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

Xh- 1-

* O X<N M < 00 1—1 zft UJ Oft V- 1- X < >- LU O* t-h-t-Q a: OOZft CC > CNJ O00 _l 00 CC 1— 00 t-* DDDUJ 0: LUm • UJ h- tt h-l-t <>_ h- »-i LU

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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