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Compucer r & Srruct un~ Vol. 22. No. 3. pp. 299-305. 1986

Printed in Great B ritain.

0045-7949186 53 00 4 .oo

0 1986 Pergamon Prc%s td.

A SIMPLIFIED PIPE FLEXIBILITY ANALYSIS PROGRAM-

STIFFNESS METHOD

R. NATARAJAN

Mechancial Engin eering Department. University of Illin ois at Chicago, Chicago, IL 60680, U.S.A.

Received 13 Augur

1984)

Abstract-Many commercial progr ams are available for pipel ine flexibi lity analysis, b ut they are all

complex and consume time in preparing data for simple problems. Also, much attention has recently

being given lo evaluating the flexibility of curved pipes more accurately. So far no consistent method

exists t o evaluate the flexibi lity factor in such cases. Hence, a need arises for a simpl ified pipe flexib ilit y

analysis program while at the same time not forgoin g the generality of the analysis. A simplified pipe

flexibility analysis program is presented and its merits are shown. This program is tested using a

comparatively simple pipeline system. Its use in obtaining consistent values for the flexibility of elbows

is also discussed.

Commercial progr ams are available for static anal-

ysis of piping systems either using the flexibility

concept or the stiffness method. These programs

are written so that complex piping systems are

solved with standard data preparation. If one re-

quires to use these programs for simple piping sys-

tems it involves extensive preparation of data and

mastering the input and output data routines. Also,

if one is interested in modifying such programs so

that, for example, special piping elements can be

included in the system, it is not easy to do so.

Recently, much work has been done to obtain

flexibility factors[l-51 of piping elbows more ac-

curately, taking into account the constraints pro-

duced by tangent pipes attached to elbows, flanges

next to elbows, etc. For such studies, one uses shell

theories in conjunction with finite difference or fi-

nite element techniques. From these analyses for

obtaining the flexibility factors, most of the authors

assume that the end cross-section of the elbows re-

main straight. It is found that such an assumption

is not correct[l]. Thus it becomes necessary to

evolve a consistent method of finding the flexibility

factor of elbows using the results obtained from the

shell analysis. It is here once again that a simplified

version of the piping flexibility program will be of

great use. Using this a consistent value for the flex-

ibility of the elbows is obtained by comparing the

deflections of the pipeline obtained from the shell

analysis and the piping flexibility[2, 31 analysis.

Further, such simple programs can be made easily

available for microcomputers. The various features

of the program are explained first. A sample piping

system is analysed using the program. Finally, the

use of this program to determine the flexibility of

the pipe elbows is described.

DESCRIPTION OF THE FLEXIBILITY PROGRAM

Several analytical methods for calculating ther-

mal stresses in high temperature piping are avail-

able in the literature[6]. The matrix method of anal-

ysis for piping system is the most widely used

procedure since it is well suited for high-speed dig-

ital computer application. It can handle complex

piping systems involving many anchors, closed

loops within loop and/or interconnecting branch

lines.

FLEXIBILITY

ND

STIFFNESS METHODS

For the purpose of development of the method,

a right-handed rectangular co-ordinate system is

specified. Consider, at any point in a deformable

structure, an applied force system causing stresses

in the structure. This is represented as a column

matrix:

(1)

An elementary volume of material of a flexible

structure which is acted upon by the force system

may experience displacements, due to distortion of

the structure, which can also be represented as a

column matrix:

(2)

An analytical method in piping flexibility analysis

as distinguished from the graphical, semigraphical

or numerical methods, will lead eventually to the

solution of a set of simultaneous algebraic equa-

299

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300

R.

NAT ARAJ W

tions. In general, these equations may be written

in the following form:

m = A =

Zh(l +

CL).

where A is the shear distribution factor.

The compliance matrix with andj as base-point

in which [Cl represents the matrix of influence can be obtained using transformation matrix. The

coefficients (usually called the compliance matrix). transformation matrix is written as

The piping flexibility analysis is concerned tvith the

solution of the redundant {F}. In practice, however,

iC0.1 = J [Cpl mv~l~

(3)

the number of equations required to solve a partic-

ular piping system differs with the various methods

where the transformation shifts the base-point p to

of analysis and essentially, it depends on the man-

P.

ner in which the compliance matrices are obtained

and manipulated. For a piping system involving

many anchors, interconnecting branches or closed

1

(6)

loops within loops, there is not only the problem of

the size of the equations, which often imposes a

T(p - P')

limitation on practical application even in the case

[

0

- -;P,

of digital computation, but also the problem of how

=

-(z, -z,.)

cp

-(Yp -Yp.)

