchor acts in

a more favourable manner at concave bends. Structural and stability analysis should follow a process similar to that described in the foregoing.

Bends, in which the direction changes simultaneously in profile and in plan, are referred to as combined (Fig. 7/93). If the angles made in profile by adjacent tangents with the horizontal are ¡h and #*, and the angle in plan is oc, the angle in space included by subsequent tangents may be computed as:

cos oco “ cos fii cos h cos a+ sin fii sin f}2. (19/93)

Penstocks laid side-by-side may in many instances be fixed more economically and practicably by a c o m m o n a n c h o r b l o c k instead of separate ones(Figs 8/93, 9/93, 10/93 and 11/93).

Fig6193. Forces acting upon the anchor block at a concave vertical bend


If the penstock tracing requires bends also in plan, the axial forces acting upon the structures in horizontal plane have to be determined. In large-angle curvatures,

415 CHAPTER 93

Fig. 7/93. Penstock with combined bends, Chanet power station, La Reuse River, Switzerland. 2=2.5 to 6.0 m3/sec, H= 73 m, AT=3300 kW. The penstock is 294 m long and of a diameter of 1.75 m. (After

a publication of the Schweizerische Wasserwirtschaftsverband)

Fig. 8j937 Anchorage of parallel penstocks by common anchor blocks. — Above: Aasgaard hydroelectric plant, Norway. (After G. G. Robb: Water Power 1949). — To the right: Pykara power plant, India.

(After R. D. Rajan: Water Power 1952)

Fig, 9J93. Lowest anchor block for the penstocks supplying the Cubatdo aboveground power station. Tiete system, Brazil. The weight of individual blocks in the line ranges from 1000 to 3200 tons.

(Photographed by the author, 1954)


namely, the resultant of the axial forces, especially that of the usually predominating hydrostatic loads, can be decisive for the design of the relevant penstock section (pipe, joint, anchor block). This aspect is important for dimensioning the lowest anchorage and the manifold too, as it will be shown here later. It also occurs that, for avoiding extremely high loads upon the anchor block, the pipe is not discontinued by a joint in the curvature, but, instead, has a sliding support upon the block. Such a solution was chosen for the penstock of the Santa Isabel plant (Bolivia), where, in order to avoid erection of aqueducts and tunnels along the rugged terrain, the 2640 m long penstock was traced with several curvatures. At one of the bends, near the powerhouse, the pipe had burst during the test filling in 1973 under a pressure head of 735 m (the design head was 880 included water hammer overpressure), and the violent jet inundated the powerhouse and the substation. The pipe with an internal diameter of 1.15 m was fabricated from 22 mm thick high-strength steel plate. The actual cause of the failure could not definitely be detected, however, it seemed probable that the wide rupture started from a brittle fracture of a weak spot of longitudinal weld in the curved section of the pipe. (The accident and repair of the failure are described by S. Jacobsen, W. P„ Jan. 1974.)

The upper part of s u p p o r t p i e r s located between anchorages should be designed to ensure the accurate location for the penstock and to keep, at the same time, the friction as low as possible. A lower frictional resistance is favourable as regards both stresses in the penstock and the moment tending to overturn the

417 CHAPTER 93

Fig. 10/93. Juan Carosio-Moyopampa power development, Peru. The 800 m long penstocks are divided by anchor blocks, respectively, expansion joints into 8 sections. Supports are spaced at 6 to 9 m. Each of the two pipes conveys a flow of 5.3 m3/sec at a clear diameter of 1250 to 1050 mm and a shell thickness of 10 to 27 mm. H—456 m, A=45,000 kW. The steep ditch for the waste water spilling from the headpond is to be seen to the right of the penstocks. (After a publication of Motor Columbus AG,

Schweizerische Bauzeitung 1953)


support. Consequently, efforts to minimize frictional resistance should be the more intensive, the higher the pipeline is above the terrain.

For penstocks of small and medium diameter the upper part of the support pier may be shaped as a saddle (or cradle) in which the penstock may slide freely.

