---

For positive AT (heating) the deformations and forces are:

At the NFP:

є - = -(a • AT - ^VN_x= -A(E ■ a ■ AT - v ■ a_p)

At the expansion zone:

Л/v ~ N_r + Л/ _R= A ■ —— + Л/p

X p r 2 *'

The friction length is calculated as the numerical value of:

A N

L = —{EaAT + (0,5-w)cr_p) + ^-

and the expansion is

F L²

EA 2

where

AT is the temperature difference related to the installation or pre-stressing temperature; AT is negative by cooling;

F is the friction force per metre of pipe, see Annex B;

A is the sectional area of the steel pipe;

стр is the hoop stress from the over-pressure (positive by over-pressure);

a_x is the axial stress (positive for tension);

N_x is the axial force (positive for tension);

N_p is the axial force from pressure at expansion;

Nr is the axial force from lateral soil reaction at expansion (N_R is normally negative).

For systems with axial compensators, Уг is left out from the last link (¹/2-v) of the expressions for L and 8.

For axial compensators where the active area is larger than the area of the run pipe 7г can be replaced by:

where

d_w is the average diameter of the bellow;

d, is the internal diameter of the steel pipe.

L can be calculated through iteration by first calculating the upper limit for L setting N_R= 0. A first estimate of N_R can then be calculated with the belonging value of 3. It should be taken into account that N_R normally gives a considerable reduction of L and 5 (75% - 80%).

For systems as Figure C.3 where fixpoints or other methods ensure that the distance I from the fixed point to an expansion facility is shorter than or equal to the friction length L, or where the distance between two expansion facilities is less than 2L, the axial stresses in the steel pipe are calculated as:

F N

(7_X=-(—/

^{x V}A ^pA ’

Expansion at the free pipe end from the partly restrained pipe (see Figure C.2) is calculated as:

8 = — (E a AT - —— + (¹/₂- w)<t + ^-)

E 2A A where

1st link is the movement from the temperature change AT;

2nd link is the "fixation" from the friction;

3rd link is the movement from the internal over-pressure;

4th link is the movement from the reaction from lateral soil pressure.

For systems with axial compensators the !4 in the link (14-v) of the expressions for a_x and £is omitted.

l<L

Key

Expansion loop

Fix point Compensator

Figure C.3 — Reduced friction length, I

When axial compensators do not have end-stops designed to take high axial forces, it should be ensured that the expansion conditions are well defined, for instance when installing fix points, in order to avoid overloading the compensators.

If expansion from the section partly restrained by friction is absorbed in L, Z or U bends expansion is ensured by means of foam or sand cushions, the resistance in the foam or sand cushions shall be taken into account when calculating the expansion loop.

Compensators and L, Z and U bends should not be mixed between two fix points.

Ageing of foam cushions throughout the service life of the pipeline should be taken into account.

See Annex В concerning the transverse horizontal movement of the pipe underground.

C.6.4 Boundary conditions for pipe systems with single use compensators (SUC’s)

Preheating in open trench can be avoided by using single use compensators.

The allowable length L_an between the single use compensators has to be calculated for the actual conditions, taking into consideration.

The tolerable axial stress⁰^" as compression has to be maximum the limit state C1 value at the max temperature T_d.

By means of the actual L (can be less than L_an) and the temperature T_d, the displacement at the single use compensator can be calculated. T₂ is the temperature for closing and has to be calculated for each compensator. After

has been reached, the compensator is welded.

Key

1. O’,a// Allowable tensile stress

2. a_{c a}ii Allowable compressive stress

3. gap at SUC in cold condition

4. NFP Natural fixed point

5. axial stress diagram at installation temperature (cold condition)

6. axial stress diagram at design temperature T_d

7. stress variation range from temperature increase (T_d- T_r)

8. axial stress at NFP, at the prestressing (closing) temperature T₂

9. axial stress at SUC , from temperature increase (T_d.- T_r)

10. allowable distance between two SUC’s

11. allowable length between NFP and SUC’s

12. и displacement of one pipe end at SUC

Figure C.4 — Allowable distance between SUCs

The stress variation range is:

Л(7 = а_гЛ.Т ■ E

The stress reserve due to friction forces from heating the pipe for the first time is:

аі=2<у_аіі-Аа

The stress, caused by restrained expansion is

:

о₂=Ло-а_аІІ

The allowable length І_ац depends on the stress reserve :

cr, • A_s /_a// = -y^

„ . . . T,

The temperature at which the SUC is welded (the closing temperature) ² can be calculated on the basis of.

