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SHOCK LOADS ON PIPING SYSTEMS
Dennis Harold Peters
NAVAL POSTGRADUATE SCHOOL
Monterey, California
THESISSHOCK LOADS ON PIPING SYSTEMS
by
Dennis Harold Peters
Thesis Advisor: R.E. Newton
September 1972
T !*
Approved ^on. puJotic. ^.eXeaie; dibt/UbiLtion imZimctzd.
Shock Loads on Piping Systems
by
Dennis Harold PetersLieutenant, United 'states Navy
B.A.E., University of Minnesota, 1965
Submitted in partial fulfillment of therequirements for the degree of
MASTER OP SCIENCE IN MECHANICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOLSeptember 1972
7z
ABSTRACT
This thesis describes the development of a Fortran IV
computer program for finding shock-induced stresses in a
three-dimensional piping system. Discretized equations of
motion are formed by the finite element method. Input
motions are support translations specified by response
shock spectra. Maximum responses in individual modes and
resulting octahedral shearing stresses at selected points
are found for shock motions in three orthogonal directions,
Extreme stresses, based on the square root of the sum of
the squares of the modal stresses, are estimated for each
input direction. An example problem is analyzed to demon-
strate the use of the program.
TABLE OF CONTENTS
I. INTRODUCTION 11
II. THEORY 14
A. MODEL LIMITATIONS 14
B. FINITE ELEMENT 15
1. Element Displacement Vector 15
2. Element Stiffness and Mass Matrices 16
C. ASSEMBLED EQUATIONS OF MOTION 18
D. MODAL ANALYSIS 19
E. GENERALIZED FORCES AND STRESSES 20
III. PROGRAM DEVELOPMENT 23
A. SYSTEM DESCRIPTION 23
B. STIFFNESS AND INERTIA MATRIX FORMULATION 24
C. FREE VIBRATION MODE SHAPES AND FREQUENCIES — 24
D. MODAL RESPONSE 25
E. STRESSES 26
F. PROGRAM OUTPUT 29
IV. PROGRAM TESTING 30
A. TEST PROBLEM 1 30
B. TEST PROBLEM 2 31
C. TEST PROBLEM 3 32
V. EXAMPLE PROBLEM 34
A. PIPING SYSTEM 34
B. DATA INPUT AND OUTPUT 34
VI. DISCUSSION 40
A. CONFIGURATION LIMITATIONS 40
B. SIZE LIMITATIONS 40
VII. CONCLUSIONS ^2
VIII. APPENDICES ^3
A. ELEMENT STIFFNESS AND INERTIA MATRICES ^3
B. VELOCITY SHOCK SPECTRA AND MODAL RESPONSE ~ ^5
C. MODE SHAPES AND FREQUENCIES OFEXAMPLE PROBLEM l\$
D. CALCULATION OF SPECIFIC WEIGHT ANDRADIUS OF GYRATION 51
E. PROGRAM 52
LIST OF REFERENCES 83
INITIAL DISTRIBUTION LIST 85
FORM DD 1^73 86
LIST OF TABLES
I. Comparison of first three bending frequenciesof Test Piping System 2 with the exact results 32
II. Data input, example problem 36
III. Mode velocities, example problem 37
IV. Mode participation factors, example problem 38
V. Octahedral shearing stresses, example problem 39
VI. Element stiffness matrix 43
VII. Element inertia matrix 44
VIII. Eigenvalues and eigenvectors, example problem 50
LIST OF FIGURES
1. Element generalized displacements 15
2. Element generalized forces 21
3. Stress location points 27
4. Positive stress convention 28
5. Test Piping System 1 30
6. Test Piping System 2 31
7. Test Piping System 3 33
8. Example piping system 35
9. Beam element coordinates 46
LIST OF SYMBOLS
A single underline on a capital letter denotes a
rectangular matrix, and a single underline on a lower case
letter denotes a column vector. Superior dots denote time
derivatives. The symbols used in the computer program are
described in Appendix E.
A Element cross-sectional area
b Mode participation factor, mode r
C A constant
D Pipe outside diameter
E Young's modulus
f Element generalized force vector
f. Components of the generalized force vector
G Shear modulus
I Second moment of the pipe cross-sectional areaabout a diameter
I_ Identity matrix
J Mass moment of inertia per unit length about pipe axis
K Assembled stiffness matrix
K Element stiffness matrix
£, Length of an element
M Assembled inertia matrix
Me Element inertia matrix
Mg.jp Element inertia submatrix, bending 1-2 plane
MR1-, Element inertia submatrix, bending 1-3 plane
M_L
Element inertia submatrix, longitudinal motion
MS Element inertia submatrix, torsional motion
m Modal mass, mode rr
nu Mass of the lagging per unit length
mp
Mass of the pipe per unit length
N Row vector of shape functions
p_Vector of principal coordinates
p Principal coordinate, mode r
Q Moment of pipe cross-sectional area lying on one sideof neutral axis
c[ Displacement vector
r Radius of gyration
s Base displacement
Te Kinetic energy of an element
t Pipe wall thickness
u Vector of absolute displacements corresponding tounit base displacement
u. Components of displacement vector
V Modal matrix
V Spectrum velocity, mode r
v Eigenvector, mode r
VL Weight of lagging per unit length
Wp Weight of pipe- per unit length
w Absolute displacement vector
z One degree-of-freedom system coordinate
V Modified specific weight
£ Dimensionless length coordinate
p Density
o Modal stress
x Shearing stress
t . Octahedral shearing stress
2ft Spectral (diagonal) matrix
u Natural circular frequency, one degree-of-freedomsystem
a) Circular frequency of mode r
Superscripts
T Transpose of matrix
(r) Mode designator for eigenvectors (r=l,2, . ..,n)
ACKNOWLEDGEMENTS
I wish to express my sincere appreciation to my advisor,
Professor Robert E. Newton, for his invaluable assistance
and constructive supervision in the preparation of this
thesis, and to Professor Gilles Cantin for his assistance
in the computer work.
I would also like to express my thanks to the staff of
the W.R. Church Computer Center, Naval Postgraduate School.
10
I. INTRODUCTION
With the greater demand for electrical power, and the
diminishing fossil fuel supply, there has been an increase
in the construction and use of nuclear power generating
plants. Correspondingly, there has been greater concern for
the safety of the power plant during an earthquake.
Also, the reduction in the size of the Navy has led to
an increased emphasis on the integrity of the internal
systems of a naval vessel under shock loads. Thus, there
has been a growing engineering interest in finding a rapid
and accurate method of analysis to determine the shock-
induced stresses in a complex continuous piping system.
This thesis describes and presents a computer program,
written in Fortran IV computer language, to find the
stresses in a piping system responding to shock.
Complex piping systems can be analyzed by using dis-
cretized models. In using the discretization technique, the
continuous piping system is modeled by finite size elements
connected at nodes. After the system has been subdivided,
the elastic and inertial matrices of each element can be
found and assembled to form the elastic and inertial
matrices of the system [1,2].