The depth of the saddle is usually about 1/3 or 1/4 of the pipe diameter. Linings of cast iron, yet more recently of steel plate, are applied to reduce friction. It may be stated that unlined concrete saddles are used very seldom and for very small diameters only. Saddles supporting pipes having a diameter larger than 0.8 m are usually lined.

In some instances support piers topped by roller-mounted bearing shoes, similar to moving bridge supports, are also applied. Supports of the rocker type have also

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419 CHAPTER 93

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been adopted for penstocks. The use of these solutions is indicated if the supports are widely spaced (e.g. self-bearing pipe bridges of smaller span) or if the piers are high and are subject to considerable forces.

Forces acting on the support pier are: the frictional force Pf given by Eq. (25/92) and acting in the centroid of the bearing surface, and the dead weight of the pier itself. Inasmuch as the frictional force Pf may be positive or negative, depending upon the change in temperature, stability conditions of the pier should be investigated for both cases.

Supports between anchorages may be further of the ring girder type, which carry the load to the piers of plain or reinforced concrete either over a rigid connection or over a rocker or roller assembly (Fig. 12/93).

A few examples for the design of support piers are given in Fig. 13/93. Supports, respectively anchors for wood stave pipes are shown in Fig. 14/93, while a view of a wood stave penstock is presented in Fig. 15/93.

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ANCHORAGES AND SUPPORTS 420ifC-- f ;. '

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M/ï#. 12/93. Rocker-mounted ring girder supports for penstock. — Above: 2.60 m dia. pipe, Owens River Gorge, Los Angeles, USA. (After P. J. Bier: Water Power). — Below: Estes Park power plant, Big Thompson development, USA. (After R. G. Baumhoff:

Water Power 1950)

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P-t*0J6dSt [kN],

(20/93)

(20a/93)

For erection of long, small-diameter penstocks it may be expedient to precast reinforced-concrete cradles and transport them, hung on the pipe sections, together to the site, as described by A. Eberhardt. In such a case the pipes have to be tied by straps to the cradles which will be placed upon footings being concreted previously.

The so-called rigid-pipelines (definition is given in Chapter 87) are supported by anchor blocks only, they are not fitted with expansion joints and, generally, they are not supported by cradles between the anchorages, viz. because of their rigidness no sliding can take place. Accordingly, the anchor blocks have to be placed nearer to each other, than it would be necessary in case of a semirigid solution. The forces acting upon the anchor blocks can be determined on basis of the previous analysis. Some forces are the same as formulated for the semirigid pipe, while other ones (friction resistances) do not exist. On the other hand, an additional axial force of considerable magnitude, owing to changes in temperature, can come into being between two neighbouring blocks. This force, according to Eqs (18/92) and (19/92) can be expressed as

421 CHAPTER 93

C = 9 0°

Fig. 13/93. Details of various supporting piers: a) steel-plate lined saddle, b) saddle made up of rolled sections, c) roller- mounted support, d) double saddle for supporting an expansion joint

where the diameter d and the wall thickness of the pipe 8 in cm and the temperature change in Celsius degrees have to be substituted.

Decrease in temperature exerts tensile forces in the pipe, while with rising temperature pressure develops. If the pipe is rigidly fixed to the blocks, these forces are transmitted to the latters. When there is no change in the direction of the penstock line at an anchor block, viz. neither in profile, nor in plan, these temperature forces balance each other, so

that the block is not exposed to additional loads. On the contrary, an anchor block supporting a

bend of the pipe is exposed to the resultant of the two forces acting from both sides (Fig. 16/93):

Z=2P, sin AilZA.. (21/93)

This force is at an angle of(22/93)


with the horizontal. A decrease in temperature exerts upon a convex bend, while a rise in temperature upon a concave bend, a force which presses the anchorage to the foundation. On the contrary, temperature rise affects a convex bend and a tem-

423 CHAPTER 93

perature drop a concave bend reversely, i,e. the resultant has a lifting affect upon the block. Naturally, all other forces, which may act in the selected case, have to be allowed for.