О₂= ^ат ^E= =^ат

to:

T₉ — 7”, ч ——

Z J r—

а_гE

... T

with ¹ as installation temperature in cold condition.

T^^{1 r}j~^>rj^{i r}j~'

The maximum displacement at the end, when the pipe is heated to ¹¹ (with ^{- 2} ) is:

F -I²

^Г

^{W — ' lall}

^lall
—

2 ■ E_T • Л

¹2 Ь

Is there a pipe system with equal distance between the compensators, the gap at the compensator has to be:

Au = 2 и

To increase the distance 2*І_ац between the compensators, foil can be wrapped around the pipe (“sliding foil”), which will decrease the friction coefficient and the friction force F.

The reduction factor should be chosen with care on the basis of local soil conditions, foil properties and trench backfill.

If use is made of friction foil (to increase distance between two SUC's, due attention shall be paid to possible effect on the displacement of nearby bends!

C.7 Calculation of stresses

C.7.1 General

The size of the stress in the component is obtained by multiplying the maximum stress (membrane stress) in the equivalent straight pipe with the stress concentration factor i_a (for the current type of impact) of the component.

The methodology presupposes that hot-spot values for i_a are used in combination with SN-curves from uni-axial tests.

The stress history is established by calculating forces and deformation in an elastic model. Stresses are calculated assuming linear elastic conditions. Hot-spot stresses are calculated by applying і-factors (stress concentration factors according to 6.7.3 - 6.7.6) or by FEM analysis.

An alternative method is to use і-factors from “the experimental method” cf. Power Piping, ASME В 31.1. In this case the matching SN-curve shall be used.

C.7.2 Simplified procedure

In project classes A and В design and installation can be performed on basis of documented generalised calculations and instructions, see 6.2.

C.7.3 Cross section analyses, steel

Membrane stresses, cr_m, are mean stresses over the wall thickness, whereas resulting stresses, a_№S, are all occurring stresses, i.e. membrane stresses plus stresses varying over the wall thickness.

Membrane stresses are positive by tension and negative by compression.

The radial stresses from internal over-pressure have not been included in the following formulas due to their limited size.

The individual design stress components are determined from the following expressions. The expressions give an upper limit for the stresses. With simplified analysis it can safely presupposed that the maximum stresses occur in the same spot. Refer to specialised literature concerning size and location of the actually occurring stresses.

Stresses and internal forces are illustrated in Figure C.5

For the calculation of areas and section modulus the wall thickness less possible allowance for corrosion shall be used. For district heating pipelines the allowance for corrosion can usually be valued at 0.

Area and section modulus are calculated by:

A = л • (d₀ - t) • t

^w =чЛгІ<-«А.-²')⁴]

32 d₀

For tees the outside diameters d_m and d_b0 and the thickness t_r and t_b for run pipe and branch, respectively, are inserted.

Table C.5 — Calculation of stresses

p is positive for internal over-pressure.

The first link in the expression for a_t represents the membrane stress from internal pressure, whereas the second link (ovalisation stress from external moments) is only included in the calculation of resulting stresses.

N_xis the resulting axial force including the contribution from pressure.

When calculating stresses for fatigue analysis, Aa_a, Aland Aa_t the values for AN, AM, AV and Ap shall used in the expressions in Table C.5. All changes of actions must be considered (cold-warm, warm-cold).

A<7₀ is the ovalisation stresses from for example soil pressure and traffic.

The effect of soil pressure and traffic actions can normally be ignored for pipe dimensions d_n< 300 mm.

For d_n> 300 mm the ovalising stresses from soil pressure and traffic action will normally not be decisive, but should be checked. Internal pressure will counteract ovalisation and thereby reduce the ovalising stresses. Therefore the ovalising stresses should not be added to the stresses from internal pressure.

Calculation of the stress concentration factors i_ap and /_a7 - i_a5 for a number of pipe components appears from C.7.4 to C.7.7.