(Xp - .t .) . (7)

the equations may be set up readily for solutions.

typ - yp.) -(.yp - .V&+)

0

hese difficulties are overcome using the stiff-

ness method of analysis. From the compliance ma-

I3s a 3 X 3 unit matrix.

trix of piping components[6], by an elegant method,

03 is a 3 X 3 null matrix.

the corresponding stiffness components are ob-

Bend. Figure 1 shows a circular bend having a

tained. The conventional stiffness method is now

bend radius

R

and central arc JI. Such a piping ele-

used for the solution of the displacements. These

ment does not obey the Euler-Bernoulli-Navier

are then used to calculate the stresses at specified

theory of bending.

points.

The cross-section is able to warp from its original

circular shape in such a way that the relationship

between moment and curvature is

COMPLIANCE MATRICES

Tmgenr.

The flexibility matrix of a tangent with

the mid-point as the base-point is available. This is

written as

where n is a factor greater than unity. The elements

[C&J =

DMG&{( m/p)lP2,

of the compliance matrix are given as

(; + mip) ,I,/(1 + I*LI}, (4)

C,, =

A C,3 = F

Cr, = Cl2 = B C,a = Ca = G

where is Youngs modulus, I is flexural 1M.I. of

the pipe, p is radius of gyration, 1 is length of the

tangent pipe and p is Poissons ratio.

Curvature = n . FI ,

C,6 = Cal = C CJ5 = Cs3 = H

C

2

=D

cu = I

8)

Cz6 = Ce2 = E CJs = CT4 = J

cg = K

CM = L,

where

A ,

etc. are given in Appendix A.

AUGMENTED STIFFSESS MATRIX

Using the flexibility matrix derived earlier, the

corresponding (12 x 12) stiffness matrix can be ob-

tained, correlating the 12 displacement components

at the ends of the element to the corresponding

force components. Thus

Fig.

I.

Circular bend with bend radius R and central

arc rL.

[K]{D) = {F).

(9)

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Pipe flexibility analysis pro gram

301

The matrix [K]c can be subdivided into four mat-

I- 5:

(10)

Where Kji is the stiffness matrix whose columns

are obtained by restraining the end j and computing

the force components at the i end for unit displace-

ment components at the end i and Kij is the stiffness

matrix whose columns are obtained by restraining

the end

i

and computing the force components at

thej node for unit displacement components at the

end j.

To obtain the submatrix Kfi the matrix [Cj]

should be inverted. Thus

Kii = [Ci]-a

11)

TR~~SFORMATlO~ TO GLOBAL CO+ORDIiVATES

The compliance and hence the stiffness matrices

for the piping elements are derived on the basis of

a local co-ordinate systems. Hence, for assembly

these matrices are transformed into chosen global

system. Thus it is written as

or

WJIW = V-2.

(131

where [L] is the transformation matrix and suffix g

denotes the global reference system.

ASSEMBLY AND SOLUTION

Equation (13) represents the force deformation

relationship of a pipe element in global direction.

A piping system has a large number of elements

consisting of tangents, bends, tees, valves, etc.

Each of these has a relation of the type eqn (13).

Summing up all such equations we get

or

The load vector gives the external loads applied on

the structure including thermal loads.

The boundary conditions for the piping system

is generally specified in terms of prescribed dis-

placements at the anchors and other restraint

points. Thus the vector {D) is split into two parts,

{D,/DK)

where

DK

corresponds to the knowns and

D,,

corresponds to the unknowns.

The solution of eqn t 19, subject to the boundary

conditions (&}, for the vector {D,,} and {F,,}. re-

suits in the complete solution of the piping system

for the displacements and reactions.

For the assembly and solution of the problem,

the front solution method is adopted. This uses

Gaussian forward elimination and back-substitu-

tion.

THERMAL LOADISG

The thermal loading problem is treated as an in-

itial strain problem. To calculate the nodal forces,

we write the initial strain as

where a is the coefficient of thermal expansion in

OC/~~crn and T is the difference in temperature in

C. The equivalent nodal forces are given as

STRESS CALCULATIONS

From the solution of tee system of equations,

the global displacements have already been ob-

tained. The internal forces and moments can be

computed easily using the equation

@=I = KIW.

15)

At any point along the length of a straight pipe,

there are moments and forces which can be re-

solved into the following components: one axial

force, two cross-shearing forces, one torsional mo-

ments. The stresses can be computed as

S, = F,iA

S, = h F,fA

S, = kitroil,

18)

where n, s, t and

b

stand for axial, shear force and

twisting and bending moments, respectively.

A

rep-

resents area of cross-section of the element. rll rep-

resents radius of the cross-section of the pipe, lP

and f represent polar and bending moment of inertia

of the pipe cross-section,

The pressure piping code recommends that the

expension stresses be based only on the combina-

tion of torsional stress S, and the bending stress S,,.

