3. Pressure Lining Part

Article 13. Loads to be Considered

f

A pressure lining part shall be so designed as to be safe against the fol­lowing loads: _r

For exposed pipes

internal pressure, self-weight of pipe, water weight in pipe, tempera­ture change, external pressure

For embedded pipes pAA internal pressure, temperature change, external pressure

Description :

The shell thickness at the pressure lining part is mainly decided by the circumferential tensile stress caused by the internal pressure, but in case of a long-span ring support type of exposed pipe, the self-weight of pipes, water weight in the pipes, temperature change, etc., give a great influence to the shell thickness, whilst in case of an embedded pipe the external pres­sure and temperature change give a great influence to the shellthickness.

With regard to the temperature change for embedded pipes, the dif­ference between the pipe temperature when installed and the water tem­perature after water is filled should be taken. It is general to take the temperature drop as 15°C to 20°C, but in a cold district a temperature rising may take place when the temperature at the time of installation is very low, thus requiring a careful determination taking weather and con­struction work conditions into consideration.

In addition to the loads listed in this Article, there are other ones, such as seismic force, wind pressure, snow load and so forth. The seismic force and wind pressure do not directly give an influence to the pressure lining part and so they should only be considered in the designing of its sup­port., (See Articles 29., 55. and 58., this Chapter.)

With regard to snow load, snow around pipes melts due to the temper­ature of the flowing water inside the pipe, and so it is considered that the snow load hardly works by an arch action. However, its influence is greater when pipes are empty and in case of a long span ring supporting type, and thus a careful study is required.

Article 14. Combination of Loads

The loads in the previous Article shall be taken_into consideration with the following combination:

For exposed pipes

1 . With water fully filled in the pipe:

internal pressure, self-weight of pipe, water weight in pipe, tempera­ture change

2. During water filling in the pipe: water weight in pipe

3 . When the pipe is empty: external pressure when drained .

For embedded pipes

1 . With water fully filled in the pipe: internal pressure, temperature change

2. When the pipe is empty: external pressure

Description :

With water fully filled in an exposed pipe, the internal pressure is the most influential load, but self-weight of pipe, water weight in pipe and temperature change should also be. taken into consideration.

During water filling, a circumferential bending moment by water weight is generated in the pipe shell. This bending moment gets largest with water ' half filled (with water filled up to 50% of a pipe section) in case of a ring support type, and gets largest with water just fully filled in case of a sad­dle support type, thus requiring a careful review and study.

When it can be assumed that a half-filled water condition cannot be generated due to a ^harp gradient pipeline, the calculation for a hal^filled water condition may be neglected.

It is required to study the difference in pressure inside and outside the pipe when drained.

For embedded pipes, it is not necessary to consider the self-weight of pipe and water weight in pipe because the pipe is supported with surround­ing concrete, and it is only required to study the internal pressure and tem­perature change when water is fully filled, and to study just the external pressure when the pipe is empty-

Article 15. Design Condition for Pressure lEining Parti

The pressure lining part shall be designed against the,loads in Article 14. in accordance with the following conditions:

1. With water fully filled in the pipe

1. Circumferential stress, longitudinal stress, perpendicular, stress to

' a pipe axis and resultant stress shall be less than ajlowable stresses of the materials used. In case that a local bending stress(secondary stress) of a pipe shell is added, it shall be permissible to increase the allowable stress up to 1.35 times.

2. The resultant stress shall be calculated from the following formula: a_g = Vct,² + a₂² — a_la₂ + 3T²

1

where a_g: Resultant stress

a₍ : Circumferential stress (tension to be positive) ct₂ : Longitudinal stress (tension to be positive) r: Perpendicular shearing stress to a pipe axis

a UvvLsj <-«,

2. During water filling in the pipe

The circumferential stress shall not exceed 1.5 times the allowable stress of the material used. <?5 LT"

3. When the pipe is empty

Buckling shall not take place by external pressure equivalent to 1.5 times the design external pressure.

Description :

In case of water fully filled in pipes, it is set to check the resultant stress provided that the stresses in every direction do not exceed the allowable stress. X-

A formula to determine the resultant stress is based on the shearing strain energy theory (Mises-Heneky-Huber Theory) in that a fracture takes place when the shearing strain energy gets equal to the strain energy at the yield point of simple tension. For biaxial stresses, the following formula can be applied to ductile materials:

CT_X² ₊ a/ + 3rxy^l

where r_xy: Shearing stress normal to a_x or a_y

a_e: Yield point Of material under simple tension

With regard to the staring stress, the values given by the formula in paragraph 9 of the Description of Article 16. should be used for the side face of a pipe.

In case of a, lon^ Span of a ring support type and a thin pipe shell plate compared with its diameter, Tcannot be neglected around a support, but t diminishes at the pipe Section midway of the span. In case of the latter, however, the resultant-stress §efc large at the pipe top and so attention should be paid to this ■fact.

The stress including local bending stress (secondary stress) in the pipe shell is specified not to exceed 1.35 times the allowable stress so as to be less than 757’ of theyleld point considering that the yield point gets high forbending stre&s and that this calculation is made under normal loading conditions.

The local bending Stress (secondary stress) at a pipe shell is generated due to restriction of nng girder, Stiffener, anchor block, etc., and the cal­culation formula of poxAgraph 3 in the Description of Article 16. is gener­ally us@L

AS the stress during vvxdet filling is transient, the allowable value is set at 1,. ytimes the cdlovrctble-stress.

As for the external pressure, it is. necessary that an external pressure of 1.5 times the design external pressure should not exceed the critical buck­ling pressure of the pipe shell itself and stiffener.

Article 16. Stress to be Considered

• As for the design specified in Article 15. of this Chapter, the following

. stresses shall be calculated respectively:

For exposed pipes

1. With water fully filled .in the pipe:

1. Circumferential stress .

1. Tensile stress due to internal pressure

2. Longitudinal stress

1. Bending stress regarding the pipe as a continuous beam

2. Local bending stress due to the restraint of the pipe shell displace- - ment by means of a ring girder, a stiffener, an anchor block, etc.

3. Stress due to pipe gradient

4. Stress due’to longitudinal components of the internal pressure acting on the reducer

5. Stress due to longitudinal component of the internal pressure acting on an expansion joint

6. Stress due to temperature change of the pipe

’ (3) Shearing stress perpendicular to the pipe axis

1) Shearing stress regarding the pipe as a continuous beam

2. During water filling in the pipe:

(1) Circumferential bending stress due to water filling

3. When the pipe is empty

1. Critical buckling pressure due to external pressure

For embedded pipes

1. With water fully filled in the pipe:

1. Circumferential stress

1) Tensile stress due to internal pressure

2. Longitudinal stress

1. Local bending stress due to the restraint of the pipe shell dis­placement by means of a stiffener/etc. .

2. Temperature stress

3. Stress due to Poisson’s effect

2. When the pipe is empty

1) Stress due to external pressure and critical buckling pressure

Description :

For calculating the stresses listed in this Article, formulae commonly used are shown below.

Nomenclatures in the formulae are shown in each paragraph, but plate thickness, corrosion allowance, diameter and radius, and longitudinal length are defined as illustrated in Fig. 1.16-1 and below.

e

Fig. 1.16-1

Dimensions used for calculation should all be the nominal dimensions when manufactured.

D_q : Internal diameter (cm)

D_o^z: External diameter (cm)

D: Internal diameter subtracting 1/2 the corrosion allowance from the internal surface of the pipe shell = D₀ + e (cm)

D' : External diameter subtracting 1/2 the corrosion'allowance from the • external surface of the pipe shell = D_o' -e'(cm)

D_m: Diameter to the center of plate thickness = 2r_m (cm)

r_m: Radius to the center of plate thickness (cm)

t₀: Shell thickness (cm)

t: Shell thickness excluding corrosion allowance = - e (cm)

e: Allowance thickness for corrosion and wear (cm)

D_a: Pipe length from anchor block to expansion joint (cm)

L : Length of pier span (cm)

/: Interval of stiffeners (cm)

For exposed pipes

I. Tensile stress due to internal pressure

PD - PD or — 2/ z 2(r₀-e)

where cr: Stress (kgf/cm²)

P: Maximum hydraulic pressure at a place to determine stress (kgf/cm²)

2. Bending stress regarding the pipe as a continuous beam

where M„: Bending moment as a continuous beam (kgf-cm) Z_p\ Section modulus of pipe’s effective section

= tD_m² (cm²)

4

3. Local bending stress due to the restraint of the pipe shell displacemen by means of a ring girder, a stiffener, an anchor block, etc.

, ' _ff=1._g2

A_r+ 1.56tJ r_mt It

• where A;. Ring’s sectional area = 2tJ\_r + bt (cm²)

Fig. 1.16-2 Ringer Girder Section

“b” used for calculation should be within the following range:

1.56jr_mr +2t_r (cm)

, According to this formula, it may be necessary to increase the shell thickness around the ring girder and a minimum range L_r to increase the shell thickness can be obtained from the following formula:

L, = 4.67 J r_mr + b (cm)

4. Stress due to pipe gradient

E (irD_nitnL_n) A

7jSin0

where

t_n: Shell thickness of the pipe included in the portion con­

cerned (cm)

L_n: Length of the portion having a plate thickness of t„ (cm) 7/ Unit weight of steel material = 7.85 x IO^-3 (kgf/cm³) 0: Angle between pipe axis and horizontal plane

A_s: Pipe shell plate’s sectional area of the portion to consider stress = irD_mt (cm²)

5. Stress due to longitudinal component of the internal pressure acting

on the reducer

where D_x\ Internal diameter equivalent to D at a place before reduc­tion (cm)

• Z)₂: Internal diameter equivalent to D at a place after reduc­tion (cm)