A considerable volume of material is available to assist
in analyzing structures and systems responding to
earthquakes [3,^,5]. Two of the techniques used in earth-
quake response analysis are modal analysis with a
11
discretized model and transfer functions [4,5]. The shock
input for earthquakes can be specified by using time
history [3]>or shock spectrum [4,5].
The Navy uses a technique called the Dynamic Design-
Analysis Method (DDAM) to determine the shock response of
shipboard equipment and systems [6]. The DDAM is a modal
analysis technique using shock spectra to specify the shock
inputs.
The shock spectrum presents, as a function of the system
natural frequency, the maximum response of a one degree-of-
freedom system responding to a shock motion. The shock
spectrum is generally presented in terms of response velocity
or acceleration.
Several graduates of the Naval Postgraduate School have
written theses on determining natural frequencies and mode
shapes for piping systems. Fink [7] developed a program to
analyze planar piping systems, including out-of-plane
bending, using transfer matrices. Baird [8] presented the
necessary theory to expand Fink's work to a three-dimensional
piping system. Rudolf [9] developed a program that deter-
mines the frequencies of a general three-dimensional piping
system. Because the transfer matrix does not yield a dis-
cretized model of the structure, there is no ready means
for using the frequency and mode shape data to find shock-
induced modal responses. On the other hand, the finite
element technique provides a consistent discretized model
12
which may be used for finding frequencies, mode shapes, modal
responses to shock inputs, and resulting stresses. For this
reason the finite element method was chosen for this thesis.
Shock spectra and mode participation factors are used
to determine the modal responses. At selected points the
modal octahedral shearing stresses are combined by scalar
addition of the distortion energy to estimate extreme
stresses
.
13
II. THEORY
Material and structural linearity and material local
homogeneity and isotropy are assumed in developing the
mathematical model.
A. MODEL LIMITATIONS
There are few inherent restrictions on the capabilities
of the finite element method to model complex details of
piping configurations. Considerations of the quantity of
input and output data, program length, core storage capac-
ity, computing time, and cost do impose practical limita-
tions. The limited time available in developing this
thesis has necessitated the following restrictions on the
model used here.
The system consists of straight lengths of constant-
section pipe (elements). Each element centerline is parallel
to one of the three mutually orthogonal global refer-
ence axes, and 90° bends are replaced by fictitious exten-
sions of the tangent sections to the intersection point of
the centerlines. Effects of added mass contributed by
external lagging are represented, but no provision is made
for added mass due to valves or fittings. Pipe hangers
furnishing uniaxial restraint in a global direction may
also be represented. Bending action is described by the
Euler-Bernoulli beam theory, neglecting shear deformation
lh
and rotatory inertia. All ends of the configuration are
treated as fixed. The shock input motion is a translation
of the base along a global axis.
B. FINITE ELEMENT METHOD
Once the system has been subdivided into elements,
the elastic and inertial characteristics are determined.
The element stiffness matrix can be determined by many
different techniques, whereas the inertial matrix is gener-
ally determined by using shape functions and integrating
along the length, [1,2].
1. The Element Displacement Vector
Consider an element with displacement components at
the nodes (ends) as shown in Fig. 1.
FIG. 1
ELEMENT GENERALIZED DISPLACEMENTS
The clamped ends of the configuration are treated asjoined to a rigid base.
15
Here, the double arrowhead vectors are positive rotations
in accordance with the right-hand rule. The displacement
vector is
Tq = [u
1u2
... u12 ]
From Fig. 1, components u, and u„ are longitudinal dis-
placements, components u~ , u_, u,-, Ug, Ug, uQ
, u,, and u,«
are bending displacements and rotations in the 1-2 and 2-3
planes, and components Ul and u..Q
are twisting rotations.
2. Element Stiffness and Inertial Matrices
Przemieniecki [2] forms the element stiffness ma-
trix by solving the differential equations of the element
displacements. Appendix A reproduces the element stiffness
matrix with Przemieniecki' s transverse shear deflection
factor (<{>) set equal to zero.
To determine the element inertia matrix, an expres-
sion for the element kinetic energy T is formed, rotatory
inertia due to bending being neglected.
el -^ .T T1 itpTe
= j Ci f q N" N q d£ = ~ q1Me
q"
where i is the element length, C is a constant,' Me is the
element inertia matrix, N is the shape function row vector,
and % is a dimensionless length coordinate.
16
Therefore
Me
= Ci f NT
N d£ (1)
Since the local element axes coincide with the cross
section principal axes, the inertia matrix may be deter-
mined from 2x2 and 4x4 element sub-matrices. Consider the
following motions of the element:
a) longitudinal:
q^ = [u]_u7
]T
, N1
= 1 - 5, N?
= e
and C = pA where A is the cross-sectional area of the
element and p is the density. Prom Eq. 1
pA£ 2 1
1 2(la)
b) twist:
qT= [ U2| u
1Q ] , N4
= (1 - 5)1, N1Q
= ?!
and C = J where J is the mass moment of inertia per unit
length about the pipe axis. From Eq. 1
£-£ 2 1
1 2(lb)
c) bending in the 1-2 plane:
2,o,3tel2
= [u2
u6
u8
u12 ] , N
2= 1-35 +2C
N6
= £(C-2C2+C
3), N
g= 3S
2-2£ 3
, N12
= ft(-€2+C
3),
17
and C = pA. From Eq. 1
-B12
pA£T20
13i
3* 2
156 22£ 5^
22£ kl2
13£
5^ 13 ^ 156 -22)1
-13* -3S-2 -22£ *U
2
(lc)
Bending in plane 1-3 results in the same numerical coeffi-
cients as Mm . Because the positive senses for the—Did
rotation angles are reversed, rows 2 and 4 and columns 2
and *J are multiplied by minus one. These submatrices are
assembled to form the element inertia matrix. This matrix
is displayed in Appendix A.
C. ASSEMBLED EQUATIONS OP MOTION
The governing equations of motion, in matrix notation,
for undamped free vibration [2] with a fixed base are
M q + K q (2)
where q is the assembled displacement vector and M and K
are the assembled inertial and stiffness matrices. M and
e eK are referred to a global coordinate system and M and K
are referred to local coordinates. The assembly process
consists of realigning the local system to the global
system and building the M and K matrices. Reference [2]
gives sample assembly techniques.
18
D. MODAL ANALYSIS
Modal analysis is the process of finding the response
motion of a many degree-of-freedom system by superposition
of the response motions in the principal (or natural) modes.
This technique is advantageous because the motions in the
principal modes are independent of one another (uncoupled).
To use the method, it is first necessary to find the nat-
ural frequencies and mode shapes for the free motions
governed by Eq. 2, i.e., to solve the eigenvalue problem
K v = w2M v.