Thus

S.&= JCSi + 4s;).

(191

While calculating the stresses in a bend, stress

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301

R. NATARAJAN

intensification factors have to be brought in. The

calculated bending moment at a point is divided into

two components, one causing in-plane bending Mbi

and the other an out-of-plane Mbo. Thus

(20)

where L,, and Li are the stress intensification factors

along in-plane and out-plane bending. z is the pipe

section modulus. Thus

SE = J[(MbjLi) + (Mb&,) + Mf]/z.

(21)

Thus the axial stresses, shear stress and the

bending stress can be calculated at a point in a

straight pipe or a bend.

EQUILIBRIUM AND COMPATIBILITY CHECK

The program has an built-in capacity to check

whether the solution obtained, namely displace-

ments at the nodal points and forces at the anchor

points, are accurate enough.

Equilibrium check. The reaction force vectors

calculated at the anchor points are summed up with

the externally applied force vector to check

whether the total force vector is zero. Further, the

moment produced by these reaction forces about

the origin is found and the check is applied to see

whether this quantity is again zero.

Compatibility check. For this, a separate anal-

ysis is done for the entire piping system by releasing

one of the anchor points but substituting the dis-

placement boundary conditions at that point by a

force boundary condition, in terms of calculated re-

action forces by the earlier analysis. The resulting

displacements at the anchor points in this analysis

should correspond to the prescribed anchor dis

placements in the original analysis.

DETAILS OF THE ALGORITHM

A flow chart for the program is given in Fig. 2.

SHORT DESCRIPTION OF THE PROCRAlf

MAIN. This calls all the subprograms, in order,

required for the analysis of the system as well as

for checking the solution thus obtained.

STFTR, FORWAD. BUFFER AND BACKWD.

These four routines assemble the stiffness equa-

tions of the elements and solve for the unknown

deformations and reactions in the entire piping sys-

tems.

STFTR. Takes the stiffness matrix of an element

and places its elements in proper places in the area

allocated for assembly of all the equations from the

assembly which will not appear again in the system.

FORWAD. Eliminates those equations from the

assembly which will not appear again in the system.

The elimination process is in fact done by the con-

ventional Gaussian elimination process. These

eliminated equations are stored in a back store in

the BUFFER routine. The sequence of calling

STFTR, FORWAD and BUFFER is done for all

the elements in the system. The stored equations

are now solved for the unknowns using back-sub-

stitution technique.

INIAL.

Here in the program as a special tech-

nique known as front solution method is used in-

stead of the conventional assembly process, the ne-

cessity arises for the calculation of a quantity,

namely, the front width. This determines the size

of the assembled matrix of the entire system, and

is evaluated in this routine.

NODE.

Here the element node connection data,

identification of the element-tangent or bend, co-

ordinates of a special point with respect to the ele-

ment useful for calculating the transformation ma-

trix and element material properties are read. Fur-

ther nodal co-ordinates for the entire system are

also read here.

PRDF.

The amount of constraints given to the

system at different prescribed nodal points are read

here.

PDATA. Reads in all different internal diameters

and thicknesses of the pipe and the different tem-

peratures encountered in the system.

TRANS.

Calculates the transformation matrix

for tangents and bends, which will be used when

obtaining global stiffness matrix from calculated

local stiffness matrix.

GEOP. From the given co-ordinates of the ends

of a bend, this routine calculates the radius and in-

cluded angle of the bend.

GLOSTF.

Calculates the global stiffness matrix

of the pipe element using the local stiffness matrix

as the input to the routine.

STIFF. With the flexibility matrix of a bend as

input, this routine augments and obtains the stiff-

ness matrix in the local co-ordinate system.

TANGT. Calculates the stiffness matrix of a tan-

gent element once again augmenting the flexibility

matrix given as input to the routine.

TLOAD.

The load on the piping element due to

increase in temperature is calculated. To this the

externally applied load, if any, is added.

STRESS. It calculates the forces and moments

at the ends of the element. Using this, axial

stresses, bending and torsional stresses are evalu-

ated. These stresses are combined according to

ASME specifications. The global forces and mo-

ments are also evaluated here at the nodal points.

MATIV. Standard routine to find the inverse of

a given matrix.

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Pipe flexibility analysis p rogram

START

PDATA

NO

I

:

GEOP

I

- STIFF r

I

YES

I

STAGE 2

NO

STOP

YES

303

Fig. 2. Details of the algori thms.

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AC/GMT. This routine is used while obtaining

stiffness matrix of an element from the flexibility

ma&is.

EQBM

Here the reaction forces at all restraint

points are calculated and summed. The moments of

these reactions are evaluated about the origin.

PRNT. Prints out the results in the desired form.