P_r: Maximum hydraulic pressure in the center of reducer (kgf/cm²)

6. Stress due to longitudinal component of the internal pressure acting on an expansion joint

•KDmtEPE

A_s

where 'P_E: Hydraulic pressure at expansion joint (kgf/cm²) t_E. Plate thickness of internal pipe of expansion joint (cm)

7. Stress due to temperature change of the pipe

In case of temperature change in pipes, the direction of frictional force may change depending upon rising or dropping temperature, so the following stress should be added so as to be on the safe side: (1) Stress due to friction between pipe and support

rr ess ■ ¹ i'■ ■ ■'

where W_s: Load due to self-weight of the pipe (kgf) W_w‘. Load due to weight of water in the pipe (kgf) f_sp: Friction factor (Refer to Article 19. of this Chapter)

2. Stress due to friction of expansion joint

/ex'* (Do + 2/g)

A_s

where f_EX: Frictional force of expansion joint

According to the Bureau of Reclamation, f_EX = 7kgf/cm

8. For longitudinal stresses with no expansion joints, it is necessary to consider longitudinal stresses due to temperature changes .and Pois­son’s effect.

9. Maximum shearing stress regarding the pipe as a continuous beam

2Scos0 . 7 =

A_s

where S: Shearing force at a support (kgf)

10. Circumferential bending stress due to water filling Bending stress due to water filling differs greatly in cases of a saddle

. support and a ring support.

,• In case of a saddle support, a maximum bending moment generates when water is fully filled and the formula shown in Article 28. is applied. In case of a ring support, the stress variation in the process of water filling is rather complicated, and the stress does not become maximum with water half filled depending on places. But in case of a steel pen­stock, different from a waterway conduit, the water partially filled is not a normal loading condition, and so it is not necessary to make cal­culation as precisely as in the case of a waterway conduit. Therefore, as for the circumferential stresses in the mid-section of a span, it is prac­tically permissible to assume the stress on the side of the pipe with water half filled to give a maximum value.

As for the longitudinal stress, it may get a little larger with water half filled than with water fully filled, but the difference is minor. There­fore, the calculation for the longitudinal stress due to water filling may be omitted.

Calculation formulae for the circumferential stress can be divided into three, depending upon the ratio of the length of the span to the diameter.

(1) L > 1329.

where + : Inside of pipe

: Outside of pipe

ap Circumferential bending stress (kgf/cm²)

p: Fluid density 1 x io^-3 (kgf/cm³)

(2) 1329 > L > ID

± + ^7r²7?²X₂] • (2)

7T (1 — V_s )

F

Xi = 2(<pj + y>4 +•••••• + <p\i)

Xj = + $4 + • + $12

&n = Cz<Pn

where v_s: Poisson’s ratio = 0.3

Table 1.16-1

n ■! *■-	■ - <v>‘	r - 1 12(1-y/)	2 C. - . n2 - 1
2	6.088	’ 0.8242	0.6667
4	0.3805	20.60	0.1333
6'	0.07517	112.2	0.05714
8	0.02378	363.5	0.03174
10	0.00974	897.5	0.02020
12	0.00470	1873	0.01399

3. ID > L > 2D

The following formula should only be used for n = 2.

(2.467^+1)²

'^f,2=’c,^₊c7«~ ⁽³⁾

Formula (1) is assumed that, taking out a ring of a unit length from a cylinder, the water weight is transmitted to the next ring by the shear­ing flow. This assumption can be applicable to a long span but the assumption of transmission by the shearing flow cannot be applica­ble as a span becomes shorter, and thus this formula is liable to give a larger stress than the actual one.

Formula (2) is assumed that both ends of a cylindrical shell be free supported, and this formula is an appropriate equation for a bend­ing stress.

In either case, as the direct stress is quite small, the calculation of stress by bending moment is sufficient.

x/0³

10

7

G

5

4

3

2

I

0

Fig. 1.16-3 Bending Stress on Side of Pipe with Water Half Filled

By using Formulae (2) and (3), bending stresses with D/t as parameters are shown in Fig. 1.16-3.

Calculated values given from Formulae (2) and (3) are plotted like

the dotted lines around the right side in the figure but in case of L/D

- 13, values calculated by Formula (1) are closer to the actual ones and so they are adjusted as shown in the figure as full lines.

11. Critical buckling pressure due to external pressure

■ (1) Without stiffeners

2E, ( ¹ V

| 2 I ! I

1 — vr \ Do /

where p_k: Critical buckling pressure (kgf/cm²)

E_s: Elastic modulus of steel (2.1 x 10⁶ kgf/cm²) '

(2) With stiffeners -

1) Pipe shell proper

When computing the critical external pressure of a pipe shell proper, formulae of Tokugawa and R.V. Southwell are frequent­ly used.

Tokugawa’s formula

_e_L_

‘Do.’ [2a* 2. J I t V

^P‘~ , , a^!l(n^! + a³)^!'⁺ 3~’ i’-VW J

where n: Number of wrinkles