2Finding eigenvalues (to ) and eigenvectors (v) is a
standard problem of numerical analysis. The first step is
a transformation to a single-matrix problem. Details of
the transformation used here, which requires a Cholesky
decomposition of the inertia matrix, are given on p. 3^9
of reference [1]. Following this, Jacobi rotations [15]
2are used to find the spectral and modal matrices (ft and V)
.
The modal matrix is normalized with respect to the inertia
Tmatrix, i.e., V M V = I, where I is the nth order identity
matrix.
The shock analysis is based on using velocity shock
spectra to characterize the input motion and mode parti-
cipation factors to determine the responses. Although both
shock spectra and mode participation factors have been
extensively used in earlier studies [6], the finite element
19
discretization necessitates a new development of working
formulas. Details of this development are given in
Appendix B.
Each shock input motion consists of base translation
parallel to a global axis. It is shown in Appendix B that
the mode participation factor b for mode r is
Tb = v
(r)M u (B.5)
(r)where v is the modal eigenvector and u is the vector of
absolute displacements corresponding to unit base displace-
ment. The maximum relative displacements due to mode r
response are
q = v(r)
b V /uj (B.6a)—max — r r r
where w is the modal circular frequency and V is the
spectrum velocity at that frequency.
Since responses to shock inputs in each of the three
global directions are separately determined, the vector u
and the mode participation factors b must be evaluated
separately for each direction. The modal masses m (see
Appendix B) are likewise different for each input direction,
E. GENERALIZED FORCES AND STRESSES
The subvector qe
of element peak modal displacements
may be found from the system vector, taking into account the
20
orientation of the local reference axes. From this the
element generalized force vector f is determined. This
force vector consists of the elastic and inertial
contributions for each element.
fe
= (Ke
- Wr2
Me
) qe
(3)
Fig. 2 shows the positive directions for the components
of the generalized force vector.
fe
= [f f ... f12 i
T
FIG. 2
ELEMENT GENERALIZED FORCES
21
The stresses at the nodal points of the element can be
determined from the generalized forces by standard methods
of solid mechanics [10].
22
III. PROGRAM DEVELOPMENT
The computer program was written in Fortran IV com-
puter language using double-precision arithmetic on an
IBM 360/67 digital computer. It consists of a main calling
program and seven subroutines. The main program calls the
subroutines, where the actual calculations occur, in the
necessary order. Information is passed between the main
program and the subroutines by the use of a common storage
core. Appendix E contains the program listing.
A. SYSTEM DESCRIPTION
After the piping system has been modeled and subdi-
vided, subroutine INPUT is used to read the data input of
the geometry and material properties of the system. The
geometry of the piping system model is determined by the
global coordinates of the nodes of the model. The global
system is a right-handed orthogonal coordinate system.
Each pipe segment must be parallel to a global axis, but
there are no restrictions on the placement of the origin of
the global system. The parameters from the subdivision of
the system are read. For each node the global coordinates
and boundary conditions are read. If the node is fixed,
the stiffness and inertial matrix components due to the
node displacements are not assembled into the system
matrices
.
23
The numbering of nodes and elements is arbitrary.
There are two nodes per element and six degrees of freedom
per node. For each element, the node numbers and pipe
group are read. A pipe group includes all elements that
have the same Young's modulus, Poisson's ratio, specific
weight, radius of gyration, and cross-sectional dimensions.
Appendix D shows the method used to calculate the effect
of lagging on the specific weight and the radius of
gyration.
B. STIFFNESS AND INERTIAL MATRIX FORMULATION
Subroutine F0RM generates the element stiffness and
inertia matrices as shown in Appendix A. Rotatory inertia
and shear deformation due to bending are neglected. The
orientation of the element is determined with respect to
the global coordinate system and the correspondence between
local and global degrees of freedom is established. After
determining the correspondence between the local and global
degrees of freedom, the element stiffness and inertia
matrices are assembled to form the system stiffness and
inertia matrices. If pipe hangers are present, the corres-
ponding (uniaxial) hanger stiffnesses are added to the
appropriate diagonal elements of the system stiffness
matrix.
C. FREE VIBRATION MODE SHAPES AND FREQUENCIES
Subroutine CHJ0M0D uses Cholesky decomposition of the
system inertia matrix and coordinate transformation to form
2i|
a single symmetric matrix for which eigenvalues and eigen-
vectors are found. Subroutine JACR0T is a modification of
a program originally coded by Professor G. Cantin. It
uses the Jacobi variable threshold method to find eigen-
values and eigenvectors of a real symmetric matrix. The
modal matrix, when transformed back to system coordinates,
is normalized with respect to the system inertia matrix.
Since the system matrices increase in size with finer
subdivisions or more complex systems, an economizing tech-
nique, to reduce the number of degrees of freedom in the
eigenvalue problem, was investigated during the program
development. The component mode synthesis method as used
by Benfield and Hruda [11] was studied. It was established
that the constrained component branch technique [11], with
interface loading, with no node suppression gave results
identical with those obtained by the direct method described
above when applied to planar vibration of a clamped-c lamped
beam represented by six elements. Despite the success of
this trial, it was concluded that the added program com-
plexity accompanying the use of component mode synthesis
would offset the potential economies. Another standard
economizer technique [14] was considered, but ultimately
rejected for the same reason.
D. MODAL RESPONSE
The number of modes to be used in the stress analysis
and the specification of the shock spectrum are input data
25
for subroutine SPCTRM. There are three choices available
to select the number of modes to be used: a designated
upper frequency limit, an upper limit which is a multiple
of the fundamental frequency, and an option for any method
the user desires. Three choices are also available for
specifying the shock spectrum: a spectrum defined by
straight-line segments; a constant velocity, then constant
acceleration shock spectrum; and an option for any method
the user desires. The shock input is restricted to a
translation of the base along a global axis and is speci-
fied by one shock spectrum and three scaling factors for
the three input directions. Subroutine M0DE modifies the
spectrum velocity for modal mass, if desired. There are
three choices available: no correction, log of the modal
weight correction, and an option for any method the user
desires. This subroutine also calculates the mode
participation factors.
E. STRESSES
Subroutine STRESS calculates stresses at the nodes of
the elements. The element peak displacements are deter-
mined and realigned from global degrees of freedom back to
the local element degrees of freedom. The element gener-
alized force vector is then calculated. Consider the
action of the generalized force vector on the element
shown in Pig. 3-
26
FIG. 3
STRESS LOCATION POINTS
At locations a,b,c, and d, the stress vector acting on the
cross-section may be resolved into normal and shearing
stresses. Fig. k shows the positive sign convention used
for the normal and shearing stresses.
The normal stress a and the shearing stress x at each
point shown in Fig. 3 are given by:
1. at location a)
a = -f-j/A - fV D/2I
x = -fjj D/4I + f2
Q/2It
27
FIG. 4
POSITIVE STRESS CONVENTION
where I is the second moment of cross-sectional area about a
diameter, D is the outside diameter, Q is the first mo-
ment, about a diameter, of the portion of the cross-sectional
area lying on one side of the diameter, and t is the pipe
wall thickness.
