####
**Continued from Flexibility Analysis of Piping (Part-1)**

####
**Flexibility in Tortion**

Let us now consider the deflection of a beam when it is

subjected to bending and tortion. In case of a beam subjected to moments as .shown in figure, the angle θ (The change in slope) is given by;

subjected to bending and tortion. In case of a beam subjected to moments as .shown in figure, the angle θ (The change in slope) is given by;

θ - Angle in radians Homent, 1b inch

M - Moment, 1b inch

I - Moment of inertia, inch4

L - Length in inches

If the same length of pipe is subjected to torsion, the rotation of one end relative to other is given by,

θ - Angle of twist in radians

T - Tortion moment, lb/inch

L - Length in inch

G - Modulus of rigidity, Ib/inch2

This result in very important consideraing 3-D layouts. It shows that a given length of pipe will given 30% more rotation if moment from adjacent leg produces torsion instead of bending. Tortion deflection alone is rare as means of obtaining flexibility, but the fact demonstrated above may influence the stress engineer in choice of alternative routes for a pipe.

#### Piping Auxiliaries

Those elements other than straight pipe which go to make up a complete piping system are described as “piping Auxiliaries”. These are important to know to the extent of knowing their individual effects on the flexibility of piping system and the stresses in it, before going for a analysis of complicated piping system. The common auxiliaries used are bends, tees, reducer flanges, etc.

**Elbows**

These are used when change in direction of pipe is required, .they can be of the type short radius, long radius, or pipe bends. Now let us consider bending of elbows.

We know in case of a straight pipe the deflection is given by;

and the maximum bending stress

The earlier analysis of piping systems containing elbows and results of experiment showed deviations. The practical piping system was found far more flexible than what the theory predicted and the discrepancy was shown to lie in the flexibility of the elbows.

The first theoretical analysis of the behaviour of pipe bends when subject to a bending moment was made by Theodore Von Karman, who showed that when curved pipe is subjected to a bending moment in its own plane, the circular cross section undergoes changes and is flattened and this results in increased flexibility. The ratio of the flexibility of a bend to that of a straight pipe havirig the same length and cross section is known as “flexibility factor” and usually denoted by letter "K".

Now let us examine how this flattening of elbow ox chanqe in cross section occurs. Let us consider a elbow with AB as neutral axis is subjected to a bending moment of M (see figure 10). The outer fibre of elbows shall be subjected to tensile stress and the inside surface to a compressive stress. Let us take a thin cross section and study in detail. The resultant tensile load on outer fibre results in inward radial load in the element. Similarly, the compressive load C on inside fibre also produces a resultant inward radial load on the element.

If we now take a slice as a cross section of pipe and draw the loading diagram for the ring which is in effect, we arrive at view (a) in Fig. 10 (iv). Under the loading, the ring flattens into an ellipse with its major axis horizontal. If we now reverse the sign of bending moment the cross section w i l l e l o n g a t e instead of flattening.

If we now consider the element in more detail, we see that the flattening produces bending moments in the ring which are maximum at the end of the horizontal diameter. These moments produce a stress which varies from tension to compression through the thickness of pipe wall. and which is circumferential in direction. If we consider the half of the ring (Fig. 10 (v», we can illustrate the same in a simplified way.

The circumferential stress in pipe wall due to moment M can be many times the value calculated as (My/I) as per ordinary bending theory for structural members. The factor by which the circumferential stresses exceed the longitudinal stresses in bend is called the “Stress intensification factor” often denoted as S.l.F.

The circumferential stress in pipe wall due to moment M can be many times the value calculated as (My/I) as per ordinary bending theory for structural members. The factor by which the circumferential stresses exceed the longitudinal stresses in bend is called the “Stress intensification factor” often denoted as S.l.F.

One of the practical manifestations of the existance of these circumferential stresses is that when an elbow is subjected to repeated in-plane bending, it ultimately develops a fatigue crack along its sides. When we take into account the elbows of a piping system, we are therefore able to claim additional flexibility due to this flattening, but at the same time we must also take into account the induced circumferential stresses by multiplying the stresses at the bends due to overall bending moment in the piping system by appropriate “stress intensification factor”.

The expression for calculating both factor and stress intensification factor given oodes such as B3l.3 etc. are as follows.