Finally, it has to be noted that, usually, the semirigid solution is chosen for buried penstocks or for exposed penstocks of small diameter.

Fig. (16/93) and Eq. (21/93) are, according to the meaning, applicable for bends in

Fig. 14/93. Support and anchorage for wood stave pipe (after A. Ludin): a) on concrete saddles(Norway), b) on anchored steel trestles (Rio and Delaware works, USA,)


plan, when the temperature forces are acting in a horizontal (or almost horizon-V

tal) plane, the anchor blocks are pressed against the foundation either to the left or to the right.

Special attention is to be devoted to the design of the l o w e s t a n c h o r b l o c k (end anchorage) located at the end of the penstock. As can be seen from Fig. 17/93, the lowest anchor followed by a bend in the penstock is subject to a significantly greater sliding component originated from the water pressure, than intermediate anchorages. The force Pt acting at the anchor block is balanced by the force P'w acting at the bend and thus the entire pressure P'w rides against the anchor block. (The force P„ is, of course, again proportionate to the static head increased by water hammer.)

425 CHAPTER 93


The component tending to displace the anchor is, under otherwise similar conditions, the greater, the flatter the slope of the preceding tangent section. It should be recalled that some of the forces transmitted to the block are not affected by the elevation of the block itself and thus proceeding downstream, the hydros-

427 CHAPTER 93a Concave bend

ñ\ \ V' a Decrease in temperature

90•- /,n'/3n"

/

b Rise in temperature

Convex bend

-Pnb Rise in temperature

I80“pn+1~($0~

o Pn+1~Pnj „ go\ I Pn+1* On, j ^J? a Decrease in

temperature

¿h

£ -180’- fîn - (90-Mill; - 90 - (JMlhUlL)

Fig. /Ć/95

tatical force due to the change of direction (Pw) becomes more and more significant as regards the stability of the anchorage. Consequently, at developments operating under very high head, the thrust acting upon the lower anchorages is hardly influenced by the spacing of the latter, and will be especially great at the lowest block if it is anchoring an almost horizontal pipe section immediately before the manifold (Fig. 18(93). Yet in spite of these considerations, the end anchorage

28 Mosonyi


429 CHAPTER 93

Fig. 19(93. Deformation of the penstockmanifold due to temperature rise

End anchorage

Pían

Fig. 17(93. Location of the lowest anchorage

Fig. 18(93. Location of the lowest anchorage


block should expediently be placed at the end of the horizontal section as close to the powerhouse as possible, to thus reduce the length of the unsupported penstock section. A manifold (header) occasionally incorporating bends and without intermediate anchorage would be subject to excessive forces that would result in impermissible deformations.

Although the arrangement according to Fig. 18/93 is more favourable for the manifold than that shown in Fig. 17/93, it still requires a careful analysis. Let us consider the system consisting of manifold and unit penstocks fixed between the lowest anchor block and the powerhouse substructure illustrated in distorted scale in Fig. 19/93. The conduit may expand more or less freely, there being only intermediate supports (and no anchorages). At mean temperature the position of the conduit system is represented by the broken line ABCDE. In case of tern-


28*

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432ANCHORAGES AND SUPPORTSI

perature rise the pipe system assumes the deformed shape ABC DE (exaggerated in the figure). The expansion of the pipe section AC due to a temperature rise t would be

Al—colt,

if it were not restricted in its movement by the unit penstocks BD and CE. However, in view of the fact that, on the one hand, the expansion due to the change in temperature corresponds to very great forces and, on the other, bending moments due to relatively small forces result in appreciable deflections of pipe sections BD, respectively CE, the compression due to forces acting at points B\ respectively C may safely be neglected. As an approximation it may therefore be assumed that the force acting in point C is of a magnitude to cause the displacement Al of the pipe section CE. The force causing a displacement Al at the end of the cantilever of length Ii is

3 £ / l " [kg], (23/93)

P Aln

and the moment induced at point E is

M=PU = [kgm]l\ (24/93)

Considering that the sectional modulus of the pipe cross-section is 2/i/di, the maximum stress at the fixed end attains, of course approximately, the following value:

[kg/cm2]. (25/93)

It is to be seen that stresses increase rapidly as the length of the unit penstock is reduced and the installation of too short branches may easily lead to rupture.