2. at location c)
a = f /A + fn D/2I
x = f1Q
D/4I - fg Q/2It
3. at location b)
a = - f1/A - fg D/2I
f .. D/JJI + f. Q/2It
28
at location d)
a = - f?/A - f
12D/2I
t = f1()
D/4I - f9
Q/2It
These normal and shearing stresses are then used to find
the octahedral shearing stresses [12], which are given
by
Toct = 1/3 (2 *
2+ 6^ )2 W
For each element the above calculations are performed for
each mode. At each of the four points, the shearing
stresses for the individual modes are then combined by
determining the square root of the sum of their squares.
This process is accomplished for each input direction.
This is done for each element in turn, starting with the
first.
F. PROGRAM OUTPUT
All input information is echo-checked. The square
of the mode frequency and the mode shape are printed for
each mode. The spectrum velocities and mode participation
factors are also printed. The octahedral shearing stresses
at points a, b, c, and d (Fig. 3) of each element are
printed for the three shock input directions.
29
IV. PROGRAM TESTING
In order to explore the capabilities of the program
and verify its integrity, a number of plane test configur-
ations were studied. These are described in the following
sections.
A. TEST PROBLEM 1
The configuration shown in Fig. 5 has three identical
uniform runs of equal length between the central junction
and the clamped edges.
/&
tFIG. 5
TEST PIPING SYSTEM 1
The mode shapes and frequencies for in-plane vibration of
this system were studied extensively with a developmental
program. The present program gave identical results for the
30
in-plane modes. The out-of-plane modes exhibited the
required symmetric or anti-symmetric form.
Additional program tests on this configuration involved
different element numbering and node numbering and six
different orientations of the global axes relative to the
structure. Mode shapes and frequencies were unaffected by
these changes.
B. TEST PROBLEM 2
This system, shown in Fig. 6, consisted of a straight
uniform pipe clamped at the ends and represented by four
elements.
%
FIG. 6
TEST PIPING SYSTEM 2
The frequencies found for the lower modes showed good
agreement with exact results. Table I shows the comparison
of the first three bending frequencies of the finite element
solution with exact results [13]. All the eigenvectors
showed the required symmetric or anti-symmetric form.
Bending modes occurred in pairs having equal eigenvalues
and with the corresponding eigenvectors representing
deflections in orthogonal planes.
31
Stresses found for this system were studied in detail.
At each node, the stresses are calculated at two points of
the cross-section. Except at the clamped ends, there are
two adjacent elements sharing each node, so that stresses
at these sections are calculated twice. Complete agreement
was found between these two sets of stress evaluations.
Also, identical stresses resulted from shocks in the two
directions perpendicular to the length.
TABLE I
COMPARISON OF FIRST THREE BENDING FREQUENCIES OF TESTPIPING SYSTEM 2 WITH EXACT RESULTS
ode Exact
(Hz)
FiniteElement(Hz)
PercentDifference
1 422.32 422.65 0.08
2 1164.15 1174.26 0.9
3 2282.20 2329.64 2.1
C. TEST PROBLEM 3
The equal- legged configuration of Fig. 7 consists of
uniform pipe throughout. Mode shapes of this system again
showed the required symmetric or anti-symmetric form.
Likewise, the shock-induced stresses exhibited the expected
symmetry properties.
32
4^
FIG. 7
TEST PIPING SYSTEM 3
33
V. EXAMPLE PROBLEM
A. PIPING SYSTEM
The example system analyzed is a model of the main
steam piping system on a modern naval vessel. The main
steam piping from the boiler to the rigid cross-connect
anchor assembly is modeled. The ends of the piping system
are clamped, and there are two hangers. The system is
three-dimensional with no branches. The pipe properties
are:
outside diameter 7.625 inches
inside diameter 6.011 inches
Poisson's ratio 0.300
Young's modulus 30x10 psi
specific weight 0.327 lb/in 3
radius of gyration 4 . 04 inches
The specific weight is a fictitious value which accounts
for the weight of five inches of lagging bonded to the pipe
Fig. 8 is a schematic of the piping system. The piping
section PI is 84 inches long, P2 is 264 inches long, P3 is
108 inches long, and P4 is 288 inches long.
B. DATA INPUT AND OUTPUT
The input data cards are punched in accordance with the
instructions contained in Appendix E. The echo-check of
the data is shown in Table II and is arranged as it would
appear at the end of the program deck. The system
34
s
*S-Pi
FIG. 8
EXAMPLE PIPING SYSTEM
eigenvalues and eigenvectors are listed in Appendix C,
Table VIII. Tables III and IV show the modal velocities
and the mode participation factors. The stress output is
given in Table V.
This problem required a storage capacity of 17^K bytes,
and 2.67 minutes of computer time.
35
TABLE II DATA INPUT, EXAMPLE PROBLEM
PROBLEM NUMBER! 1
NUMBER OF PROBLEMS TO BE SOLVED
TOTALELEMENTS
12
TOTALNODES
DEG OF FREEDOMPER NODE
DEG OF FREEDOMPER ELEMENT
12
TOTALGROUPS
SYSTEM TOTALDEG OF FREEDOM
66
NODE X COORD Y COORD Z COORD NODAL BOUNDARYINCHES INCHES " INCHES- CONDITIONS
1 0.0 0.0J2 42.00 .0 0.0
3 84.00 0.04 84.00 66.005 84.00 .0 132.006 84.00 3 198.037 84.00 264.008 138.00 264.009 192.00 264.00
10 192.00 72 .00 264.0011 192.00 144 00 264.00
n 192.00 216 00 264.00192.00 288 .00 264.00
ELEMENT LEFT RIGHT- GROUPNUMBER NODE NODE NUMBER
1 1 2 I
2 2 3 .
3 3 4 .
4 4 5'
.
5 5 66 6 77 7 68 8 9 ;.
9 9
\l
.
10 1011
12 12 13
OUTSIDE INSIDE POISSONS SPECIFIC YOUNGSDIAMETER DIAMETER RATI MODULUSIN. IN. LB/CU.IN PS I
SHEARMODULUSPSI
7.625 6.Q11 0.300 3.000D 07 1.1540 07
PIPEGROUP
RADIUS OF GYRATIONINCHES
1 4. 04
NUMBER OF HANGERS:
GLOBAL DOF IADOF HANGER STIFFNESSLB/IN.
711
33
9.817477D 059.8174770 05
SPECTRUMTYPE
1
FREQUENCYTYPE
BREAK FREQUENCYHERTZ
CONSTANT VELOCITYINCHES/SEC.
50.00 100.00
SHOCK INPUTX
FACTORSY Z
1.000 1.000 1.000
MODE WEIGHT CORRECTION TYPE
36
TABLE III MODE VELOCITIES EXAMPLE, PR(
MODE SPECTRUM VELOCITYINCHES/SEC.