The flexibility characteristic

stress intensification factor

#### Effect of Pressure on Stress Intensificaition Factor and Flexibility Factor

Some of the piping codes give fomulas for correcting the values of SIF and flexibility factor for elbows and bends. When the pressure effects are considered, SIF values are lower thus actually reducing the value of thermal stress. However, the terminal forces increase because of reduced flexibility at elbows. Pressure can affect significantly the rnagnitude of flexibility factor and SIF in case of large diameter and thin wall elbows. The correction factor CKF for flexibility factor due to preseue on elbows is given by,

The correction factor CFI for SIF

where,

T - Nominal wall thickness of the fittings for elbows and miter bends, inches

r2 - Mean radius of matching pipe, inches

R1 - Bend radius, inches

P - Gauge pressure in psi

EC - Cold modulus of elasticity, psi

T - Nominal wall thickness of the fittings for elbows and miter bends, inches

r2 - Mean radius of matching pipe, inches

R1 - Bend radius, inches

P - Gauge pressure in psi

EC - Cold modulus of elasticity, psi

#### Stresses in a Piping System

The equation for expansion stress S is given by equation using as installed modulus of elasticity Ea;

(1)

The equation for resulting bending stress is given by

(2)

And the torsional stress

For branch connections, the resultant bending stress requires a bit more attention as the section modulus Z for header and branch is slightly different.

The pipe wall thickness has no significant effect on bending stress due to thermal expansion but it affects the end reactions in direct ratio. So overstress cannot be remedied by adding thickness; on the-contrary, this tends to make matter worse by increasing the end reactions.

The values of S calculated shall not exceed the values calculated by equation

SE = f ( 1.25 SC - 0.25 SH)

or

SE = f [1.25 SC - 1.25 (SH - SL)]

Refer to the in-plane and out-plane bending moment sketch for elbows.

**FOR Elbows**

where,

ii - In plane intensification factor

io - Out plane intensification factor

Mi - In plane bending moment

Mo - Out plane bending moment

z - Section modulus of pipe

ze - Effective section modulus

for branch = ðr2

2TS

r2 - Mean branch cross section area

Ts - Effective branch wall thickness

(lesser of tb and (io) Tb)

Tb - Thickness of branch pipe

#### Cold Spring

A piping system may be cold spring or prestressed to reduce anchor forces and moments. Cold spring may be cut short for hot piping and cut long for cryogenic piping. The CDt short is accomplished by shortening overall length of pipe by desired amount but not exceeding the calculated expansion. Cut long is done by inserting a length (making a length longer than required).

The amount of cold spring is expressed as percentage of thermal expansion. Credit for cold spring is no.t allowed for stress calculation. Different codes state the same.

#### Code

- Sets forth the engineering requirements deemed necessary for safe design and construction of pressure piping

- Safety is the main consideration

- The above alone will not govern the final specification for any piping installation.

- Code is not a designs hand book.

- It does not do away with the need of designer or competent engineering judgment.

· basic design principles

· formulas

· supplemented by specific requirements to assure uniform

application of principles and guide selection of piping materials code prohibits designs and practices known as unsafe and contains warnings where caution, but not prohibition is warented.

- Safety is the main consideration

- The above alone will not govern the final specification for any piping installation.

- Code is not a designs hand book.

- It does not do away with the need of designer or competent engineering judgment.

· basic design principles

· formulas

· supplemented by specific requirements to assure uniform

application of principles and guide selection of piping materials code prohibits designs and practices known as unsafe and contains warnings where caution, but not prohibition is warented.

#### Code Section Includes

(a) Reference to acceptable material spec. and compo. std including dimensional std 8 pr./ temp rating.

(b) Requirement of or design of component, assemplies, supports

(c) Requirement and data for evaluation and limitation of stresses, reactions and movements. Dug to pressure changes, temp. changes and other forces.

(d) Guidance and limitation on selection of materials

(e) Req. of fabrication, assemblies, and erection of piping

(f) Requirement for examination, inspection and testing. of piping.

#### ASME B 31 Code for Pressure Piping

B-31.1 - Power piping.

(1998) - Electric power generation station.

(1998) - Electric power generation station.

· Geothermal heating systems

· Central district heating systems.

· Central district heating systems.

B-31.2 - Process piping

(1999) - petroleum refineries

(1999) - petroleum refineries

· chemical

· pharmaceutical

· textile

· paper

· cryogenic plants

· pharmaceutical

· textile

· paper

· cryogenic plants

B-31.4 - pipeline transportation system for liq.

(1998) hydro car-bons and other liquids

(1998) hydro car-bons and other liquids

B-31.5 - Refrigeration piping for refrigerants and scondey coolants

(1992)

(1992)

B-31.8 - Gas transportation and distribution piping.

B-31.9 - Building services piping

(1996) - industrial

(1996) - industrial

· institutional / commercial

· public buildings

· public buildings

B-31.11 - Slurry transportation piping system

(R1998) - aqueous slurries

(R1998) - aqueous slurries

1926 - American standards association initiated project

March B.31 at req. of a.s.m.e.

1935 - American tentative standard code for pressure piping

To accommodate current development in piping design, welding, stress computation new dimensional standards and specification, and increase in the severity of service condition, revisions, supplements and new edition of code was published as

1942 - ASA-B-31.1

1955

1952 - A new section of code for gas transmission and distribution

1955 - Decision was taken to develop and publish separate code sections for other industries

1959 - To supersede section 3 of B-31.1

ASA B-31.3 was published Revisions in – 1962 – 1966

1967-1969 - American standard association became united states of America standards institute then American standards institute. And code became. American national standard code for pressure piping.