If a more exact analysis is deemed necessary, the method illustrated for the extremely simple case in Fig. 20/93 may be applied with careful judgement to any given pipeline. In case of unrestricted movement the displacement of point C would be co 11. However, the resisting force P acting at point C in the direction AC results

ait

Fig. 20/93

433 CHAPTER 93

Ï


PIndôE'

(26/93)

435 CHAPTER 93

m a compression

On the other hand, the deflection of the element BC due to opposite of force P is

Pll3EIx ’

and since the latter displacement is equal to the resultant of the first two values

pii , PI—— ~ (D It-------- r,3EJi ndôE

the unknown force P can be computed.Of course, the bending moment acting on the manifold due to the elongation of the

unit penstock should also be taken into consideration. Although the force is usually smaller, the lever arm / is relatively long.

A simple inspection of Fig. 21/93 will reveal that for the same temperature changes an obtuse-angled branching is considerably less favourable than a rectangular one since, in the former case, the displacement of the corner point is significantly greater. Although the use of a hydraulically round bend having a long radius is permissible the axis of the branch pipe should be perpendicular, or at least approximately so, to the manifold.

End anchoraoesFig. 22/93. Lateral penstock arrangements

Ï


The installation of a distributing pipe system fixed between the lowest anchor block and the powerhouse substructure cannot be avoided unless

437 CHAPTER 93

i»

Ï


♦

Anchor block

Expansion joints

13 Anchorages \ i

Expansion jowls

»kr E nd ancnor blockPouemousc

Substructure acting as end anchorage

mm wiwz mÎ 1 - V,

IPowerhouse End

anchorages

biot*ioni011

^rvc|l0r " ; «in*5

^,0#439 CHAPTER 93

Fig. 24/93. Various solutions for the direct connection of the penstocks to the powerhouse

a) the penstocks run directly towards the powerhouse and are (in plan) at right angles, or at least approximately so, to the longitudinal axis thereof, andb) each generating unit is supplied by a separate penstock and therefore no manifold is necessary.

I

Fig. 25/93. Direct connection of the penstock to the powerhouse. Papanasam hydroelectric plant, Thambrapami River, India. H~ 101 m, N= 18,000 kW. 1.75 m diameter penstocks are supported at 6 m

centres. (After R. Dorai Raj cm, Water Power 1952)

Ï


Direct connections to the powerhouse were formerly not favoured and, in spite of the difficulties involved, one of the lateral arrangements shown in Fig. 22/93 was

ANCHORAGES AND SUPPORTS 441 v . Ar. *

. r

Fig. 26/93. Lowest anchorage of the penstocks immediately before the powerhouse. (Vcntavon power

station, after A. Ludin)

employed. The main reason for this was the effort to safe-guard the powerhouse by all means against flooding in case of occasional penstock bursts (Fig. 23/93). Since, however, on the one hand, these arrangements alone do not guarantee full safety, and, on the other, reliable automatic devices have been developed for the rapid closure in case of penstock failure (see Chapter 89), in recent times direct connection is gaining in popularity. Advantages of direct connections are as follows (Figs 24/93 and.25/93):

a) No distributing pipe system is necessary.b) The end anchorage block may be arranged immediately before the powerhouse and consequently the rigidly fixed pipe section may be very short (Fig. 26/93).c) In some instances the lowest anchor block may be omitted entirely if the substructure alone is capable of resisting the thrust transmitted from the penstocks. The lowest anchorage may be economically united with the machine foundation, respectively with the powerhouse substructure, even if the latter alone is insufficient for anchoring purposes (Figs 27/93 and 28/93).

d) Exclusion of bends and bifurcations diminishes hydraulic losses.

It would be mistaken, however, to assume that because of the preference for direct


connections, lateral arrangements have been discarded entirely for recent developments. Lateral penstock connections may be encountered even at very up-to-date stations, the layout of which is the result of careful studies and considerations. Regardless of the above advantages of the direct connection, site conditions may decide in favour of the lateral arrangement. Indications therefore, however, various they may be, can be traced back to the relative position of the hillside selected for the penstock and of the potential powerhouse site. The method of connecting the penstock to the units is ultimately governed — with due regard

443 CHAPTER 93-* L _______

Fig. 27/93. Loch Sloy power plant» Scotland. The lowest anchor block of the directly connected penstocks is designed to form a unit with the substructure (see also Fig. 28/93). H- 260 m, the diameter of the 460 m long penstocks varies from 2.14 m at the upstream end to 1.93 m at the bottom. Erection sections were 7.3 m in length. The support saddles are steel-lined and spaced at 14.6 m centres. The

anchor blocks are from 1700 to 3600 tons in weight. (Water Power 1950)

to the considerations listed above — by the condition whether it is possible to locate the powerhouse at right angles to the line of fall and thus to the horizontal projection of the penstock, or whether it is more expedient to set the powerhouse with its longitudinal axis parallel to the conduit. The former situation calls obviously for a direct connection, while the latter for a lateral arrangement.

In order to avoid misunderstanding, it should be stressed that the endeavour to achieve direct connection does not contradict the principle expounded in Chapter 87, i.e. to apply as few penstocks as possible (eventually one). By choosing high- capacity generating units the number of unit penstocks can be reduced, and a favourable bifurcation obtained by direct connection as well (Fig. 11/95).

Conditions may frequently impose limitations on the location of the powerhouse relative to the hillside, and may require the modification of the predetermined method of connection. A direct solution preferred for the afore-mentioned structural as well as hydraulic considerations may prove impractical if other conditions call for an


arrangement parallel to the penstock. The orientation of the machine hall may be governed by the availability of space required for the transformers and the switchyard. The practical alignment of the tailwater tunnel, or at least of the initial section thereof, is sometimes an important aspect, moreover the entire

445 CHAPTER 93Anchor

LOCH •L OhOND

Fig. 28/93. Anchor block built in unit with the powerhouse substructure. Loch Sioy power station. Each of the four penstocks is connected to a Francis turbine //=260 m, N—4 x 32,500 kW, «=428 rpm.

(Water Power 1950)

station layout may be influenced by the space available to carry the wheel discharge away from the powerhouse sometimes in a very narrow valley or river bank.

Details of a modem power station with the machine hall arranged parallel to the penstock are shown in Fig. 29/93. The Cimego powerhouse of the Alto Chiese development in Italy featured at that time (in 1957) the largest horizontal-shaft Pelton units in the world. (Double-wheel Pelton turbines having a capacity of 150,000 HP, while the electrical output of the generator is 110 MW.) The two large generating sets installed at the station are supplied through a laterally connected banded (hooped) penstock. Four


unit penstocks convey the plant discharge of 34 m3/sec to the wheels. A separate small penstock is connected directly to the third machine installed, a vertical-shaft Francis turbine having an output of 9 MW. The small penstock is supplied from the Ponte Murandin, while the large one draws on the Malga Boazzo reservoir. Each of the high-capacity units can discharge 17.75 m3/sec under a net head of 721 m. The large penstock is 1220 m in length and is

447 CHAPTER 93


500,011MW Jim 0 0 6 0 0 0

Control room. % cz Z) . j■ 1 ■

■

0It

l6 h c -J • 4 460,40

W00t

.iff * V » • I 4

^TESSXSSSXttCBStSSBCHISt^^SSSL

. # m 4 + « «

•*k * * p M » * «

4?W

N.r-rr-n

479,6

0

479,5

0Fig. 29(93. Cimego power station with lateral penstock arrangement. .4/to Chiese scheme, Italy. Three generating sets are installed at the powerhouse: two double-wheel horizontal-shaft Pelton units and a vertical-shaft Francis machine supplied from a separate penstock. Each of the high-capacity units generates a power of 110 MW, while the smaller set driven by a Francis turbine has a capacity of 9 MW

only. The power capacity of the whole station totals 229 MW. (Water Power 1957)