1 100.002 100.003 10 3.004 100.005 100.006 10 0.007 100.008 100.009 100.00
10 88.8711 84.4912 75.1013 71.3914 58.8815 53.4616 47.1117 42.7518 42.1919 32.5220 30.3021 29.9222 29.2323 27.0724 26.0825 25.7626 25.4327 22.0728 21.7629 17.2930 17.0431 16.1032 15.1833 14.3434 14.1435 13.8436 12.2437 12.1638 12.0839 10.6740 10.2941 10.0342 9.6543 9.3244 8.8545 8.3546 8.3147 8.0248 7.8249 7.535 3 7.1451 6.9152 6.6153 6.5154 5.9455 5.8056 5.4857 5.2758 4.7459 4.5860 4.4761 4.3062 3.9163 3.6864 3.4365 2.9766 2.73
37
TABLE IV MODE PARTICIPATION FACTORS, EXAMPLE PROBLEM
MODE MODE PARTICIPATION FACTORSX Y Z
1 2.42DD DO -5.433D-01 -2.376D-012 8.881D-01 1.936D 00 -8.579D-033 6.749D-01 -2.151D-01 1.366D 004 1.983D-01 -8.6390-01 -2.739D-015 1.813D-01 3.681D-02 4.449D-016 1.0D2D JJ 2.950D-02 -4.5250-017 -2.002D-01 8.847D-01 -5.983D-028 1.342D-01 2 .9090-02 -4.1320-019 3.281D-01 7.394D-02 -1.452D-01
10 -5.372D-01 2.087D-02 -2.2340-0111 -1.760D-01 -2.7650-02 3.701D-0112 -1.461C- 01 1.7600-02 -4.3580-0113 -2.436D-02 -2.057D-02 -2.5810 0014 2.101D-01 6.4130-01 -2.771D-0215 2.120D-01 3.6170-02 6.2730-0216 2.414D-01 -4.2060-01 -5.3080-0217 -1.557D-01 -1.836D-01 -8.9290-0318 -2.999D-01 2.7700-01 -9.8480-0219 -2.523D-03 -1 .5750 00 2.7830-0220 3.7280-02 3.9690-01 4.6230-0121 -7.001D-02 -7.5210-01 1.8320-0122 • 5.6800-02 -2.8310-03 3.3900-0123 2.730D-02 3.0590-02 1.122D-0124 1.372D-02 -1.4560-01 -7.496D-0225 2.O96D-01 3.593D-02 5.776D-0226 -5.284D-02 1.7100-01 -2.951D-0327 3.018D-01 1.8490-02 -2.0460-0128 6.857D-02 -4.139D-01 3.6120-3229 1.502D-01 1.9650-02 2.480D-0130 1.343D-01 -3.7780-02 5.7400-0231 1.741D-01 -5.5900-02 -1.286D-0132 8. 7360-02 -7.8550-03 -2.3340-0133 3.208D-01 -3.3680-02 -1.143D-0134 2.O57D-01 1.0940-01 -7.9530-0235 -1.832D-02 1.4720-03 -1.7040-0136 -5.653C-02 -2.6610-01 3.9710-0237 -2.768D-01. -2.8490-02 -1.585D-0138 -1.338D-01 4.8110-02 1.0380-0139 5.2820-31 1.843D-02 -1.1140-0140 -3.848D-01 -1.9360-01 -4.5490-0241 3.217D-01 -4.5550-01 1.7640-0342 -9.8130-02 -2.3620-01 5.5900-0243 1.810D-01 9 .5520-02 8.5750-0244 -4.923D-31 -6.4730-03 -6.5880-0245 -1. 517D-01 4.6710-03 -9.275D-0246 4.5190-02 2.3120-01 6.8950-C347 -3.715D-32 8.8880-04 -4. 6700-0248 -7.110D-02 2.7250-02 -1.0660-0149 -8.744D-02 4.0900-02 2.3070-0253 -5.489D-02 -6.571D-02 -1. 5690-0251 3.389D-02 4.8980-02 -3.5650-0252 8.851D-02 -1.2770-01 5.838D-0253 -5.329D-02 -6.4960-02 -1.231D-0154 5.C54D-02 1 .6870-01 8.028D-0355 6.4810-34 -2.6370-02 -1.6450-0256 -2.0470-02 2.8140-01 -7.3650-0357 -2.748D-03 -5.7720-02 -3.583D-0358 1.960D-02 -2.5910-02 -1.7730-0259 -1.5410-C1 1.049^-02 8.5290-0260 -1.5480-01 -1.8900-02 1. 1470-0161 -8.0590-03 2.6220-02 4.9190-0262 1.948D-03 -1.546D-01 2.3570-0363 8. 5170-05 -1.3750-01 1 .0660-0464 -3.405D-02 -5.9190-03 -3.0 030-0265 5. 4040-03 -7.4280-03 6.0630-0366 2. 524D-01 4. 6170-35 3.8080-02
38
TABLE V OCTAHEDRAL SHEARING
ELEMENT 1
9.271550 03 6.508980 03 8,
6.674260 03 1.687530 04 6.
6.609590 03 6.198620 33 2.
4.588570 03 8.645530 03 3.
STRESSES, EXAMPLE PROBLEM, PSI
jl ELEMENT 7
408350 03 ^ 1.055150 04 3.19562D 03 7.845000 03
48256D 03^ 4.203340 03 5.39532D 03 2.892360 03
61721D 03 5.88687D 03 3.140880 03 3.869280 03
095510 03 4.372620 03 5.55352D 03 3.82004D 03
ELEMENT 2
6.609590 03 6.198620 03 2.
4.58857D 03 8.64553D 03 3.
1.879580 04 7.527770 03 7.
4.96003D 03 7.392360 03 1,
jELEMENT 8
617210 03 j 5.886870 03
095510 03 4.37262D 03
71406D 03 2.656300 03
04808D 03 7.424390 03
3.14088D 03
5.55052D 03
3.213450 03
1.318900 04
3.869280 03
3.82004D 03
3.567750 03
4.40520D 03
ELEMENT 3
5.764590 03
1.88813D 04
4.458580 03
1.114120 04
7.613430 03
6.471280 03
5.76172D 03
4.91469D 03
1.77620D 03
7.52389D 03
3.03388D 03
2.67322D 03
ELEMENT 9
7.59211D 03
2.46920D 03
5.77644D 33
4.85717D 03
1.27242D 04 4.36537D 03
2.81124D 03 3.50809D 03
1.01342D 04 3.201110 03
2.06310D 03 8.55130D 03
ELEMENT 4
4.458580 03
1.11412D 04
4.561880 03
7.49948D 03
5.761720 33
4.91468D 03
9.146560 03
4. 300960 33
3.03388D 03
2.673220 03
3.568610 03
6.18924D 03
ELEMENT 10
5.77644D 03
4.857170 03
7.21896D 03
9.86817D 03
1.01042D 04 3.20111D 03
2.06310D 03 8.55130D 03
6.12863D 03 4.520530 03
3.488530 03 1.570740 04
ELEMENT 5
4.561880 03
7.499480 03
4.543860 03
6.25280D 03
9.146560 03
.4.300960 03
7.59655D 03
4.33059D 03
3.56861D 03
6.18924D 03
4.76524D 03
7.081600 03
ELEMENT 11
6.94094D 03
9.868170 03
1.04300D 04
3.535350 03
6.083360 03 4.586790 03
3.488530 33 1.573740 04
4.37922D 03 2.33575D 03
2.30537D 03 9.55394D 03
ELEMENT 6
4.543860 03
6.252800 03
3.61390D 03
1.086730 04
7.59656D 03
4.33059D 03
4.921470 03
4.381410 03
4.765240 03
7.08160D 03
2.47225D 03
7.711130 03
ELEMENT 12
1.34303D 04
3.535340 03
1.769770 04
6.153500 03
4.379220 33 2.33575D 03
2.30537D 03 9.553940 03
1.275160 04 6.00477D 03
2.751500 03 1.630620 04
39
VI. DISCUSSION
Some of the limitations of the capabilities of the
program developed above are considered here.
A. CONFIGURATION LIMITATIONS
The restrictions that pipe axes must be parallel to a
global axis and that finite radius bends are not represented
provide the principal limitations on the configurations that
can be modeled. There are no inherent limitations on allow-
able topological complexity.
Certain additional features of real piping systems that
are not represented in the present modeling include added
mass due to fittings and pipe contents, and partial fixity
or elastic restraint at ends or intermediate points.
Despite these limitations, it is believed that a signi-
ficant fraction of current piping systems can be adequately
modeled using the present program.
B. SIZE LIMITATIONS
For the present purpose, the appropriate measure of
system size is the number of degrees of freedom of the
system model. This number n, which bears no direct relation
to physical size, determines the core storage requirements
and execution time of the program. For approximate estima-
2tion, storage requirements are proportional to n and
execution time is proportional to n . Using the data from
the example problem, one can estimate that a 100
40
degree-of-freedom system would require about 400K bytes of
core storage and about 9 minutes execution time.
Increasing the problem size also increases the round-off
errors. Because double-precision arithmetic (56 bit mantissa)
is used throughout, observed round-off effects have been
found negligible (a maximum of 1 unit in the 6th significant
digit for n 66) in all applications to date. The very
small errors detected are attributed to the residual eigen-
vector errors present when the Jacobi rotations are
terminated.
In view of the foregoing considerations, it is believed
that the practical upper limit on problem size (with the
IBM 360/67) is about n = 110 and is determined by core
storage capacity.
Ml
VII. CONCLUSIONS
It is concluded that an effective program has been
developed for determining shock-induced stresses in piping
systems. The principal limitations of the present version
can be removed by adding features that are clearly within
the current state-of-the-art. Recommended additions are
listed below:
1. Remove the pipe axes orientation restriction so thata general piping system can be analyzed.
2. Develop, element stiffness and inertia matrices forbends.
3. Modify the program so that partial fixity and elasticrestraint at the ends or intermediate points may beincluded in the boundary conditions.
4. Include the mass effects of valves, fittings, andpipe contents.
5. Replace the Jacobi rotation method by a more efficienteigenvalue-eigenvector algorithm.
42
APPENDIX A
ELEMENT STIFFNESS AND INERTIA MATRICES
TABLE VI Element Stiffness Matrix
EA
12EI
,3
12EI
*3
2GI
-6EI 4EI
I2 I Symmetric
6ei
I2
4EI
EA EA%
12E1
*3-
6ei 12EI
,3
12EI
.3
6EI
£2
12EI
.3
2GII
2GII
6EI".3
2EI 6EI
I2
4EI
6EI
I2
2EI 6EIo ¥
43
en
<
IJOJ
f< CM
CM
<Q.
OCM
OJ cOJ
CMo*
<Q.OJ
O CM
trCM
<CO
<
oCM
Cl=T
:hc~! CM<ClVDLT\
oCM <
Q.O CM
oCM
orH CM
<^ CM<QVD
OCM
O o c}
<Q.
CM
OCM
rH CM
< o<o. cm
< oa cm cn=r
t-i
h
SI
44
APPENDIX B: VELOCITY SHOCK SPECTRA AND MODAL RESPONSE 1
A shock spectrum exhibits the maximum response dis-
placement of a single degree-of-freedom system whose base
is subjected to the shock motion. Consider a single degree-
of-freedom whose base has a shock displacement s, and whose
displacement relative to the base is z. The equation of
motion is
z +. 0) z = - s (B.l)
where u is the natural circular frequency of free vibration
with the base fixed (s = 0). If z represents the ex-max
treme value of z in response to the shock motion s, then the
displacement shock spectrum for that shock motion is a plot
of z versus w. Equivalent information can be given inmax
the form of a velocity shock spectrum. In this form, the
spectrum velocity V is related to the maximum displacement
zmaxby
max
To develop the equations for finding modal response
from shock spectrum data, let w be the vector of absolute
This Appendix is based on material presented by ProfR.E. Newton in the course ME4522, September 1971 -
45
displacements in the system degrees-of-freedom. For a
uniaxial translation s of the base, this may be expressed
as
C[ + us (B.3)
where q is the vector of displacements relative to the
base, and u is a vector of constants. To illustrate the
meanings of these vectors, consider the beam element of
Fig. 9.
\\\ Inerital Reference \\ \\
FIG. 9
BEAM ELEMENT COORDINATES
m mFor this example, q = [q
n q2 q^ q^] , w = [q 1
+s q 2 q^ + s q^] ,
and u = [1 1 0]. It can be seen that u represents the
46
absolute displacements resulting from unit base
translation.
The equations of motion of an n degree-of-freedom
system with base motion may be written
M w + K q_ =
Using Eq. B.3, this becomes
Mqt+K^= -Mus
Using the substitution q = V p_ where p is the vector of
principal coordinates, this may be rewritten as
VT MV£+VT KVp= -VT Mus
or
p_+fi2
p = -bs (B.4)
Twhere b = V M u. Eq. B.4 is equivalent to the scalar
equations
Pr+ "
r2
Pr= " b
rs , (r=l,2,...,n) (B.4a
where p and b are the rth components of p_ and b,
respectively, and w is the circular frequency of mode r
47
The component b Is called the mode participation factor
for mode r. It is given by
Tbr
= v(r)
M u (B.5)
where v is the eigenvector for mode r (rth column of
I)-
Comparing Eq. B.4a with Eq. B.l, and using Eq. B.2,
the maximum response in mode r may be expressed as
(p ) = b V /oi (B.6)Vi r ; max r r r
where V is the spectrum velocity at frequency u . Usingr
Eq. B.6, the corresponding extremom for the relative
displacement vector q_ is
q = v(r)
b V /u) (B.6a)^max - r r r
Eq. B.6a is used for calculating peak modal responses in
order to find stresses.
In shipboard shock studies, it has been found that
massive equipment may partially suppress the base motion
[6]. This may be taken into account by appropriate modifi-
cation of the spectrum velocity V . Such a correction re-
quires apportionment of the system mass among the modes.
The modal mass for the rth mode is taken to be
m = br r
2 (B.7)
U8
APPENDIX C: EXAMPLE PROBLEM EIGENVALUES AND EIGENVECTORS
Table VIII shows the first six eigenvalues and the
corresponding eigenvectors of the example problem. The
eigenvalue (square of the circular frequency) appears at
the head of a column followed by the components of the
eigenvector grouped by nodes. The remaining eigenvalues
and eigenvectors are omitted. Spacing of the eigenvalues
remains approximately uniform throughout the remainder ofo
the set. The largest eigenvalue is 1.32 x 10 .
49
TABLE VIII
EXAMPLE PROBLEM EIGENVALUES AND EIGENVECTORS
3.20210D 02 1.768220 03 4.283940 03 7.284420 03 1.366430 04 1.93694D 04
3.734190-052.41956D-034.49829D-032.61317D-04-9.722550-051. 227360- 04
1.34663D-051.09003D-021.04163D-03•1.29764D-032.247990-053.964450-04
2. 084720-042.65694D-021. 106640-024.11111D-04-2.55767D-041.134310-03
-1.14030D^04-1.28407D-01-5. 213800-031.33441D-031 .123570-04
-5.43247D-03
4.651040-048.53335D-02
•1 .59767D-021.670380-043.099150-043.61351D-03
9.202400-045.243660-022.80391D-023.87608D-04•5.48334D-042.206790-03
7. 4 6 83 00- 051.09429D-021.65671D-035.226180-045. 072280-042.90708D-04
2.693140-052.303660-023. 865930-042.59488D-031.175200-046.000790-05
4. 16900D-048.44930D-021.211330-038. 219070-041. 109210-031.50032D-03
-2.28018D-044.007280-011.910950-032.66708D-035.85658D-04-6.91251D-03
-9.29892D-042.67855D-011.14491D-023.336680-042.05621D-034.70215D-03
1.83959D-031.628400-011.906140-027.738690-043.551720-032.82479D-03
7.586450-02-3.27225D-02-1.502070-037.81717D-041.68048D-036.80364D-04
1.66544D-022.268140-01-3.50837D-04•3.29915D-033.47592D-04-1.36944D-03
1. 461590-011.37883D-01-8.567620-045.61288D-042.712830-03-1.50438D-03
7.41273D-02-5. 514730-011.733630-039.10543D-041.33431D-03-6. 234990-03
•2.396660-012.05302D-011.086020-021. 877200-03-3.91045D-034.76584D-03
935970-0126126D-02804080-0271397D-03854290-03827700-03
2.077980-018.98682D-021.34738D-039.335880-042.214390-031.06992D-03
4.113490-024.263500-013.150070-04-2.55823D-033.624590-04-2.79781D-03
2.873100-01-1.35861D-015.01745D-047.25499D-041. 113250-03•1.505580-03
-1.40471D-01-4.479070-011.55477D-03-4.14818D-03-4.794290-045.5373 5D-03
3.879600-012.414480-021.025300-023.312280-03-4.39684D-054.800720-03
5.383140-014.12832D-021.697730-021.745820-032.271960-032.80638D-03
3.55428D-01-1.54172D-011. 192630-031.005280-032.171520-031.45933D-03
6.127200-025.49921D-01•2.791070-041. 16814D-032.32548D-04-4.223990-03
2.50525D-01•3.75462D-021.46464D-04•2.229600-032.408720-031. 503930-03
1.256530-014.37906D-021.37451D-03•7. 483740-038.999310-044.821860-03
-2.50044D-011.78932D-019.62857D-032.470720-033.612420-034.806570-03
1.421630-011.13515D-011.58733D-023.151950-04-8. 321710-032.761020-03
4. 84437D-01-2.219640-01•1.037840-031.050640-031.67614D-031.84853D-03
7. 136440-025. 922980-01-2.43147D-04-3. 187010-048.26620D-05-5.64686D-03
•3.727910-021.513220-012.088940-04•3.438590-03•6.26385D-03-1.49942D-03
-3.99987D-024.44113D-011.19302D-036.623490-031. 465560-03-4. 090820-03
-5.137650-02-2. 818790-018.98798D-038.15115D-041.02998D-034.783360-03
3.036070-01-9.597510-021.473150-02-5.005620-04-3.163490-032.69201D-03
4.844510-01-1.159680-01-7.857740-021.11158D-031. 227640-032.06739D-03 -5
7.13490D-022.76427D-012.70496D-031.87342D-041.566670-05781560-03
3.73090D-027.10119D-023.940520-014.58183D-037. 87601D-03-1.448660-03
•3.98607D-022.25377D-017.170640-02•3.766930-031. 203200-034.12090D-03
-5.06787D-02-5.032210-027.37544D-023.146250-06
-2. 844290-033.23151D-03
•3.04055D-012.482560-023.788760-02•7.11803D-063.160610-031.32777D-03
4.844510-015.323830-051.372120-011. 172420-039.79064D-042.211200-03
7.13231D-021.192160-03-2.735590-03-5.58841D-05-6.817080-06-4.079800-03
•3. 732580-022.462120-048. 116120-01-5.71925D-03-7.355270-03•1. 10736D-03
-3.969890-024.44083D-04-1.282270-01-9.02245D-049.235360-04-4.11453D-03
•4.99239D-022.334730-032.579220-018.08835D-04-3.532320-03-2.12272D-03
-3.040190-01-1.372800-03-2.65755D-014.863670-044.543460-03-2.71595D-03
3.223770-01-3.993090-05-5.555850-021.023740-037.344430-042.254460-03
2.42176D-018.943870-04-2.962940-033.44072D-05-5.118530-06-7.87964D-04
1.77887D-021.847940-043.684660-016.240510-03•5.531030-03•4.405480-04
2.072240-013. 334770-04-1.034650-011.293160-036.95771D-04-2.32091D-03
3.027290-011.75514D-031.527010-01-2.121270-032.67172D-03-5.61151D-03
6.34815D-02•1.03302D-03•1.749090-012.211040-033.44875D-03•6.110440-03
1.67489D-012. 662160-056.52906D-044. 239510-044.89697D-041.98080D-03
2.105480-015.963870-04-8.49562D-052.89245D-05-3.415200-061.42427D-03
2.93284D-021.232610-045.64067D-03•3.012020-03•3.694300-038.409130-05
2.586930-012.22516D-04-2.71326D-039.958980-044.653360-048.950440-04
5.251100-01-1.17205 0-033.42056D-03-1.42807D-031.79190D-033.088030-04
3.673960-01-6.90308D-04•4.391680-031.73252D-032.31889D-03•1.28743D-03
4.818110-021. 331100- 057.315500-03-9.92107D-052.448690-041.24990D-03
8.04306D-022.98232D-044.820690-04-6.354490-06-1.708410-061.840420-03
1.346290-026. 164940-05•5.242910-026.96480D-041. 84924D-032.856850-04
1.17881D-011.113170-041.714040-02-2.21904D-042.331150-042.488810-03
2.76985D-01•5.866 110-042.559000-023.24666D-048.991850-045.464770-03
2.298260-01•3.45644D-043.17866D-023.929870-041.16538D-034.226120-03
50
APPENDIX D: CALCULATION OF MODIFIED SPECIFIC WEIGHT
AND RADIUS OF GYRATION
There is negligible effect on the stiffness of the
pipe element from the lagging on the pipe; however, the
lagging does contribute significantly to the mass of the
element. By appropriately modifying the pipe element
specific weight and radius of gyration, the mass effect of
the lagging can be included. The modified specific weight
is
y = (w- + WT)/A (C.l)m p j_i
where y is the modified specific weight of the pipe ele-
ment, and the sum (W + WT ) is the combined weight of thep L
lagging and pipe per unit length. The radius of gyration
is given by the relation
J + JT
= (ni + mT
) r2
(C.2)p L p L
where J and J T are the mass moment of inertia per unitp L
length of the pipe and lagging, respectively, (m + mT ) isp L
the combined mass of the pipe and lagging per unit length,
and r is the radius of gyration.