1973 - ANSI B-31.3 adenda through 1975.

1974 - code for chemical plant piping (B-36.6) was ready for approval.

1976 - B-31.3 was published to combine req. of B-36.6 and published as chemical plant and petroleum refinery piping.

Addenda upto 1980.

Dec 1978 - American national standards committee

B-31 was reorganized as asme code for pressure piping, B-31 committee. All addenda and new addition was developed as ANSI / ASME B-31

1980 - New edition ANSI / ASME B-31.3

Published

1981 - Code for cryogenic piping (B-36.10) was ready approved. Addenda of B-31.3 1980 was published to cater for B-36.10

1984 - Chapter for cryogenic piping added.

1987 - New edition B-31.3

26th October 1990 ANSI/ASME was adopted.

#### Design Pressure

Shall not be less than the pressure at most severe condition of coincident internal / External and temperature min / max expected during service.

#### Design Temperature

The coincident temp. at severe condition.

Consider

- fluid temp.

- ambient temp

- heating or, cooling medium.

- fluid temp.

- ambient temp

- heating or, cooling medium.

#### Design Minimum Temperature

Lowest component temp during service.

Design Temperature of Uninsulated Piping

(a) fluid temp. below 100o F (38OC) component. Temp. as fluid temp.

(b) fluid temp. above 100OF (38OC)

unless lower temp. is determined by test or heat transfer calculation.

(a) values pipes, B N fitting - 95% of fluid temp.

(b) flanges (except lap joint) - 90% of fluid temp.

(c) L.T. flanges - 85%

(d) Bolting - 80%

**Externally Insulated Piping**

Fluid temperature

**Internally Insulated Piping**

To be determined by heat transfer calculation limitation of calculated stresses due to sustained load and displacement strains.

(a) Internal pressure stresses:

The selected pipe thickness – T

The mill tolerance 12.5%

Min. thickness = T – mill tol. > t + C

The mill tolerance 12.5%

Min. thickness = T – mill tol. > t + C

C - Sum of mechanical allowance (thread) + corrosion allowance.

t- Pressure design thickness.

where,

P - Internal design pressure gage.

D - Outside diameter

S - Stress value for material. From table A-1

P - Internal design pressure gage.

D - Outside diameter

S - Stress value for material. From table A-1

E - Quality factor table A-1A / A-1B

Seamless pipe E = 1.0

ERW E = 0.85

Furness butt welded E = 0.6

Electric fusion welded E = 0.95

(double butt.)

100% radio graphed E = 1.0

Y - Coefficient from table 304.1.1 t < D/6

Seamless pipe E = 1.0

ERW E = 0.85

Furness butt welded E = 0.6

Electric fusion welded E = 0.95

(double butt.)

100% radio graphed E = 1.0

Y - Coefficient from table 304.1.1 t < D/6

t ≥ D / 6 d – inside dia. (max.)

function of material and design temperature 0.4 to 0.7

#### Thickness Cal.

**Pipe Bends**

And at side wall on bend center line I = 1.0

The – thickness req is at the midspan.

**Miter Bends**

Angular offset more than 3 deg are req. to be checked

**Branch Reinforcement**

Figure 2.14

L4 min of 2.5 (Thk of pipe – Mill tol. – Corr.AL) for header

Or

2.5 (Thk of pipe – Mill tol. – Corr.AL)branch + Tr

d1 = OD of branch – excess thk in branch.

**Required Reinforcement Area**

A1 = th d1 (2 – sin â)

d1 = Effective length removed

= [Db – 2 (Tb – C)]/ sin â

th/tb – thickness req. for external pressure.

**Available Area**

A2 – Area resulting excess thick ness on run pipe (header)

= (2d2-d1) (Th - th- C)

d2- half width of reinforcement zone

= d1 or (th-C) + (th-C) + d1/2 which ever is greater but no case

more than Dn

A3- Area resulting from excess thickness on branch

= 2 L4 (Ta-tb-C)/ sin â

Tr. bs min thick of branch reinforcement

L4- 2.5 (Th-c) or 2.5(Th-C) + Tr

A4 – Area of rein forcement

### Appendix D

Flexibility and Stress Intensification Factors

**Longitudinal Stresses (SL)**

Due to pressure, weight and other sustained loading, sl, shall not exceed sn for calculation of sl. Will be based on nominal thickness – mechanical, corrosion and erosion allowance.

**Allowable Displacement Stress S**

**h**

SA = ƒ (1.25 Sc + 0.25 Sh)

When SH > SL

SA = ƒ{(1.25 sc + 1.25 sn) – SL}

ƒ

**f****. Stress Range Reduction Factor**#### Flexibility in Tortion

**Inbending**

Guided cantilever method

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