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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