51
APPENDIX E: PROGRAM LISTING
ft
ft
*
QUJ1-
00UJ
a:_J UJ
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LIST OF REFERENCES
1. Zienkiewicz, O.C., The Finite Element Method inEngineering Science . McGraw-Hill, 1971.
2. Przemieniecki, J.S., Theory of Matrix StructuralAnalysis , McGraw-Hill, 196~B~:
3. Housner, G.W. , "Behaviour of Structures During Earth-quakes," ASCE Journal of Engineering Mechanics Division ,
EM4, October 1959, pg. 109-129.
4. ASME, Shock and Structural Response , Papers presentedat the Shock and Structural Response Colloquium at theAnnual meeting of the ASME, November I960.
5. Newmark, N.M., and Rosenblueth E. , Fundamentals ofEarthquake Engineering , Prentice-Hall, 1971.
6. Naval Ships System Command, Shock Design of ShipboardEquipment , Dynamic-Design-Analysis Method, NAVSHIPS250-423-30, May 1961.
7. Fink, G.E., Vibration Analysis of Piping Systems , MSThesis, Naval Postgraduate School, Monterey, Calif.1964.
8. Baird, W.S., Vibrational Analysis of Three DimensionalPiping Systems with General Topology , MS Thesis, NavalPostgraduate School, Monterey, Calif. 1965.
9. Rudolf, CD., Ill, Vibrational Analysis of ThreeDimensional Piping Via Transfer Matrices , MS Thesis,Naval Postgraduate School, Monterey, Calif. 1971.
10. Timoshenko, S., and Young, D., Elements of Strengthof Materials , Van Nostrand, 1968"]
11. Benfield, W.A., and Hruda, R.F., "Vibration Analysisof Structures by Component Mode Synthesis," AIAAJournal , Vol. 9, No. 7, pg. 1255-1261, July 1971.
12. Sines, G., Elasticity and Strength , Allyn and Bacon, 1969
83
13. Thomson, W.T., Vibration Theory and Applications ,
Prentice-Hall, 1965.
14. Irons, B. , "Eigenvalue Economisers in Vibration Problems,"Journal of the Royal Aeronautical Society , Vol. 67,pg. 526-528, August 1963.
15. Meirovitch, L. , Analytical Methods in Vibrations ,
Macmillan, 1967.
Qh
INITIAL DISTRIBUTION LIST
No. Copies
1. Defense Documentation CenterCameron StationAlexandria, Virginia 2231^
2. Library, Code 0212Naval Postgraduate SchoolMonterey, California 939^0
3. Prof. Robert E. Newton Code 59NeDepartment of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 939^0
4. Prof. John E. Brock Code 59BcDepartment of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 939^0
5. Prof. Gilles Cantin Code 59CiDepartment of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 939^0
6. Lt. Dennis H. Peters, USNUSS FLINT (AE-32)FPO San Francisco, California 96601
7. Department of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 939^0
8. Robert 0. BelsheimShock and Vibration Information CenterNaval Research LaboratoryWashington, D.C. 20390
85
Security Classification
DOCUMENT CONTROL DATA -R&DSecurity c las si I ic at ion ot title, body of abstract and indexing when the overall report la claaaltled)
ORIGINATING ACTIVITY (Corporate author)
Naval Postgraduate SchoolMonterey, California 939^0
Za. REPORT SECURITY CLASSIFICATION
Unclassified26. GROUP
IEPORT TITLE
Shock Loads on Piping Systems
DESCRIPTIVE NOTES (TYpe ot report and. inc I us i ve date*)
Master's Thesis; September 1972J. AU THORlS) (First name, middle initial, laat name)
Dennis Harold Peters
«. REPORT DATE
September 1972
7a. TOTAL NO. OF PAGES
877b. NO. OF REFS
15la. CONTRACT OR GRANT NO.
b. PROJECT NO.
0«. ORIGINATOR"* REPORT NUMBER(S)
»b. OTHER REPORT NOISI (Any other number* that may be meelgnedthla report)
10. DISTRIBUTION STATEMENT
Approved for public release; distribution unlimited,
II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
Naval Postgraduate SchoolMonterey, California 939^0
13. ABSTRACT
This thesis describes the development of a Fortran IV computer
program for finding shock-induced stresses in a three-dimensional
piping system. Discretized equations of motion are formed by the
finite element method. Input motions are support translations
specified by response shock spectra. Maximum responses in individual
modes and resulting octahedral shearing stresses at selected points
are found for shock motions in three orthogonal directions. Extreme
stresses, based on the square root of the sum of the squares of the
modal stresses, are estimated for each input direction. An example
problem is analyzed to demonstrate the use of the program.
DD ,
f
r."..1473 lPAGE "S/N 0101 -807-681 1 86 S«curity Clarification
Security Classification
Modal Analysis
Finite Element
Shock Analysis
Piping Stresses
E v WORDSlOLE WT ROLE W T
DD ,
Fr..1473 i back
807-6S3 I 87 Security Classification
23U 27
,;hes i
s
p382 P«ters
5
hock ,oads on pipi"9
systems.
2 3U27170CT7*
Thf ! * 138066P3§2 Peters
Shock loads on ofoin^systems.
c.l
thesP382
Shock loads on piping systems.
3 2768 001 98013 9
DUDLEY KNOX LIBRARY