AN ANALYSIS OF

A SAiMPLE PROBLEM

A three-dimensionai piping system (Fig. 3) with

29 elements consisting of tangents, elbows in dif-

ferent planes, supports with various constraints and

ends with external loading applied, is analysed to

show the applicabiIity of the present programe. A

brief summary of the displacements at specific

nodal points is given in Table 1. The stresses ob-

tained are not presented here. It is found that the

deformation obtained here compares well with

those obtained from a commerical package.

Table I. The displacements at specific nodal points

Node

X-DISPL

im.m)

Y-DISPL

(m.mJ

z-RISPL

(m.m)

2 Il.9 3.85

0.0

6 32.84 12.66

- 13.75

IS 32.84 12.66

13.75

20 49.03 0.0

0.0

29 128.18 -5J.21

30.5 I

CALCULATION OF FLEXIBILITY OF AN ELBOW

It is explained here how the present piping anal-

ysis program is utilised to obtain a consistent value

for the flexibility factor of an elbow.

As an example, it is required to evaluate the fiex-

ibility factor of a 30 elbow when its ends are con-

strained by tangent pipes from the results obtained

from a tinite element analysis. Figure 4 below

shows the layout of the piping system.

Table 2. Results for the 30 elbow

Trial flexible

X-DISPL at

coefficient

Node I

Y-DISPLat

Node I

X-DISPLat Y-DISPLat

Node 3

Node 3

F.E.M.

-0.0013 -0.0174

-0.0012 -0.0051

12.0

-0.001 I -0.0169

-0.0011 - 0.0038

13.0

-0.001 I -0.0179

-0.001

I - 0.0040

13.5

-0.001 I -0.0184

-0.001 I -O.OWI

Y

Fig. 3. A three-dimensional piping system.

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M =686x10

Pipe flexibility analysis program

3 5

Fig. 4. Layout of the piping system.

An assumed value for the flexibility factor of the

30 elbow is input into the present piping program.

The deformations at the free end and at the elbow-

tangent junction are compared with those obtained

from finite element analysis. This iterative process,

of assuming the flexibility factor from the piping

program with those obtained from the finite element

analysis, is continued until satisfactory results are

obtained.

Table 2 shows the results obtained for the 30

elbow. Hence 13.0 is accepted as a consistent flex-

ibility factor.

COXLUSIONS

The description of a simplified piping analysis

program is given here. Its use in solving moderately

simple piping system is also shown. Further its ap-

plicability to evaluate a consistent value for flexi-

bility of an elbow is also explained. Thus this type

of simple program is of great value for practicing

engineers. In addition. efforts are on the way to

implement this program

for microcomputers such

as Apple.

REFERENCES

I . Natarajan and J. A. Blom tield, Stress analysis of

curved pipes wit h end restraints. Inr. J. Compur. Stnrc-

fit res 5, 187-196 (1975).

2. R. Natarajan and S. blir za. Stress analysis of curved

pipes with end restraints subjected to out-of-plane mo-

ment . F 2/8, Proc. 6th SMIRT Conf ., Paris (1981).

3. R. Natarajan and S. Mirza, Effect of intern al pressure

on

flexibility factor in pipe bends with end constraints.

Danerno. 83 WA/DE-l I. Proc. ASME. Bosto n C983).

4. k.Thomas, Stiffening effects of thin-killed pi&g ii-

bows of adjacent piping and nozzle constr aint. PVP-

Vol. 50, St ress In di ces and Str ess In tensifi cati on Fac-

tor s of Pressure Vessel and Pipi ng Componenrs.

pp.

93-108, ASME.

5. E. C. Rodabaugh and S. E. Moore, End effects on el-

bows subject ed t o moment land ing s. PVP-Vol. 56, Ad-

vances i n Design and Anal ysis M ethodol ogy for Pres-

sare Vessels and Pi pi ng,

pp.

99-123. ASME.

APPENDIX A

A = n 2lL - sin ZJl)f ,, + (1.55 + 0.525 sin 2b)f7

B = -n(l + cos 2Jl)f6 + 0.525t I - cos 2 )f7

C = n(l - cos 4j.f~

D = n(2JI + sin 2Jl)f6 + (I.55 II, - 0.525 sin 2*Jf7

E = -n sin fj

F = 4tl + CL) f6 + 2.6 ti f7

G = -(I + p)(I - cos Jl)f~

H =

(I + p)sinJIfj

I = [2(l + tk + n)JI - (I + p - n) sin hL]fr

J = -(I + p - nhl - cos 23r)fj

K = [2(l + F + n)l + (I + p - n) sin 2ti1f4

L = 4n3rf4,

where

f, = Rl4EI

f5 = RIEI

f6 =

Rf4EI

f7 = r RIEI

r

is mean radius of pipe cross-section