449 CHAPTER 93


451 CHAPTER 93

To Fig, 29j93

*


453 CHAPTER 93

*


455 CHAPTER 93

fixed by ten anchor blocks. Free movement is provided for by expansion joints arranged immediately below each anchor block. The inital diameter of 3.3 m is reduced at the lower end to 2.9 m, consequently the flow velocity increases from 4.0 to 5.2 m/sec before reaching the point of bifurcation. Fig. 32/93 presents a clear view of the lower end of the penstock revealing the wye sections heavily reinforced by stiffening clamps.

It has been pointed out in the preceding chapter, in connection with Figs 7/92, 8/92,9/92,10/92 and 11/92, that considerable reinforcement is necessary at bifurcations. The reinforcement required is the heavier, the smaller the angle of bifurcation, yet acute-angled branchings are sometimes unavoidable. Usually, however, the additional cost of reinforcement far outweighs the savings attainable by the reduction of head losses through the choice of small angles at bifurcations. Hydraulic considerations are consequently sacrificed and wyes are in most cases designed with angles between 35 and 45 degrees. A clear picture of the arrangement of anchor blocks may also be gained from Fig. 30/93. The lowest block rests against the substructure of the powerhouse.

In keeping with the pledge made at the end of Chapter 90, a few informative dataa

will be given for the determination of head losses at bifurcations. These losses are defined as the difference between the piezometric heads immediately before and immediately following the bifurcation (elevation of the centreline + manometric head at the centreline). Piezometric head loss in the main penstock and in the branch pipe is obviously involved. Both of them are related to the velocity head downstream of the bifurcation, i.e.

Ahm= :*-£=- and Ahb=& (27/93)2g 2g

where the suffixes m and b refer to the main penstock and to the branch pipe, respectively. For calculating these velocities cross-sectional areas following the

*


bifurcation, respectively the wye itself should be allowed for. The discharge delivered by the main penstock to the point of bifurcation is Q = Qm + Qb. The above notations have been used in compiling Table 1/93 containing values of the loss coefficient for various Qb/Q, respectively Qm/Qb ratios and for bifurcations under 90 degrees and under acute angles. It should be noted that, as revealed by accurate investigations, the main-line loss, i.e. the quantity Jhm may, in case of small discharge ratios, assume negative values as well, indicating that the piezometric head in the main penstock may under certain circumstances show a slight increase. Of course, the “energy law” is not violated thereby inasmuch as this increase originates from the velocity diminution, so that a portion of the kinetic energy of the main flow is converted into pressure increment. In view of the slightness of these increments, negative loss coefficients have not been entered into the table and are denoted by zero.

TABLE 1/93The coefficient of head loss at bifurcations for various discharge ratios

and different angles

Discharge ratio for The loss coefficient for bifurcations under

2=2», + Q b 90 degrees acute (30 deg.) angles

2*2

Q m

Q b(« & Cm b

G.10 9 0.03 0.89 0 0.780.25 3 0 0.88 0 0.630.50 1 0 0.91 0 0.440.75 1/3 0.18 1.06 0.16 0.360.90 1/9 0.28 1.20 0.26 0.41

CHAPTER 94

dCidd

¿€2 dd

8yHd 0.1 Hd2<ra 2crt

[cm],G —y x

i t d Sl1000

y 1

7c0.1 H a l 1 0 0 0 x 2 c r .

[kg],

1.2 x 7.85 x ft H d 2 l 20,000 oa 0.0015 HdTl (2/94)

THE ECONOMICAL PENSTOCK DIAMETER

Different diameters d may be considered for a penstock required to carry a given discharge Q. Although the weight and thus the first cost of the penstock increase parallel with increasing diameter, the output in electrical energy is also increased owing to the reduction in frictional head loss. The economic diameter may be determined on the basis of the following considerations: the economically justified diameter for a penstock required to carry a given flow Q is the one at which annual costs due to the greater investment do not exceed the annual value of the resulting increment energy output. The governing criterion is thus to regain economically the last increment kilowatt-hour made available by reducing the head loss through using a larger diameter. Mathematically this

criterion of economical nature may be expressed by the relation

(1/94)

where Ci is the annual cost due to the investment for a pipe of any diameter d, and C2 is the value of energy that can be produced at the same diameter d.

With H, metres, denoting the design head for the penstock section under consideration, the necessary shell thickness is from Eq. (9/92)

and the weight of the penstock section having a length / cm iswhere y i is the specific weight of steel: 7.85 kg/dm3. Considering that the installed weight of the penstock is higher by about 20 per cent due to the weight of joints, rivets,

CHAPTER 94etc., we may write

432 CHAPTER 94

(3/94)

X 100 s /g 2

12.1 d5

N—NQ~~9.%T } Q

Ah_100

Denoting the specific average first cost for the penstock Co [S/kg] and the annual operating charges including the depreciation, maintenance by a per cent, the average annual specific cost of the penstock is

The annual cost of the penstock having a diameter d is thus:

Cl = 0.0015 -1- [$/year].GQ

Let us now consider the influence of the changes in the diameter d on the annual energy Output and its value.

Using Eqs (10/90) and (11/90) the resistance of the pipe having a diameter d cm and a length / cm, and delivering Q m3/sec may be written as

(Factor 100s is introduced since ¿/has been substituted in metres into the original expression.)

The power output may be considered as the difference of an initial No and the power loss due to frictional resistance and written into the form:

(Reference is made to Eq. (1/11) derived in Volume /, according to which the power output is obtained from the expression 9.Sr}QAh in kW, if the discharge is introduced in m3/sec and the head in m units.)

Assuming the overall efficiency of the development as 0.77 and denoting the average annual duration of operation by t hours, the annual energy output is, after reductions and rearrangements,

E=Nt = (N0- 0.075 Q Ah) t [kWh/year],

THE ECONOMICAL PENSTOCK DIAMETER 433

0.075 x100 s XIQ 3

12.1 d5[kWh/year].

G 2 ËC2No—6.2 x 107

/ ct [S/year]. (4/94)

which, when substituting, the expression of Ah assumes the final forms

If the value of power at the generator terminals is c% S/kW/z, the annual energy produced may be represented by the sum:


0.003 î-d£6.2xl0 7 x5Oa

10

Cl H ’d£ 71011 X<JatC2 Q

Cl

H ’

¿^100X O g t C 2

g 3 1000 ci H

(5/94)

Differentiating Eqs (3/94) and (4/94) with respect to d and substituting into Eq. (1/94),

we obtainRearranging terms and rounding offand

wherein

X “ the friction coefficient, values of which have been given in Chapter 90 (may be taken as 0.02 for preliminary estimates), aa — the allowable stress for steel in kg/cm2, t = the annual duration of operation in hours, c2 — the value of one kWh at the generator terminals,Q — the discharge conveyed by the penstock in m3/sec, ci = the annual cost of 1 kg weight of the penstock in the same currency units as c2, andH = the design head in m.

It should be pointed out that the investigation is more involved if the utilized discharge is subject to considerable fluctuations, inasmuch as under such circumstances both Q and t are to be determined by auxiliary computations before being entered into Eq. (5/94).

As can be seen from Eq. (5/94) the economic diameter depends upon the head thus upon the elevation of the pipe section under consideration. Theoretically, the

r

relationship indicates the necessity of gradually reducing the penstock diameter towards the lower end but this, and even the use of small decrements, is not feasible for practical


reasons. When applying therefore Eq. (5/94) to any proposed installation, the head pertaining to the central section of the investigated pipe length, i.e. the mean head should be introduced. Reductions in diameter are practicably carried out at anchorages. Decrements of at least 50 to 100 mm are commonly used, the shell thickness is specified in mm. Neglecting slight variations in X with the diameter, each step in the latter can be computed in keeping with Eq. (5/94)

29 Mosonyi

436 CHAPTER 94from the relationship

(6/94)

where H, respectively Hi are the mean heads pertaining to the compared sections. Constituents of the annual charges are:

Depreciation for a useful life 50 to 33 years . . . . 2 to 3%Annual maintenance............................................................3 to 5%Other charges (cost of money, etc.)......................................2 to 4%

Thus coefficient “Û”.........................................................7 to 12%

In estimating the cost co it should be considered that the cost of shop-welded pipe sections is about 1.5 to 2.0 times that of steel plates delivered from the mill. The higher coefficient applies to longer sections designed for higher heads. Erection may be taken as from 20 to 25 per cent of the cost of fabrication, but may be considerably higher. (E.g. if longitudinal joints must be prepared at the site, i.e. if half cylinders only can be transported.)

The method of investigation described above and referred to on several occasions in the literature should, of course, not be accepted without criticism, inasmuch as the validity of the basic considerations may justly be questioned. It is not settled yet whether the diameter at which an infinitely small additional investment is balanced by the increment return, should actually be accepted as the most economical. The cost of the penstock represents an item of varying significance relative to the total cost of the development and it is the mean unit cost of energy produced that is eventually of decisive importance. In some instances the unit cost of energy production may be very low and in order to increase energy output a relative investment higher than the limit defined in the foregoing may be justified for the penstock without raising the mean unit cost of energy above the permissible value. This consideration will become clear if it is remembered that, under exceptional conditions, the investment required for the penstock (or penstocks) may be as low as from 3 to 5 per cent of that of the entire development. Under average conditions the cost of the penstock amounts to about 5 to 10 per cent of the total investment, yet in some cases the share of the penstock was from 15 to 20 per cent. As

f437 CHAPTER 94

i

demonstrated by A, W. K. Billings, the total cost of the penstock is at high-head developments in many cases equal to the aggregate cost of all mechanical and electrical equipment (including the switchyard and the transformer station).

In connection with the above deduction the following should be remembered: the treatment of the head H as a constant value is an approximation, since changes in the pipe diameter involve changes in the flow velocity as well as in the dynamic pressure component caused by water hammer.

For determining the economic diameter the graph shown in Fig. 1/94 has been proposed by G. Ferrand. The graph represents a function of two variables

THE ECONOMICAL PENSTOCK DIAMETER 438Diameter [m]

e<bO)O

Uvir

Q

25 3.0Head [ mjFig. 1J94. The mean value of economical penstock diameter plotted against discharge and head. (After

G. Ferrand)

£

in the form of a family of curves d~<p(Q) for different H= const, values between 150 and 2000 m. The graph provides

information as to the mean diameter.

In connection with the economic analysis of the penstock, it should be noted that by dividing the discharge to 2, 3, 4, 5 penstocks of equal diameter, their cost is increased to 1.10,1.17,1.22,1.26 times that of a single pipe, respectively. (After W. Bauersfeld and A. Schoklitsch.)

It should be emphasized in conclusion that any diameter determined by the method described above, or on the basis of other considerations of an economic character, is not

f439 CHAPTER 94

i

feasible unless requirements of fabrication, handling, transport and installation as well as limitations as regards the velocity of flow are complied with.

For a more accurate analysis of the economical penstock diameter the reader is referred to the exact mathematical transaction of J. O. de Mello Flores (see bibliography).

29*


During the last three decades, i.e. since the time that the second edition of this book was published, several experts dealt with this problem (G. S. Sark aria, E. J. Low, T. Sungur, D. I. H. Barr). Lately, Sarkaria, with reference to his earlier suggestion, recommended a simple empirical formula (expressed in the foot-pound system) to be applied for a quick estimate on the economic penstock diameter. An instructive table is attached to this paper, listing 38 plants and penstocks, respectively, and containing a

i

f441 CHAPTER 94

comparison between the estimated and the actual diameters.

APPENDIX

ANCHORAGES AND SUPPORTS

Documents

intermediate anchor

axial forces

location of penstock

resultant of forces

various kinds of anchor

sign indicataes forces

adjacent penstock sections

semirigid types of penstock