Figure C.5 — Stress components and internal forces

The reference stress is calculated from the above-mentioned stress components (calculated with sign) by Tresca or by von Mises’ formula:

<7₁ -tr₂

cr₂ -cr₃ cr₃ - (Т,

o-₇ = 7 a² + - О-_Й ■ ст, + 3т²

The above calculation method is approximate, as it presupposes that all peak stresses are referred to the same point and added although they are not necessarily located at the same point of the cross section.

C.7.4 Straight pipes

Straight pipes

cr_m (membrane stresses) and Cf_res (resulting stresses):

Table C.6 — Stress intensification factors for straight pipes

Іа4 Іа5 0

C.7.4.1 Butt welds

к = 0,9 + 2,7

max min

2t„

Table C.7 — Stress intensification factors for butt welds, same actual wall thickness (see Figure C.6)

к is min. 1,4 and max. 1,9

NOTE 1 Factor 1,3 is for normal weld control (5, 10 and 20% in project classes A, В and C). If weld control is extended to 10, 20 and 100% in project classes A, В and C the factor can be reduced to 1,1.

Figure C.6 — Butt weld misalignment

Butt welds at changes in wall thickness without after-welded root pass. The joint fulfils the requirement specified in 8.5.8.

Table C.8 — Stress intensification factors for butt welds, different wall thickness (see Figure C.7)

к = 1,3 + 0,0036 — + 3,6

tn 2f„

к is max. 1,9

NOTE 2 Factor 1,5 is for normal weld control (5, 10 and 20% in project classes A, В and C). If weld control is extended to 10, 20 and 100% in project classes A, В and C the factor can be reduced to 1,3

.

Figure C.7 — Butt weld at change in wall thickness

C.7.5 Bends

The methology below for bends is based on present practice. Recent work shows that the plastic strain in larger diameter pipe bends is larger than in smaller diameter bends. The proposed limit state for low cycle fatigue does not take this into account.

The і-factors are valid for in-plane and out-of-plane bending.

Table C.9 — Stress intensification factors for bends

The maximum stress is an ovalising stress, сг₍, and for in-plane bending it is at the side of the bend

(Ф = 0°and 180° in Figure C.7).

For in-plane bending the maximum axial stress, tr_a, is found about 70° from the symmetry plane of the bend (Ф- 20°, 160°, 200° and 340°). The maximum axial stress at the side of the bend (Ф = 0° and 180°, where the maximum ovalising stress is found) can be evaluated by using the stress concentration factor 0,5 i_a2.

For out-of-plane bending and combinations of in-plane and out-of-plane bending the maximum values of tr_a and a_t must be used when calculating the references stresses, or the more exact method below must be used.

When a pipe bend is connected to a flange at the one end, k_b, i_a2 and /_a6 shall be multiplied by h^1/6. If there are flanges at both ends of a bend k_b, i_a2 and i_aS shall be multiplied by h^1/3

where i_a2 and /_a5 shall be valued at minimum 1,0 after the multiplication.

If the bracing effect of over-pressure has been considered when calculating the flexibility of the pipe bend, i_a2 and /_a5 shall be divided by:

5

2f?p ^d_m)

1 + 3,25 —f—Y E I 2t

i_a2 and i_a5 shall be valued at minimum 1,0 after the division.

For more exact calculation of stresses and location of stresses the formulas below can be used.

These formulas include the effect of rerounding and shall only be used if the flexibility factor, k_b, is reduced due to over-pressure; cf. C.6.2. Alternatively bending stresses shall be calculated for p = 0 in the formulas below.

Figure C.8 — Local co-ordinate system for bends

Membrane stresses cr_m:

^A°_a

zlrr, = /

^AN_X.

-+ і A

і LA ^ap2t

ay

AM . AM

+ 'az

И/ W

A ■ ^AMX

AT - 'mi

™ 2W

2^i_vyAV_y+i_n■ AV₂

Resulting stresses <T_res:

^A<?a

Лет, = /

AN ■

- + /

A '

: P ' ^dі

^ap2t

ay

AM .

-+ 'az

W ^aZ

. AM

^l,yW ⁺

AM ₂

W

• ^AMz a

/_ft-+ ^A&t

^bw ¹

л ■ ^AM>

At = i

^mx2W

■zlV'y + /

_vz ■ JV/

A

Stress intensification factors: