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NSF /RA-760716
L~'? 274
THEO~ETICAL STUDY ON
EARTHQUAKE RESPONSE OF A
REINFORCED CONCRETE CHIMNEY
Prepared by
T. Y. Yang, L. C. Shiau, and Hsu Lo
School of Aeronautics and Astronautics Purdue University
West Lafayette, Indiana
Submitted to
THE NATIONAL SCIENCE FOUNDATION
June 3, 1976
820 r~ I
BIBLIOGRAPHIC DATA SHEET
4. Title and Subtitle 1
1. Report No.
NSF /RA-760716 3., Recipient's Accession No.'
5. Report Date
Theoretical Study on Earthquake Response of a Reinforced ConcretE June 3, 1976 Ch i mney 1-6:-. ----.:....-~------l
7. Author(s)
T. Y. Yang, L. C. Shiau, Hsu Lo 9. Performing Organization Name and Address
Purdue University School of Aeronautics and Astronautics West Lafayette, IN 47906
12. Sponsoring Organization Name and Address
Research Applied to National Needs (RANN) National Science Foundation Washington, D.C. 20550
15. Supplementary Notes
16. Abstracts
8. Performing Organization Rept. No.
10. Project/Task/Work Unit No.
11. Contract/Grant No. GI41897
13. Type of Report & Period Covered
14.
A detailed dynamic analysis, presented in a series of reports, was conducted on the seismic response and structural safety of key subsystems (steam generator, high pressure steam piping, coal handling equipment, cooling tower, chimney) of Unit #3 of TVA at Paradise, Kentucky in order to: (1) determine for the key components the natural frequencies below 50 Hz and the corresponding normal modes; (2) determine response of plant to seismic disturbances; (3) verify through full scale tests results, obtained in (1) and determine estimates of damping needed in (2); (4) determine potential failure modes of major structural components; and (5) determine a spare parts policy for a power system so that outages due to damage from seismic disturbances are minimal. Analytical and experimental methods are used. Here, the chimney is modeled and its ~ dynamic responses analyzed by modal superposition. Three assumed values of damping coefficients are applied to its two reinforced concrete shells, yielding data for various stress areas after application of 1940 El Centro earthquake parameters. Stress distribution around flue openings at the worst time is determined from quadrilateral sheet finite element data. 17. Key Words and Document Analysis. 17a. Descriptors
Earthquake resistant structures Earthquakes Earth movements Safety Hazard Safety engineering 17b~ Identifiers/Open-Ended Terms
17c. COSATI Field/Group
18. Availability Statement
NTIS
Chimneys
/
FORM NTIS-a!! (REV. 10-73) ENDORSED BY ANSI AND UNESCO.
19., Security Class (This
Re~~~tfT A""TFTFn 20. Security Class (This
Page UNCLASS1FIED
THIS FORM MAY BE REPRODUCED
21. 'No. of Pages
cr~
USCOMM-DC 8265- P 74
THEORETICAL STUDY ON
EARTHQUAKE RESPONSE OF A
REINFORCED CONCRETE CHIMNEY
By
T. Y. Yang, L. C. Shiau, and Hsu La
Any opinions, findings, conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
I '
/1
Summary Report
Prior to 1974 there has been no detailed dynamic analy-
sis of the seismic structural response and safety of large
fossil-fuel steam generating plants. In March, 1974, under
NSF Grant GI4l897, a detailed dynamical analysis was begun
on the seismic response and structural safety of key sub-
systems
(steam generator,
high pressure steam piping,
coal handling equipment,
cooling tower,
chimney)
of Unit #3 of TVA at Paradise, Kentucky to accomplish the
following objectives:
a) Determine for the key components the natural frequencies below 50 Hz and the corresponding normal modes.
b) Determine response of plant to seismic disturbances.
c) Verify through full scale tests, where possible, results obtained in a), and determine estimates of damping needed in b).
d) Determine potential failure modes of major structural components.
e) Determine a spare parts policy for a power system so that outage due to damage from seismic disturbances are minimal.
' .. II f
Analytical and experimental methods are used.
The attached Reports present what has been accomplished
to date.
Before making a few summarizing remarks on the indi
vidual Reports, some comments must be made in order to pro
vide perspective on the study.
Paradise, Unit #3 of TVA was selected for study because
near-by mine operations provide excitation (due to blasting)
for the plant, and TVA was willing to cooperate in the con
duct of the study. It should be pointed out that this plant
was not designed to resist earthquakes. However, it was
felt that this disadvantage was outweighed by the experi
mental possibilities.
The key components selected for study are critical for
operation of the plant and would cause significant outage
if damaged. All components can be studied using similar
types of analyses. These are the basic reasons for includ
ing in this study only the stearn generator, high pressure
piping, coal handling equipment, cooling tower, and chimney.
Basic data for the analyses were obtained from drawings
provided by TVA and Babcock-Wilcox. In addition to these
data, a number of assumptions had to be introduced into the
analyses. These assumptions refer in the main to the nature
of the connections among elements of known properties, the
i V
fixity of columns, the properties of hanger elements, etc.
Choices were made based on physical as well as computational
reasons.
The analyses were confined to the linear range. After
such a study, it is possible to assess at what level of
excitation parts of the structure become nonlinear.
structure-foundation interaction was neglected. Unit
#3 of Paradise rests on excavations in limestone. It is
assumed that there is little interaction. However, experi-
mental studies will be made on this point.
It was decided at the start that all computations would
be carried out with an existing computer program. SAP IV
was chosen. Some program modifications have proved necessary,
but these have been relatively minor. To obtain familiarity
with the program it was necessary to study a number of
special cases of the actual structure to ensure that it was
functioning properly. For example, substructures within the
stearn generator support were considered seperatelYi assumed
values of viscous damping coefficients were used in generat-
ing time histories*; etc. We found the program execution
* It should be noted that the magnitude of the response with zero damping must be interpreted with some caution as systems with slightly different frequencies can exhibit significantly different magnitudes of response.
v
time slow in some respects which indicates that some of its
internal subroutines, such as eigen value solution, could
be improved. It is beyond the scope of this project, how
ever, to improve existing programs.
The experimental part of the study has proved much more
di£ficult to conduct than anticipated. TVA has been most
cooperative. However, the sheer physical size of the units,
the weather, etc. have caused a number of difficulties that
were not easy to foresee. Progress is gradually being
achieved.
Interest in simple models stems from their possible use
in design studies. It was decided to develop a methodology
for constructing simple models. At present, our simple
models are in the embryonic stage. It is hoped that a£ter
the study of two more plants a useful methodology can be
obtained. Simple models developed could have been used
for one component under study; however, timing made this
impossible.
No recommendations will be made or conclusions drawn
at this time, except in special situations. The partial
examination of one plant does not provide a sufficient basis
for such actions. At the completion of the study conclusions
and recommendations will be presented.
Vj
A number of factors of some importance have not been
considered so far. For example, the steam generator's
internal elements can move with respect to it, the steam
piping exerts dynamic forces on its supports, dynamic
stresses in steam piping are just part of its stress system,
many different seismic excitations are available, plus many
more. Also a spare parts policy was not considered. As
additional progress is made, we shall consider some of these
problems. However, it must be recognized that it is pos-
sible to consider in this study only those factors of major
importance. A spare parts policy involves economic consid-
erationsi it may not be possible to acquire the information
needed to address this point.
Contact with industry ~n this country and Japan clearly
indicates that the current detailed study is of great inter-
est.
An Advisory Committee consisting of
Carl L. Canon - Babcock & Wilcox Product Design Supervisor for Structural Steel and Design
William A. English - Tennessee Valley Authority Head Civil Engineer
Clinton H. Gilkey - Combustion Engineering, Inc. Manager, Engineering Science
Richard F. Hill - Federal Power Commission Acting Director, Office of Energy Systems
, , Vi {I
R. Bruce Linderman - Bechtel Power Corporation Engineering Specialist
D. P. Money - Foster-Wheeler Corporation Supervisor of Stress Analysis
R. D. Sands - Burns & McDonnell Chief Mechanical Engineer
Erwin P. Wollak - Pacific Gas & Electric Company Supervisor, Civil Engineering Division
has been formed to provide a forum for an interchange of
practical and conceptual views on various aspects of the
study. The aim is to ensure that what is developed (in
simple models) will be of practical use to industry. The
Advisory Committee has met twice and reviewed plans and
the progress of the investigation.
Contact is also maintained with the following firms:
Mitsubishi Heavy Industries
Babcock-Hi tachi
Ishikawajima Harima Heavy Industries
Kawasaki Heavy Industries
Taiwan Power Company
The initial visit provided considerable information on the
me'l:hods they have used In seismic response studies conducted
by the research groups in each organization and plant ex-
perience under seismic disturbances.
Comments from the Advisory Committee and reviewers have
been most helpful and encouraging. Many of the comments
have been considered. However, it is not possible to take
account in our studies of all points that have been brought
to our attention.
, .,
Five professors, 8-10 graduate students, 2 technicians,
and a secretary devoted part time to the study. A great
deal of effort was devoted to acquiring information and
equipment. The cooperation of TVA and Babcock-Wilcox was
most helpful and deeply appreciated. Progress was excellent
when it is remembered that education of students is a major
function of a University.
This research project was sponsored by NSF through
Grant No. G141897.
The Reports in this series are as follows:
Dynamic Behavior of the Steam Generator and Support Structures of the 1200 MW Fossil Fuel Plant, Unit #3, Paradise, Kentucky, by T.Y. Yang, M.l. Baig, J.L. Bogdanoff.
The High Pressure Steam Pipe, by C.T. Sun, A.S. Ledger, H. Lo.
Coal Handling Equipment, by K.W. Kayser and J.A. Euler.
Theoretical Study of the Earthquake Response of the Paradise Cooling Tower, by T.Y. Yang, C.S. Gran, J.L. Bogdanoff.
Theoretical Study on Earthquake Response of a Reinforced Concrete Chimney, by T.Y. Yang, L.C. Shiau, H. Lo.
A Simple Continuum Model for Dynamic Analysis of Complex Plane Frame Structures, by C.T. Sun, H. Lo, N.C. Cheng, and J. L. Bogdanoff.
A Timoshenko Beam Model for Vibration of Plane Frames, by C.T. Sun, C.C. Chen, J.L. Bogdanoff, and H. Lo.
ABSTRACT
The 822 feet tall chimney is modeled using Bernoulli-Euler beam
finite elements and its dynamic responses are analyzed using the method
of modal superposition. The chimney includes two reinforced concrete
shells, each with a pair of flue openings. Three assumed values of
damping coefficients are considered. Results include time history plots
of tip deflections, spacings between the two shells, bending moments,
1
and shearing forces at the base, etc. of the responses from the horizontal
and vertical components of E1 Centro 1940 earthquake, The diagrams for
the distributions of moment, shear, and deflection at several instants
are shown. The stress distributions around the openings due to the
combined effects of axial force, shearing force, and bending moments at
the worst time are found by using quadrilateral shell finite elements.
Introduction
The chimney of the TVA Steam Generating Power Plant, Unit 3, at
Paradise, Kentucky is studied. The 832 feet tall chimney includes two
reinforced concrete shells with a 4-1/2 feet minimum air space. The
chimney is analyzed with the use of eight pipe-type finite elements for
mulated on the basis of Bernoulli-Euler assumptions. The stiffness co
efficients for the particular element that includes the flue openings
are obtained by the use of quadrilateral shell finite elements.
Natural frequencies for the inner and outer shells both with and
without flue openings are found. The frequencies are then used to find
the time-history responses due to the north-south horizontal as well as
the vertical components of the El Centro earthquake for the first 30
seconds. The method of modal superposition is used. Six flexural modes
2
are used for the horizontal earthquake input and two vertical modes are
used for the vertical input. The 30 seconds are divided into 1500 time
steps. Several assumed values of damping coefficients are included in
the analysis.
The results are represented by time history plots of the maximum
deflection (at the top) and the maximum bending moment and shearing force
(at the base). The time-history plots of the variation of the spacing
between the outer and inner shells at the top is given. The effects of
damping are shown by plots of tip deflections, base shearing force, and
base bending moment versus the value of damping coefficient, respectively.
The diagrams for the distributions of deflection, bending moment, and
shearing force along the chimney at several instants are presented. The
stress concentrations around the flue openings due to the combination of
axial and shearing forces and bending moment at the worst time are found
by the use of quadrilateral shell elements and represented in the form
of contour plots of surface stresses.
Description of the System The chimney is composed of two slender cylindrical reinforced con-
crete shells as shown in Fig. 1. The inner shell serves as a liner and
has 2" fiber glass insulation on its outer surface. The inner shell also
has a stainless steel cap at the top covering the gap between the inner
and outer shells. However, there is no significant structural connection
between them. There is a 4 1 -6 11 ~inimum air space between the two shells.
The top surface of the chimney foundation is at elevation 390 feet.
The earth fill extends to the elevation of 422 feet. The height of the
chimney is 832 feet above the foundation. The outer diameter for the
outer shell varies from 71.8 feet at the base to 38.5 feet at the top.
3
The thicknesses for the two shells also vary with their maximum values
around the flue openings.
Each of the two shells has a pair of side flue openings. They are
rectangular in shape with dimensions of 28 feet by 14 feet. The base
lines of the openings are 73.5 feet above the base. The circumferential
distance between the center lines of the two openings ;s 50.0 feet for
the outer shell and 38.0 feet for the inner shell. Each opening at the
inner shell is connected to the opening at the outer shell by steel framed
flue duct. The concrete around the openings are heavily reinforced.
Finite Elements
Two types of finite element are used for the analysis of the
chimney: the beam element and the plate element.
The finite element used in the analysis of the gross chimney behavior
is a three-dimensional beam finite element with constant hollow circular
cross section. The element has six degrees of freedom at each node:
three displacements in the three Cartesian coordinate directions and three
rotations about the three axes. The Bernoulli-Euler assumptions for beam
are used in the derivation of the stiffness matrix. The mass matrix is
based on the lumped masses. Since the element is assumed to have constant
cross section, this element can only provide a step representation of the
tapered chimney.
For the local analysis of the chimney, a three-dimensional
quadrilateral plate finite element is used. The element is composed of
four triangular elements with the four common vertices met at near the
centroid of the quadrilateral. Each triangular element is a superimposed
version of a constant membrane strain element (Ref. 1) and an HeT
4
flexural plate element (Ref. 2).
The three-dimensional quadrilateral plate elements serve for two
functions. They are first used to model the particular chimney beam
finite element that has two flue openings so that the 12 influence stiff
ness coefficients for an equivalent homogeneous beam element can be found.
After the end displacement and force vectors for each chimney beam element
are found in the dynamic response analysis, the plate elements are used
to find the detail distributions of stresses within the element, especially
around the openings.
Assumptions
The following basic assumptions underlie this study:
(a) The base of the chimney is fixed,against rotation and lateral
displacement. The chimney foundation is buried into the excavated
limestone rock.
(b) The base circles of the two shells have no relative movement
during earthquake. The two shells are unconnected elsewhere.
(c) The Bernoulli-Euler beam theory is used. No shear deformation
and rotatory inertia are considered.
(d) Thermal effect and wind effect are not considered in this study.
(e) The behavior of the chimney is linearly elastic.
Results and Discussions
As described previously, both inner and outer shells have two 28x14
square feet flue openings at 36.1 feet above the ground. It is of interest
to know how these openings affect the dynamic behaviors of the two
shells. For this reason, most of the present analyses are performed for
5
both cases--with and without the flue openings.
(1) Free Vibration Analysis Without Dampin£
Free vibration analysis is first performed for the outer shell with
out the effect of flue openings. To find out how many beam finite
elements are sufficient to model the chimney, four different kinds of
modeling are used: 4, 8, 12, and 16 elements, respectively. For each
mode, the frequency ratio (the frequency value obtained by using n elements
divided by the one obtained by using 16 elements) is plotted against the
number of elements (n) as shown in Fig. 2. It is seen that the eight
element modeling gives approximately 1.2, 1.8, 3.8, and 5.2 percent dis
crepancies as compared to the 16 element solutions for the first, second,
third, and fourth modes, respectively. It appears that the eight element
model is sufficiently accurate for the purpose of the present dynamic
response analysis. It is thus determined to use eight elements to model
both the outer and inner shells, with and without flue openings. The
modeling is shown in Fig. 3.
For the case with flue openings, the first beam element in Fig. 3
must be modified to an equivalent homogeneous beam finite element. This
is done by first modeling the element by 70 quadrilateral plate finite
elements and then finding the equivalent axial area and moments of inertia
for the equivalent beam element. This is a rather laborious process.
The results for the first 12 natural frequencies and periods are
shown in Table 1 for the outer shell and in Table 2 for the inner shell.
It is seen in both tables that the flue openings have little effect on
the values of frequencies. It is noted that for the outer shell, the
5th, 8th, and 11th modes are longitudinal modes while the others are
6
flexural modes. For the inner shell, the longitudinal modes appear at
5th, 9th, and 11th modes.
To validate the present assumptions and method, the frequency values
obtained must be evaluated. This is one of the purposes for which an
experimental program is conducted in this research project. In the ex
periment, accelerometer assemblies are installed on the foundation and
three upper levels of the outer shell to measure accelerations in the
three space coordinate directions. The set up includes nine units, 17
transducers, and 14 data recording channels. Primary excitation is due
to wind and blasting of a nearby coal mine. The experimental results
are given in a separate report by K. W. Kayser who conducted the experi
ment. Three natural frequencies have been found so far and the comparison
is given in Table 2. The agreement strongly supports that the present
assumptions and method are sound.
The results for the first, second, and third mode shapes are plotted
in Fig. 4. The fourth, fifth, and sixth mode shapes are plotted in
Fig. 5.
(2) Response to the Horizontal Component of an Earthquake.
The record of El Centro earthquake occurred on May 18, 1940 is
selected to analyze the time-history dynamic response of the chimney.
The sQuth-horth horizontal component of the acceleration record is shown
in Fig. 6. Since the magnitude of acceleration becomes considerably
smaller after 30 seconds, only 30 seconds are considered here. The record
in Fig. 6 shows that the acceleration oscillates at the frequency of
approximately 3 to 7 cycles per second. The results in Tables 1 and 2
for natural frequencies show that the seventh frequency (sixth flexural
7
frequency) is approximately 9.25 Hertz for outer shell and 8 Hertz for
the inner shell, respectively. Thus the first six flexural modes are
used in the modal superposition to simulate the dynamic response of the
chimney to the horizontal component of this earthquake.
The time interval for the modal analysis is set as 0.02 seconds
which corresponds to 50 time points for each second and 1500 points for
the 30 second period. The central processing time required by a CDC 6500
computer with the use of eight elements, six flexural modes, and 1500
time points is approximately 25 seconds. In this entire analysis, the
results for the time history response are plotted by the computer with
the time interval of 0.05 seconds. This means that each subsequent time
history curve is connected among 600 points in the 30 second period.
i. Outer and inner shell with no flue opening.
The time history response for the deflection at the tip of the outer
she11 (with the assumption of no flue opening) is shown in Fig. 7. The
maximum tip deflection is seen to be 47.7 inches at 26.6 seconds. The
late arrival of the maximum tip deflection suggests the possibility of
occurrence of an even larger tip deflection at after 30 seconds. The
results for the bending moment and shearing force at the base are shown
in Figs. 8 and 9, respectively. The maximum bending moment occurs at
the time of 13.8 seconds with magnitude of 44.7xl06 in-kips. If the
concrete is assumed not to crack under tensile stress, this bending
moment produces a maximum compressive stress of 4147 psi in concrete and
a maximum tensile stress of 33.2 ksi in reinforced bar. If, however,
the concrete is assumed to be incapable of carrying tensile stress, the
maximum compressive stress in concrete becomes 16.53 ksi and the
8
maximum tensile stress in steel becomes 136 ksi. The maximum shearing
force occurs at the time of 26.95 seconds with magnitude of 17.08xl06
pounds. This results in a maximum shearing stress of 667.6 psi in the
concrete.
After the outer shell with the assumption of no flue opening has
been studied, the inner shell with no flue opening is investigated. The
results for the time history responses of tip deflection,base bending
moment, and base shearing force are shown in Figs. 10, 11, and 12, re
spectively. It is seen that the magnitudes in Figs. 10-12 for the inner
shell are considerably smaller than those shown in Figs. 7-9 for the
outer shell. The maximum tip deflection is seen to be -24.85 inches at
28.8 seconds. The maximum bending moment occurs at the time of 18.3
seconds with magnitudes of 10.15xl06 in-kips. If the concrete is assumed
not to crack under tensile stress, such bending moment produces a maximum
compressive stress of 2220 psi in concrete and a maximum tensile stress
of 17.8 ksi in reinforced bar. If, however, the concrete is assumed as
incapable of carrying tensile stress, the maximum compressive stress in
concrete becomes 8730 psi and the maximum tensile stress in reinforced
bar becomes 72.5 ksi. The maximum shearing force occurs at the time of
18.3 seconds with magnitude of 4.75xl06 pounds. This corresponds to a
maximum shearing stress of 365.7 psi in concrete.
In view of the large magnitudes of deflection at the top of both
outer and inner shells (with no flue openings), it is desirable to find
how do the two shells approach each other at the top during a simultaneous
response to the subject earthquake. The results for the tip deflections
obtained in Figs. 7 and 10 are compared from point to point (for 600 points)
9
and the net differences are plotted in Fig. 13. It is noted that the
minimum air space between the two shells is designed as 54 inches. Fig.
13 shows that the tip spacings are everywhere greater than zero. It is
interesting to see that the spacings are quite close to zero at the time
of 14.2 seconds and 21.7 seconds.
ii. Outer and Inner Shell with Flue Openings
As described earlier in the section of Description of the System,
each shell has two openings. Their effect must be accounted for by
modifying the stiffness and mass matrices of the first beam finite element
that contains the openings. The equivalent homogeneous stiffness properties
are obtained by first modeling the first beam finite element by 70 quadri
lateral shell finite elements and then finding the influence stiffness
coefficients through applying unit value of each degree of freedom at
each nodal point of the equivalent element. The process is straight
forward but laborious. The effect of the openings on the lumped mass
matrix of the equivalent beam element is accounted for by simply neglecting
the masses due to the cut-outs.
For the equivalent beam element, the bending rigidity is not the
same in every direction. The weakest direction is along the diameter
that passes through the midpoint between the centers of the two openings.
The strongest direction is perpendicular to the weakest direction. In
this study, the earthquake acceleration is assumed to move in the weakest
direction of the chimney.
The time history responses for the tip deflection, oase bending
moment, and base shearing force for the outer shell with two flue openings
are shown in Figs. 14-16, respectively. When comparing Fig. 14 with
10
Fig. 7, it is seen that the tip deflections for the outer shell with no
opening are generally slightly larger than those for the outer shell with
two openings. When comparing Figs. 15-16 with Figs. 8-9, it is seen that
the magnitudes of bending moment and shearing force at the base for the
outer shell with no openings are generally noticeably larger than those
for the outer shell with no opening. Although the cut-outs decrease the
stiffness of the first finite element, they decrease the mass too. It
is interesting to see that the openings affect the gross dynamic behavior
of the chimney noticeably.
The time history responses for the tip deflection, the base bending
moment, and the base shearing force for the inner shell with two flue
openings are shown in Figs. 17-19, respectively. The three sets of
curves are almost identical to the ones for the inner shell with no
opening. ihe effect of openings is thus seen to be insignificant.
The time history of the net differences in tip deflection between
the inner and outer shells with openings is shown in Fig. 20. The results
are quite similar to those for the shells with no opening.
The distributions of deflection, bending moment, and shearing force
for the outer shell with openings are plotted in Figs. 21-23, respectively,
at three different instances. The instances are chosen at 22.5 seconds
when the maximum base shearing force occurs, 25.2 seconds when the maxi
mum base bending moment occurs, and 28.3 seconds when the maximum tip
deflection occurs. The distributions of deflection, bending moment, and
shearing force for the inner shell with openings are plotted in Figs.
24-26, respectively, at three instances. The instances are chosen at
26.5, 28.8, and 29.9 seconds when the maximum base shearing force, maxi
mum base bending moment, and maximum tip deflection occur, respectively.
11
For both shells, it is seen that the maximum base shearing force occurs
first, the maximum base bending moment second, and the maximum tip
defl ect ion 1 as t.
(3) Damping Effect on Responses to Horizontal Component of Earthquake.
The effect of viscous damping is considered in this study. It is
assumed that each of the six flexural modes has the same damping coef
ficient. When including the damping effect, only shells with flue open
ings are considered. Three different values of viscous damping coefficient
are assumed: 4 percent, 7 percent and 10 percent of its critical value.
The time history responses for the tip deflection, base bending
moment, and base shearing force for the outer shell with the three
different damping coefficients are shown in Figs. 27-35. Similar presen
tations for the results for the inner shell are given in Fig. 36-44. It
is seen that all the physical quantities reduce progressively as the
damping coefficient increases.
The time history response for the spacing between the top of the
two shells is shown in Figs. 45, 46, and 47 for the damping coefficient
equal to 4 percent, 7 percent, and 10 percent of its critical value,
respectively. It is seen that when the damping is included, the tops of
the two chimneys are quite far apart at all times.
The effect of damping on the tip deflection, base bending moment,
and base shearing force for both shells is summarized by the plots shown
in Figs. 48, 49, and 50, respectively. It is seen that at 4 percent of
critical damping the tip deflection, base bending moment, and base shear
ing force all reduce by more than 50 percent of their original undamped
values. The further increase in damping coefficient does not seem to
12
decrease much the three physical quantities. Figs. 48-50 serve to provide
a basis for interpolating the effect of damping for any given coefficient
value.
(4) Response to the Vertical Component of an Earthquake.
The vertical component of the acceleration record of E1 Centro
earthquake is shown in Fig. 51. Only 30 seconds of the record are con
sidered simply because the acceleration nearly vanishes after then.
The time interval of 0.02 seconds is used in the present modal analysis.
The outputs of time history response are plotted at a time interval of
0.05 seconds.
The first three natural frequencies of vertical modes are given in
Tables 1 and 2 as approximately 5.0, 10.5, and 17.4 Hertz for the outer
shell and 4.5, 11.3, and 17.7 Hertz for the inner shell, respectively.
Since the vertical component of the acceleration oscillates at the
frequency of around 8 to 9 Hertz, it is felt that two vertical modes may
be sufficient to simulate the response behaviors. Thus two modes are used.
The time history responses for the tip axial displacement and
the base axial force for the outer shell with no openings are shown in
Figs. 52 and 53, respectively. The magnitudes of both tip displacement
and base axial force decrease after 15 seconds. The time history responses
for the tip axial displacement and base axial force for the inner shell
with no opening are shown in Figs. 54 and 55, respectively. Unlike in
the case of outer shell, the magnitudes for the tip axial displacement
and base axial force for the inner shell remain quite constant during
the entire 30 second period.
For the case of outer shell with two flue openings, the results for
tip axial displacement and base axial force are shown in Figs. 56 and 57,
13
respectively. The variations of tip axial displacement and base axial
force are seen to be more steady in the case of shell with openings than
in the case of shell without opening. For the case of inner shell with
two flue openings, the results for tip axial displacement and base axial
force are shown in Figs. 58 and 59, respectively. Both the tip displace
ment and the base axial force are seen to decrease slowly but steadily
with increasing time.
(5) Stress Concentration Around Flue Openings.
The stress concentrations in the region around the two flue openings
in each shell are studied by performing local analysis of the first beam
finite element of each shell. The first beam finite element is modeled
by 244 quadrilateral shell finite elements. Both ends are subjected to
the simultaneous actions of bending moment, shearing force and axial
force caused by both the horizontal and vertical components of the earth
quake. An inspection of the results given in Figs. 15, 16, and 57 finds
that for the outer shell the most critical time is at 25.2 seconds when
the base bending moment = 32.05xl06 in-kips; the base shearing force
= 11,810 kips; and the base axial force = 217.7 kips. The results of
the local analysis is shown in Fig. 60 by a contour plot of axial stress
in the outer surface of the outer shell. Because of symmetry, only one
half of the shell segment with one opening is shown. The iso-stress
curves varies from -2500 psi to zero and then to 3000 psi along the
circumference at the base. Such distribution shows the dominant effect
of a bending moment. The stress concentrations due to the openings are
clearly presented in the figure.
For the inner shell, the most critical time appears to be at 28.8
14
seconds when the base bending moment = 11.28xl06 in-kips (see Fig. 18);
the base shearing force = 4,557 kips (see Fig. 19); and the base axial
force = 150.5 kips (see Fig. 59). The contour plots of the resulting
axial stresses in the outer surface of the inner shell are shown in
Fig. 61. The stress concentration around the opening and the dominant
bending effect at the base are clearly seen. The patterns of stress
distribution for both shells are quite similar.
Concluding Remarks
The south-north horizontal and vertical components of the El Centro
earthquake are used to perform the time history response analyses of the
subject chimney. The two 30-second acceleration records are each broken
down into 1500 time points with a time interval of 0.02 seconds. The
results are all plotted for 30 seconds by 600 time points with a time
interval of 0.05 seconds.
A thorough investigation of many aspects of the dynamic responses
of the subject chimney is carried out and the results are presented in
ovel~ 50 figures. Besides providing the comprehensive information about
the dynamic behaviors of the chimney, this study also suggests the
following conclusions.
(1) A full scale experiment shows that the free vibration analysis
based on the present method and assumptions is valid.
(2) The effect of the flue openings appears to reduce the magnitudes
of tip deflection, base bending moment and base shear (see
Figs. 7-9 and 14-16).
(3) The damping coefficient with 4 percent of its critical value
reduces the maximum deflection, shearing force, and bending
moment by more than 50 percent of their undamped values. The
15
effect due to the increase in damping beyond this value is,
however, no longer so pronounced.
(4) The stress distribution in the chimney is dominated by the
bending action. The effect due to vertical component of the
El Centro earthquake is little.
(5) For the present design, the two shells do not collide with
each other under the El Centro earthquake.
16
References
[1] Clough, R. W., liThe Finite Element in Plane Stress Analysis,"
Proceedings of the 2nd ASCE Conference on Electronic Computation,
Pittsburgh, Pa., Sept. 1960.
[2J Clough, R. W. and Tocher, J. L., "Finite Element Stiffness Matrices
for the Analysis of Plate Bending," Proceedings of the First Con
ference on Matrix Methods in Structural Mechanics, Air Force Flight
Dynamics Lab., TR-66-80, Dayton, Ohio, 1965.
17
Table 1. The Natural Frequencies and Periods for the Outer Shell With or Without Flue Openings.
Mode Without Opening With Openings
: ! Frequency Period Frequency Period
Number (rad./sec.) (Seconds) (rad./sec.) (Seconds
2.003 3. 1371 II 1.969 3.1911
2 7.149 0.8789 1 7.009 0.8965 I
3 16.775 0.3746 I, 16.573 0.3791
4 29.184 0.2153 I 29.015 0.2166
5* 31.846 0.1973 I 31.387 0.2002
6 43.475 0.1445 43.430 0.1447 ,
7 : (
58.063 0.1082 58.119 0.1081
8* 66.298 0.0948 il 65.275 0.0963 ,
9 71.364 0.0881 [
71 .436 0.0880 l !
10 i. 81.740 0.0769 I 81 .811 0.0768 ' ,
11* 109.990 0.0571 II
109.300 0.0575
12 145.550 0.0432 145.750 0.0431 ~ i
1
I
* Longitudinal modes
18
* Longitudinal modes
Table 3. Comparison of the First Three Natural Frequencies for the Outer Shell With Two Flue Openings.
Mode
Number
8-finite element solution
Full Scale Experiment*
i t
I! i
Ii ! I , I ! i
II
First Second
(rad./sec.) (rad./sec.)
1.969 7.009
1.32 6.35
* Details are given in a separate report by K. W. Kayser.
19
Third
(rad./sec.)
16.573
15. 1
Fi g. 1.
Fi g. 2.
Fi g. 3
Fig. 4.
Fi g. 5.
Fi g. 6.
Fi g. 7.
Fi g. 8.
r- • a r 1 g. -'.
20
Figure Captions
Description of the Chimney in the TVA Steam Generating Power
Plant at Paradise, Kentucky.
The Convergence Evaluation for the First Four Natural Frequencies
Obtained by Different Modelings for the Chimney Outer Shell.
The Eight-Element Modeling of the Chimney.
The First, Second, and Third Mode Shapes for the Chimney Outer
Shell.
The Fourth, Fifth, and Sixth Mode Shapes for the Chimney Outer
Shell .
The Record of the South-North Acceleration Component of the
El Centro Earthquake on May 18, 1940.
Tip Deflection vs. Time Curve for Outer Shell Without Opening.
Bending Moment at the Base vs. Time Curve for Outer Shell
Without Opening.
Shearing Force at the Base vs. Time Curve for Outer Shell
Without Opening.
Fig. 10. Tip Deflection VS. Time Curve for Inner Shell Without Opening.
Fig. 11. Bending Moment at the Base vs. Time Curve for Inner Shell
Without Opening.
Fig. 12. Shearing Force at the Base vs. Time Curve for Inner Shell
Without Opening.
Fig. 13. Time History Curve for the Spacing Between the Tops of Outer
and Inner Shells, Both Without Openings.
Fig. 14.
Fi g. 15.
Tip Deflection vs. Time Curve for Outer Shell with Two Openings.
Bending Moment at the Base VS. Time Curve for Outer Shell
with Two Openings.
Fig. 16. Shearing Force at the Base vs. Time Curve for Outer Shell
with Two Openings.
21
Fig. 17. Tip Deflection vs. Time Curve for Inner Shell with Two Openings
Fig. 18. Bending Moment at the Base vs. Time Curve for Inner Shell
with Two Openings.
Fig. 19. Shearing Force at the Base vs. Time Curve for Inner Shell
with Two Openings.
Fig. 20. Time History Curve for the Spacing Between the Tops of Outer
and Inner Shells, 80th with Two Openings.
Fig. 21. The Deflection Shapes of the Outer Shell with Two Openings at
Three Instances (Maximum Tip Deflection Occurs at 28.3 Seconds).
Fig. 22. The Distributions of Bending Moment for Outer Shell with Two
Openings at Three Instances (Maximum Base Moment Occurs at
25.2 Seconds).
Fig. 23. The Distributions of Shearing Force for the Outer Shell with
Two Openings at Three Instances (Maximum Base Shear Occurs at
22.5 Seconds).
Fig. 24. The Deflection Shapes of the Inner Shell with Two Openings at
Three Instances (Maximum Tip Deflection Occurs at 29.3 Seconds).
Fig. 25. The Distributions of Bending Moment for the Inner Shell with
Two Openings at Three Instances (Maximum Base Moment Occurs at
28.8 Seconds).
Fig. 26. The Distributions of Shearing Force for the Inner Shell with
Two Openings at Three Instances (Maximum Base Shear Occurs at
26.5 Seconds).
22
Fig. 27. Tip Deflection vs. Time Curve for the Outer Shell with Two
Openings and with Damping Coefficient Equal to 4 percent of
its Criti ca 1 Value.
Fig. 28. Tip Deflection vs. Time Curve for the Outer Shell with Two
Openings and with Damping Coeffi ci ent Equal to 7 Percent of
its Critical Value.
Fig. 29. Tip Deflection vs. Time Curve for the Outer Shel1 with Two
Openings and with Damping Coefficient Equal to 10 Percent of
its Critical Value.
Fig~ 30. Base Bending Moment vs. Time Curve for the Outer Shell with Two
Openings and with Damping Coefficient Equal to 4 Percent of its
Critical Value.
Fig. 31. Base Bending Moment vs. Time Curve for the Outer Shell with
Two Openings and with Damping Coefficient Equal to 7 Percent
of its Critical Value.
Fig. 32. Base Bending Moment vs. Time Curve for the Outer Shell with Two
Openings and with Damping Coefficient Equal to 10 Percent of
its Critical Value.
Fig. 33. Jase Shearing Force vs. Time Curve for the Outer Shell with
Two Openings and with Damping Coefficient Equal to 4 Percent
of its Critical Value.
Fig. 34. Base Shearing Force vs. Time Curve for the Outer Shell with
Two Openings and with Damping Coefficient Equal to 7 Percent
of its Critical Value.
Fig. 35. Base Shearing Force vs. Time Curve for the Outer Shell with
Two Openings and with Damping Coefficient Equal to 10 Percent
of its Critical Value.
Fig. 36. Tip Deflection vs. Time Curve for the Inner Shell with Two
Openings and with Damping Coefficient Equal to 4 Percent of
its Critical Value.
Fig. 37. Tip Deflection vs. Time Curve for the Inner Shell with Two
Openings and with Damping Coefficient Equal to 7 Percent of
its Critical Value.
23
Fig. 38. Tip Deflection vs. Time Curve for the Inner Shell with Two
Openings and with Damping Coefficient Equal to 10 Percent of
its Critical Value.
Fig. 39. Gase Bending Moment vs. Time Curve for the Inner Shell with
Two Openings and with Damping Coefficient Equal to 4 Percent
of its Critical Value.
Fig. 40. Base Bending Moment vs. Time Curve for the Inner Shell with
Two Openings and with Damping Coefficient Equal to 7 Percent
of its Critical Value.
Fig. 41. Gase Bending Moment vs. Time Curve for the Inner Shell with
Two Openings and with Damping Coefficient Equal to 10 Percent
of its Critical Value.
Fig. 42. Base Shearing Force vs. Time Curve for the Inner Shell with
Two Openings and with Damping Coefficient Equal to 4 Percent
of its Critical Val.ue.
Fig. 43. Base Shearing Force vs. Time Curve for the Inner Shell with Two
Openings and with Damping Coefficient Equal to 7 Percent of its
Critical Value.
Fig. 44. Base Shearing Force vs. Time Curve for the Inner Shell with
Two Openings and with Damping Coefficient Equal to 10 Percent
of Its Critical Value.
24
Fig. 45. Time History Curve for the Spacing Between the Tops of the
Outer and Inner Shells with Openings and with Damping Coefficient
Equal to 4 Percent of its Critical Value.
Fig. 46. Time History Curve for the Spacing Between the Tops of the
Outer and Inner Shells with Openings and with Damping Coefficient
Equal to 7 Percent of its Critical Value.
Fig. 47. Time History Curve for the Spacing Between the Tops of the
Outer and Inner Shells with Openings and with Damping Coefficient
Equal to 10 Percent of its Critical Value.
Fig. 48. Variation of Tip Deflection with Damping Coefficient for Both
Shells with Openings.
Fig. 49. Variation of Base Bending Moment with Damping Coefficient for
Both Shells with Openings.
Fig. 50. Variation of Base Shearing Force with Damping Coefficient for
Both Shells with Openings.
Fig. 51. Th~ Record of the Vertical Acceleration Component of the El
Centro Eqrthquake on May 18, 1940.
Fig. 52. Vertical Tip Displacement vs. Time Curve for Outer Shell Without
Opening.
Fig. 53. Axial Force at Base vs. Time Curve for Outer Shell Without
Opening.
Fig. 54. Vertical Tip Displacement vs. Time Curve for Inner Shell
Without Opening.
Fig. 55. Axial Force at Base vs. Time Curve for Inner Shell Without
Opening.
Fig. 56. Vertical Tip Displacement vs. Time Curve for Outer Shell with
Two Openings.
25
Fig. 57. Axial Force at Base vs. Time Curve for Outer Shell with Two
Openings.
Fig. 58. Vertical Tip Displacement vs. Time Curve for Inner Shell with
Two Openings.
Fig. 59. Axial Force at Base vs. Time Curve for Inner Shell with Two
Openings.
Fig. 60. Contour Plot of the Axial Stress (ksi·) in the Outer Surface
of the Outer Shell Around one Flue Opening at the Most Critical
Time (25.2 Seconds) with Bending Moment = 32.048xl09 in-lbs;
Shearing Force = 11.3lxl06 lbs; and Axial Compression
= 21.77xl04 lbs at the Base.
Fig. 61. Contour Plot of the Axial Stress (ksi) in the Outer Surface
of the Inner Shell Around One Flue Opening at the Most Critical
Time (28.J Seconds) with Bending Moment = 11.28xl09 in-lbs;
Shearing Force = 45.57xl05 lbs; and Axial Tension = l5.05xl04 lbs
at the Base.
, " 0.0.:: 56.20 t=18 1.0. =40.37' " t=93/4
, " 0.0. =62.60 f=r9
OPENINGS
0.0. =71. eo' t = 19,_'1 __ -J,i.:L-~t...--____ .....L...~:.xl4-2.L----'--=.!..!...iU:.~_~..&.>~~
Fig. 1. Description of the Chimney in the TVA Steam Generating Power Plant at Paradise, Kentucky. (Not drawn to scale)
26 )
I 1.00 ------------~~~
It ___
- ~r· III -c:: .95 Q.J
E Q.J
Q.J
(0
~- fourth mode 3 .90 .........
0 third ~ode r-
second mode c::t cr .85 first mode >-u
z w :::)
OJ w .80 a:: LL
.75+--------Ti---------r,--------~Ir--------,-----o 4 8 12 16
NUMBER OF ELEMENTS
Fig. 2. The Convergence Evaluation for the First Four Natural Frequencies Obtained by Different Modelings for the Chimney Outer Shell.
9 r--
@
~
(J)
7
@
6
@
5
@
.!1
@ I I
~
®
2
<D
J //// ./// /// / / //
-.... 10 I'; N o
-.... o o
-.... o N
Fig. 3. The Eight-Element Modeling of the Chimney.
27)
\' '
\ \ \ \
/ /
/', first mode~\
\
'/ i\
/ ':~ third mode
, \ second mode ----~
\ \ ' '\ ,'\ . ' \
'. \
:\ \ , . , " \ I , I , . I
" \1
Fig, 4. The First, Second, and Third Mode Shapes for the Chimney Outer Shell,
28)
.... ... ~...-- -......
/ ......
" ".-( // , / (--- __
fourth mode ---11-"'1 , .....
(
....... • "'C... .......
"
... / ./ , .
,
/ I I
...... ".
/
I .' I . , \/
(\ \ \ ~ .,
~----- --;.."..:-"
.:----fifth mode
--' . -""
..... / ~"',
\ \
\ J
I ./ /'
:.---<-..... '. , . .. )
I
/
\
) /
sixth mode
Fig. 5. The Fourth, Fifth, and Sixth Mode Shapes for the Chimney Outer Shell.
29)
160
100
- ~ 50
~ ~ - .....,. a ~.~
Illijl~
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Fig
. 6.
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e R
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e S
outh
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th A
ccel
erat
ion
Com
pone
nt o
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e El
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arth
quak
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M
ay
18,
1940
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11
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12
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TI
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Fig
. 13
. Ti
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tory
Cur
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Bet
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15
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hell
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Two
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16
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ll
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19
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r S
hell
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Fig.
20
. Ti
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tory
Cur
ve
for
the
Spac
ing
Bet
wee
n th
e To
ps
of O
uter
and
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oth
with
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~ ~
DEFLECTION
I I I I I
.. /. I
AT 22.5 SEC _-1-laol
AT 25.2
I \ \ \ \ \
SE C ----"'"' \
I -20
\ \ ~
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45)
SHAPE
800 ft
600
AT 28.3 SEC
400
200
o
Fig. 21. The Deflection Shapes of the Outer Shell with Two Openings at Three Instances (Maximum Tip Deflection Occurs at 28.3 Seconds).
~
Z
l.IJ ~
-.
o ""
'0
~
)(
II)
~
Co
z :x
0
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Z
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w -
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T
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25.2
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EC
AT
22
.5
SEC
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T
28
.3
SEC
..
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o 2
00
4
00
6
00
8
00
HE
IGH
T
(fe
et)
Fig.
22
. Th
e D
istr
ibut
ions
of
Ben
ding
Mom
ent
for
Out
er S
hell
with
Two
Ope
ning
s at
Thr
ee
Inst
ance
s (M
axim
um B
ase
Mom
ent
Occ
urs
at 2
5.2
Seco
nds)
. >1:
>0 0'
1
4
-r···
··· <D
-..
. -.
--------
Q
2
~."
)(
(I)
0 .Q
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-2
/
W
/ U
-t--
AT
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2.5
S
EC
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2
8.3
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.
/ -6
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/
Z
//~AT 2
5.2
S
EC
cr
-8
~
/ W
r
-10
(f
)
-12
,---
---r
-~·-l
o 2
00
4
00
6
00
8
00
HE
IGH
T
(fe
et)
Fig.
23
. Th
e D
istr
ibut
ions
of
Shea
ring
Fo
rce
for
the
Out
er S
hell
w
ith
Two
Ope
ning
s at
Thr
ee
Inst
ance
s (M
axim
um B
ase
Shea
r O
ccur
s at
22.
5 Se
cond
s).
.t::>.
"-.I
AT 29.3 SEC
AT 28.8
(in) I -40
DEFLECTION
\ \ \ \
'\. / •• 1
I . I I I I I I I I \ \ \ \ \
SEC -----'
I -20
\ \ \ \ \
I o
SHAPE
AT 26.5 SEC
I 20
48)
800 ft
600
400
200
o
Fig. 24. The Deflection Shapes of the Inner Shell with Two Openings at Three Instances (Maximum Tip Deflection Occurs at 29.3 Seconds).
.... z w
~,..
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o -
~
)(
\I)
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0
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.8
SE
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/ A
T
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C
......
.
-----
~ AT
2
6.5
SE
C
40
0
60
0
80
0
HE
IGH
T
(fe
et)
Fig.
25
. Th
e D
istr
ibut
ions
of
Ben
ding
Mom
ent
for
the
Inne
r S
hell
w
ith T
wo O
peni
ngs
at T
hree
In
stan
ces
(Max
imum
Bas
e M
omen
t O
ccur
s at
28.
8 Se
cond
s).
~ '"
4 ..-
. \D
0 2
)(
CI)
.c
0
UJ
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u ~
0 -4
I.J
..
l.!)
-6
z a:::
<
t -8
W
:r
(/
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....
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./
--////~AT
28
.8
SEC
o 2
00
4
00
HE
IGH
T
(fe
et)
/A
T 2
6.5
SE
C
=:::::-
- --~
--=
-=
---:
-
. .'"
....
60
0
80
0
Fig.
26
. Th
e D
istr
ibut
ions
of
She
arin
g Fo
rce
for
the
Inne
r S
hell
w
ith
Two
Ope
ning
s at
Thr
ee
Inst
ance
s (M
axim
um
Bas
e Sh
ear
Occ
urs
at 2
6.5
Seco
nds)
. In
o
qO
(J)
20
IJ.J ::t:
(J
Z - '-oJ Z
o I"
l
I I
l L
\ /
::\ l
~ l
TIM
E( S
EC )
to
)
iJ
(I
J I
L
( I
\ I
\ i
_L
-.-.. (J
IJ.J
....J
lL.
IJ.J
Cl
a..
-20
-I- -40
Fig.
27
. Ti
p D
efle
ctio
n vs
. Ti
me
Cur
ve
for
the
Out
er S
hell
w
ith T
wo
Ope
ning
s an
d w
ith
Dam
ping
C
oeff
icie
nt
Equa
l to
4 P
erce
nt o
f it
s C
riti
cal
Val
ue.
lJl
i--'
lJO
- (f)
20
w
:t:
U
Z - '-J Z
o I,
J
J }
\ /
\ t:.
:-=-.
..I>
'"
I \
J TI
ME(
SEC
)
t:J
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i )
I j
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C
I C
VV
\:
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_L
-..... U
W
....I
lL..
W
0 Cl..
-2
0 .....
. .....
-liO
Fig.
28
, T
ip
Def
lect
ion
vs.
Tim
e C
urve
fo
r th
e O
uter
She
ll
with
Two
Ope
ning
s an
d w
ith
Dam
ping
C
oeff
icie
nt
Equa
l to
7 P
erce
nt o
f it
s C
riti
cal
Val
ue.
U1
tv
40
- fS 20
:c
u z - ...., Z
o
I" /
(
\ t
\ l
t.. \
q ...
c.::.
e:;
f'j
~ /
TIM
E( S
EC )
€
)
) \
\ \
I J
\ 7~
os
'\ /
C
J _L
-t- U
W
-'
lL.
W
a a...
-20
-t- -40
Pig.
29
. T
ip
Def
lect
ion
vs.
Tim
e C
urve
for
the
Out
er S
hell
w
ith
Two
Ope
ning
s an
d w
ith D
ampi
ng
Coe
ffic
ient
Eq
ual
to 1
0 P
erce
nt o
f it
s C
riti
cal
Val
ue.
U1
W
If 3
C- o
2 - f
t-1
-4
z !
1 U
J 2
: 1
-4
D
2:
C!)
U
J o
l<:>
d\J
PI I I
Ii Ii
1\ I \
I \J
\ I
' ~~
'"" r,
11 I
( '/
'
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C"~
... r
04
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) L
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7'(
7 ) '
117 ~,
1 'c
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.0\'
) I
} C
TIM
E( S
EC )
zen
-a
:
ClC
D
-1
zo
U
JU
J
CDx .....
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L -2
t- a
: -3
-If
Fig.
30
. B
ase
Ben
ding
Mom
ent
vs.
Tim
e C
urve
fo
r th
e O
uter
She
ll
with
Two
O
peni
ngs
and
with
D
ampi
ng
Coe
ffic
ient
Eq
ual
to 4
Per
cent
of
its
Cri
tica
l V
alue
. U
1 ,j:
>.
--,
----
--
--
--
--
----
--
---
.. 3 ,...
..
""0
2
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ffi I
1 ~ ~
to
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\
lip
J 'C
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r.J
TI
ME(
SEC
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e Y
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C~"I
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o I
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r\
f
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11
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r "
c::::
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C
D
U,J
zen
-e
x:
v C
lCO
I
Za
-1
lL
JU,J
(O
x -u.. -
2 f- a::
-3
-Ii
Fig.
31
. B
ase
Ben
ding
Mom
ent
vs.
Tim
e C
urve
fo
r th
e O
uter
She
ll
with
Two
O
peni
ngs
and
with
D
ampi
ng
Coe
ffic
ient
Eq
ual
to 7
Per
cent
of
its
Cri
tica
l V
alue
. U
l U
l
... s
C"- o
2 -
~ ~
~ I
1
~
.... E
) ~
C!)
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o I-=
":ld )
j) l)
l \
f\
1\
l H
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f\
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zen
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i\(\
ro \
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~£V'\V"V·Ob,,.DVA\]A
A
0 ~
Ih
D
c=-C
.....
cr:
TIM
E( S
EC )
20
V
4>
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sb
ClI
D
Z
lLJ
Cl
-1
a::l
lLJ
X .....
lJ...
-2
t- cr:
-3
-II
Fig
. 32
. B
ase
Ben
ding
M
omen
t vs
. Ti
me
Cur
ve
for
the
Out
er S
hell
w
ith
Two
Ope
ning
s an
d w
ith
Dam
ping
C
oeff
icie
nt
Equa
l to
10
Per
cent
of
its
Cri
tica
l V
alue
. In
0'
1
15
10
,... ., 0 - •
5 IJ
../
!e u -
0.:::
x to
u.
. C
!)IJ
../
o k=
:/"l
r~
1\ ! \
~ \ q
I \
I I P
pll
1\&
/1 J
I (I
no
rl ~
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• 11
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zen
1
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44
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....
.. 0
: a:::
:CC
O
:Q
IJ
../IJ
../
:.t:
x
-5
en
......
u..
t- o:
-10
-15
Fig.
33
. B
ase
Shea
ring
Fo
rce
vs.
Tim
e C
urve
fo
r th
e O
uter
She
ll
with
Tw
o O
peni
ngs
and
with
D
ampi
ng
Coe
ffic
ient
Eq
ual
to 4
Per
cent
of
its
Cri
tica
l V
alue
. U
1 -...
.J
....... ., o ....
15
10
• W
~
5 U
1
-4
0:::
::.::
~-
CD
W
0
I cJ
\ I
' ~
II \ ~
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\
I \
l\ A
~ ~\
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1\
1\
n TIf
1 /I
J\ A
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~fI
0 A
ll II
I'
ll M
AA
'f"-
• A
IV
TI
ME(
SEC
)
z (J
)<
:e:;»
!
\ I tIl I
i
r"\
lit )F
\ (
\
( \ r
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T' ?
V\ I
t ;:
rCA
_ ~~V
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YV
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_L
.......
a:
0:::
CD
a:O
~W
X
en....
... -5
I..L
..
f- a:
-10
-15
Fig.
34
. B
ase
Shea
ring
Fo
rce
vs.
Tim
e C
urve
fo
r th
e O
uter
She
ll
with
Two
Ope
ning
s an
d w
ith D
ampi
ng C
oeff
icie
nt
Equa
l to
7 P
erce
nt o
f it
s C
riti
cal
Val
ue.
1Jl
co
15
10
~ 0 -
LU ~
5 u
.... a::
; ~
~-
C!)
LU
o J
tV \
I'
\ I \
I
\ ,,1
\ I
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( \"
A t""
n 1\
I
\ cJ
\ (Vb
0
1\
b ,/
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<
>
Art
1\
0
A Iv
> • (
tr\ to
, O
Ac
TIM
E(
SEC
) zen
...
.. 0
: =
0
J \ /
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I'lf 'T
tF
VV
iJV 09
V\(
V
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vv<F
V 2b
' V '
VV
4j
'y
OY
'V'
v"V
3b
a::;
CD
O:a
L
UL
U
:J:
X
-6
en
.....
lL.
t- o:
-10
-15
Fig.
35
. B
ase
Shea
ring
For
ce v
s.
Tim
e C
urve
fo
r th
e O
uter
She
ll
with
Two
Ope
ning
s an
d w
ith
Dam
ping
C
oeff
icie
nt
Equa
l to
10
Per
cent
of
its
Cri
tica
l V
alue
. U
1 1.
0
IJO
(J)
20
w
::z::
u z f-4
....,
Z
o I"
l
/, \
/ lL
v I
I -
:::. L
'::..L
~ t.
~ c.
. ~t::::~
:=:A
\ TI
ME(
SEC
)
0 I
\ '-
I \J
i'
"
V
\ ::>
--
\ \
_L
......
t- U
W
-1
u...
IJ.J Cl
(L
-2
0 .....
. t-
-IJO
Fig.
36
. T
ip
Def
lect
ion
vs.
Tim
e C
urve
fo
r th
e In
ner
She
ll
with
Tw
o O
peni
ngs
and
with
D
ampi
ng
Coe
ffic
ient
Eq
ual
to 4
Per
cent
of
its
Cri
tica
l V
alue
. 0'
1 o
...... ~ u z ~
'-'
z t:)
~
t U
LLJ
-I
lL..
IJ.J o a..
......
t-
'to
20
I (
\ f
\ I
V '-
\ C
>,c
, I
'J ~/
\ <=
;> /'
:-:,
.r:
~
, 0
'\J
\
/' V
\
-'-7 V~7
~n
\ r:
7'"
\
30
TIM
E( S
EC )
-20
-!f0
Fig.
37
. T
ip
Def
lect
ion
vs.
Tim
e C
urve
fo
r th
e In
ner
She
ll
with
Two
Ope
ning
s an
d w
ith
Dam
ping
C
oeff
icie
nt
Equa
l to
7 P
erce
nt o
f it
s C
riti
cal
Val
ue.
0'1
f--'
40
- en
20
UJ
::t:
U
Z
t-
4
o..
..J
Z
a I
I D
)
I )
l -v
1
("--
---.
..,,
I ~
~.<>-:.
~
t-4
V
\
Ii V
\
lb!
<h
\]
i )
t-
TIM
E( S
EC )
"C
=\J
20
'\J
=V7"
\
sb
u UJ -'
lL.
UJ
Cl
0-
-20
.....
t-
-40
Fig.
38
. T
ip
Def
lect
ion
vs.
Tim
e C
urve
fo
r th
e In
ner
She
ll
with
Two
O
peni
ngs
and
with
D
ampi
ng
Coe
ffic
ient
Eq
ual
to 1
0 P
erce
nt o
f it
s C
riti
cal
Val
ue.
"" t\.)
4 -
3
I""- 0
2 ~
..... ~ ....
z :If
1
w
z :.t
: ....
D
:.t:
......
C!)
w
o ,"
'''' 'l
l!
I I
( v
\
I I
I \
,,\
A
Mo
/\
zen
if
V
\JV
V 'V
V
"y+
:J
O~ •
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C"\
I>
c"
A
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rO
a
0 A
. Q
UA
i 4
TIM
E( S
EC )
i
v ""
'V
'1:0 "
-"f
-if
' ()
OO
v
· =
20
V
'V
\? O
J 30
......
a:
Cl
CD
z -1
w
o
co
w
X .......
IJ..
-2
.....
a:
-3
-1J
Fig
. 39
. B
ase
Ben
ding
Mom
ent
vs.
Tim
e C
urve
fo
r th
e In
ner
She
ll
with
Two
Ope
ning
s an
d w
ith
Dam
ping
Coe
ffic
ient
Eq
ual
to 4
Per
cent
of
its
Cri
tica
l V
alue
. 0'
1 W
\f -
3
-C
"-o
2 ....
I-~ -
z i
1 w
:t
: -
10
:t
:'-'
C
!)w
a I _
.. /\ N
~
f\
/'v..
L:
' Q
.c:
::.
zen
tr V
4T
V
v ...
v'·
"fct""'O
= AC
'>
A
C'"
A
2b
O'
. v 0
=<
J
cu.~
fV\
AL'
. TI
ME(
SEC
)
~
a:
o V
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> 'i'
CJO
C
'VV
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<
v a
c;;
i 30
ClC
O
z -1
w
a
co
w
>< .... IJ.
.. -2
I- a:
-3
-If
Fig.
40
. B
ase
Ben
ding
Mom
ent
vs.
Tim
e C
urve
fo
r th
e In
ner
She
ll
with
Tw
o O
peni
ngs
and
with
D
ampi
ng
Coe
ffic
ient
Eq
ual
to 7
Per
cent
of
its
Cri
tica
l V
alue
.
0"1
.t:.
'+ 3 .....
...
C'-o
2
....
I-~ .....
z
I 1
w
L .....
ID
L
-C
!)
W
a zen
,
<>
C"\
. b
c::>
-1
b _
..
c>
<>
i
_..."
<><>
2h
ad
v '"
"V
9 fY
'\,.
.6 'V
'5
3b
T I M
E ( S
EC )
1-tC
I:
aeo
z
-1
wa
co
w
x I-t
LL
-2
l- e:
-3
-Li
Fig.
41
. B
ase
Ben
ding
Mom
ent
vs.
Tim
e C
urve
fo
r th
e In
ner
She
ll
with
Tw
o O
peni
ngs
and
with
D
ampi
ng C
oeff
icie
nt
Equa
l to
10
P
erce
nt o
f it
s C
riti
cal
Val
ue.
0'\
lJ
1
15
10
- II) c " • 5
LLJ
CIl
Q..
U
.... a:::
:c
~-
C!)L
LJ
o j
01
1 /If
1\
1\/
\ ,I
\.
1'd
.J\
IIIL
f\
M
M
4
M
A,
c.I\
AI"
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A
___
AA
A
1\
... ~
AJ
j
/\tJ.
Qf\
A
A
r\.
TIM
E( S
EC )
Z
W
'20'
v v'
q;
if;
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,
r y t
o<J
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_
a::
o
V
V 'V
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vv <
TV
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IV'
0 Q
O
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0
0
v'
=
a::::
CD
a:
: a
LLJL
LJ
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x
-5
cn
_
lL. ....,
a:
-10
-15
Fig
. 42
. B
ase
Shea
ring
For
ce
vs.
Tim
e C
urve
fo
r th
e In
ner
She
ll
with
Two
Ope
ning
s an
d w
ith
Dam
ping
C
oeff
icie
nt
Equa
l to
4 P
erce
nt o
f it
s C
riti
cal
Val
ue.
0"\
0
"\
15 -
10
,.... .., 0 9"4
W
c! 5
u a.
. .....
0:::
)C
~-
C!)
W
o j
",,~l
lIn
0
/\/\
!~.,-,
A
A
zW
...
.. CC
V
rv 'V
'll" i
/wro
-."p
. 'th
W· ~V
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os J
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q r
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t
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~ A
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0
TIM
E( S
EC )
o:
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2b
v
v V
9
> V
<
JV
f 0
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"'V
'::;
:'>""3
6-
CCo
W
w
Xx
en
....
. -5
\J.
..
I- 0:
-10
-15
Fig.
43
. B
ase
She
arin
g Fo
rce
vs.
Tim
e C
urve
fo
r th
e In
ner
She
ll
with
Tw
o O
peni
ngs
and
with
D
ampi
ng
Coe
ffic
ient
Eq
ual
to
7 P
erce
nt o
f it
s C
riti
cal
Val
ue.
0\
-...J
15 -
10
.......
en 0 -
UJ ~
5 u
.... 0:
:: :li
e: E
)
u..
-C
!)L
Ll
o t
""C
l fir
.:::.
f'l",
I'.:;
;. <
::;l
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-
-•
tC"
• 0
i:::.
...::,
Z(J
)
AO('~ .
,'V'
<'
6/\
.. C
;;
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:
,jV
c;
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v
v v
10 =.0
v
IP 0
,,;0
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4V
V •
'"
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r c
V
<><:/
.....
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TI
ME(
SEC
)
0:::
CD
20
e:a
LL
lW
:J:
X
-5
CI"
J_
u..
l- e:
-1
0
-16
Fig.
44
. B
ase
Shea
ring
Fo
rce
vs.
Tim
e C
urve
fo
r th
e In
ner
She
ll
with
Tw
o O
peni
ngs
and
wit
h D
ampi
ng
Coe
ffic
ient
Eq
ual
to 1
0 P
erce
nt o
f It
s C
riti
cal
Val
ue.
0'\
0
0
IJ.J
:J::
.......
I-
8 z.
IJ
.J IJ
.J ....
~
I-
"-"
IJ.J
(O(f
.)
...J
IJ.J
...J
UU
J
a::::
r:
0...
en
en
to
o...:
x -I-
I-
60
54
30 0
-30
-60
16
r--
2b
-,--
-sb
TI
ME(
SEC
)
Fig
. 45
. Ti
me
His
tory
Cur
ve
for
the
Spac
ing
Bet
wee
n th
e To
ps
of t
he O
uter
and
In
ner
She
lls
with
Ope
ning
s an
d w
ith
Dam
ping
C
oeff
icie
nt E
qual
to
4 P
erce
nt o
f it
s C
riti
cal
Val
ue.
m
'-0
W
:I:
-.....
.
8 z w
W
to
1 :x
....
......
W
ccen
...
J W
-l
uw
ct:
::r:
o.
..cn
U)
to
0..
.3:
........
... .....
.
60 -
~
30 o r
-so
-60
-~~J\J\;;'~~A~~~\~-
16
2b
3"t
-TI
ME(
SEC
)
Fig
. 46
. Ti
me
His
tory
Cur
ve
for
the
Spac
ing
Bet
wee
n th
e To
ps
of t
he O
uter
and
In
ner
She
lls
wit
h O
peni
ngs
and
with
D
ampi
ng
Coe
ffic
ient
Equ
al
to
7 P
erce
nt o
f it
s C
riti
cal
Val
ue.
'-I
a
IJ.J
:c ..-!
z.
LLJ
IJJ
.... 3
: ...... -
LLJ
CIl
CrJ
..
J L
LJ.
.J
UL
LJ
a::
c
a..
(f)
en
e:>
Q..
.3:
~I--
......
60
5'f
30
V\--
-----
-~l\~~'-~~/~-
I o
+-10
2b
3
0
TIM
E( S
EC )
-30
-60
Fig
. 47
. Ti
me
His
tory
Cur
ve
for
the
Spac
ing
Bet
wee
n th
e To
ps
of t
he O
uter
and
In
ner
She
lls
with
Ope
ning
s an
d w
ith
Dam
ping
C
oeff
icie
nt E
qual
to
10
Per
cent
of
its
Cri
tica
l V
alue
. .
--.J
i--'
- (!) z 1.
0 it
~
z «
0 0
I-0
u Z
lL
J -
...J
Z
u..
lLJ
0 a
I- u 0
.5
e:1~
I-
u..
lJ.J a (l.
I-
... \ '" /O
UT
ER
S
HE
LL
~:"'.
L'NNE
R SH
ELL
"'o~. O~1C o===
====
=~
1 Of
:5 I
2 4
-7 10
%
OF
CR
ITIC
AL
DA
MP
ING
Fig
. 48
. V
aria
tion
of
Tip
D
efle
ctio
n w
ith
Dam
ping
C
oeff
icie
nt f
or
Bot
h S
hell
s w
ith O
peni
ngs.
"-
.I
I\..
)
--- ~ 1.
0 0::
~
« ~o
Zo
W
z
~o~
~Z
w
<.!)
I~
zO
-~
o Z WI<.
!> m
z
0.5
6\ I O
UTE
R
SH
EL
L
\''''1.
.. IN
NE
R
SH
ELL
o~ ~~
__
__
__
o~ o~, __
_
----
0 ---=
====
== ~
1 a z W
Q
)
~ ... ~
I 10
0/0
OF
CR
ITIC
AL
DA
MP
ING
Fig.
49
. V
aria
tion
of
Bas
e B
endi
ng M
omen
t w
ith
Dam
ping
C
oeff
icie
nt f
or B
oth
She
lls
with
O
peni
ngs.
--.J
W
- ~ 1.
0 a..
~
<t
o o
0 lL
. Z
-~
w
Zu
-
Q::
~
0 0
,5
UJ
lL.
:r:
U')I~ z Q
:: <t
UJ :r:
U')
INN
ER
S
HE
LL
\0
/ +~
OU
TER
SHEL
L
'~~~6~~
o·
.5
I 2
4 7
10
0/0
OF
CR
ITIC
AL
DA
MP
ING
Fig.
50
. V
aria
tion
of
Bas
e Sh
eari
ng
Forc
e w
ith
Dam
ping
Coe
ffic
ient
for
Bot
h S
hell
s w
ith
Ope
ning
s.
" ~
150
100
- ~
60
HtlNWl~~,t-;;Ji
TIM
E( S
EC )
~ t-4
O"
"'y
- Z
to
.....
. I- a.:
-6
0 0:
::: LU rd u u a.:
-100
-150
Fig.
51
. Th
e R
ecor
d of
the
V
erti
cal
Acc
eler
atio
n C
ompo
nent
of
the
El
Cen
tro
Ear
thqu
ake
on M
ay 1
8,1
94
0.
--.J
lJ1
0.6
- (f)
.26
W
:I:
u Z -.....
...
...
..J a:
o Ijl
lllll~
rulldl
~~I~~l
mml~lr
n~II~~
mlllli
llmlll
illlll
llilil
llllJl
lllmmH
fiHruI
llIIlH
lmn1II
1H11H1
IIUlnI
lIlUlW
m!lll!
lm~III
III!Ul
llIUWI
II~IUl
~~ftll
ll\~UI
~II\I~
Ml~I~M
lmAW~~
l\ijJJ
jJJ\MM
M~AN}N
ll.
zz
E:)-
_0
....
. j::
u
-W
eD
..
Jz
LL-
c w
..J
0
......
Q..
-.2
6
-t--0
.6
Fig.
52
. V
erti
cal
Tip
Dis
plac
emen
t vs
. Ti
me
Cur
ve
for
Out
er S
hell
W
ithou
t O
peni
ng.
TIM
E( S
EC )
~
0'\
10
- II) 5
0 .... «I Q..
W
....
U ~
0::
:-to
u
..w
o
J I~m~
rtf!"'~/
IVU IflmW
Un!llrfl
lilm~~ml
lmmmm~m.
!lI1iIl1
~lmll~lm
lmlmlm~l
rI!mmmll
ll~lm~r~
~1~1ItII
~~lmII_m
tlIIIIII
M.IVIIII
II~~MIIl
rRtlmllI
WWvWt,Wi
l-T
I ME (
SEC
)
fJ)
-1
a:
a:
ED
.... xc
a
::w
x .... u.
. f-
-5
a::
-10
Fig.
53
. A
xial
Fo
rce
at B
ase
vs.
Tim
e C
urve
fo
r O
uter
She
ll W
ithou
t O
peni
ng.
'-l
'-l
0.5
.....
(IJ
.25
IJJ ::r:
U z -,..
., ...
...
...J
z~
o l.l
~mll!~
llIllI
IlllIl
lIIllI
llIIll
lIllll
lllllm
~[IIII
l~IIIW
~!ulIl
llltll
IIlllm
ll!l~n
lIllll
Ul[lII
llllll
lllBJl
Jlllll
lillll
llilll
RWllWl
lllill
Llllll
UllIlU
Jlll_I
IlmU�l
��lr_-
-TI
ME
( SE
C )
t!.J
~4
_0
......
g
u-
w
C,!)
-1~
lL..
..J
W
C
I
0...
-.2
5
...... .....
-0.5
Fig
. 54
. V
erti
cal
Tip
D
ispl
acem
ent
vs.
Tim
e C
urve
for
In
ner
She
ll
With
out
Ope
ning
. -..
,J
co
10
-., 5
0 ~ ~
IJJ
....
U
:.:
0::
:-to
L
J..L
U
0 ...J~
a:r
o
i "fl
lm~~
IIIJ
mllm
mmmf
mUmJ
mmmm
mmmm
mm~m
m~nm
~J1J
Wmlf
ml1!
111l
1i1l
M1l1
m111
lt1f
lMJ1
1J~l
ffll
lltm
lB;-
T I M
E ( S
EC )
,.
.".,
~lTI
x "'"'"
lJ...
I-
-5
a:
-10
Fig.
55
. A
xial
Fo
rce
at B
ase
vs.
Tim
e C
urve
fo
r In
ner
She
ll
With
out
Ope
ning
.
-...J
\.0
80)
V! 0)
s:: ..... s:: (J) p-o 0 3 I-
-C +l ..... 3 ~
r--(J)
-C (/)
S-cu
+-> ::::s
0
S-o 4-
cu > S-::::s u (J)
E ..... l-
V! >
+-> s:: (J)
E (J) () ro
r--0.. V!
Cl
0..
I-
r--ed U
+l S-(J)
>-&0
&0 . . 9 <.0
o
( ltJNl0fllISNlJl ) L!') . 0)
u.. ( S3HJNI ) NQIIJ31d30 dIl
10
......
., 6
0 .... ~ W
....
. U
:I!
I:: O
:::-
..J
\\D 'k
LU
D J"
~mIR~rullrul
llnn
w~~m
;ulm
mmmr
uwoo
lloo
ln~~
ilm~
~~k~
' TI
ME(
~C
J tJ,
_J
C.b
a:w
-xo
a:w
x -IJ.. t-
-5
a:
-10
Fig.
57
. A
xial
Fo
rce
at B
ase
vs.
Tim
e C
urve
fo
r O
uter
She
ll
with
Tw
o O
peni
ngs.
00
f-J
0.6
.......
(J)
.26
UJ
:I:
U :z
.....
o-J -'
0:
o 1 J
U!Hmll
l\l\\l
~iml\l
\I\ll\
\lllll
llllHl
1IlllH
llllll
Hlllll
lllU\l
llllll
llllll
llllll
ll!!l!
ll!lll
llllll
llllll
tillll
1llUlU
lillll
U11Ul1
U.llIL
Ulllll
~I1lIl
liJj_\
jlllU!
UUU\_~
~~ T
I ME (
SEC
)
zz
~H
-§
i-
-&=
-u
-w
ro
...J
Z
IJ..
.D
UJ-'
0
....
Il...
-.2
5
....-
J
t--
-0.5
Fig.
58
. V
erti
cal
Tip
Dis
plac
emen
t vs
. Ti
me
Cur
ve f
or I
nner
She
ll
with
Two
Ope
ning
s.
co
N
(£OtlfScll)1 ) 3St:§9 03X 1.:1 1(;
3J~Q.:J 1I:JIXH
Q -I
83)
V)
0'1 s:::
Or-s::: (l) Cl..
0
0 3: I-
..r:: +.l Or-
3:
r--.--(l)
..r:: (/")
S-(l)
s::: s:::
I--i
S-o 4-
(l)
> S-~
U
(l)
E or-
l-
V)
> (l) V)
rtl CCl
+.l rtl
(l) U S-o
lJ...
r--rtl 0,... X
c:::C
Q) LO . 0'1
Or-lJ...
•
Symmetry Line
I
I
I
I
~ I \~
I /7'
\ ~
I
(,J
0
BASE
Symmetry Line
(---1 s/------l
\
\
\ .... u.
0
\ (\!
10 ~
Fig. 60. Contour Plot of the Axial Stress (ksi) in the Outer Surface of the Outer Shell Around one Flue Opening at the Most Critical Time (25.2 Seconds) with ~ending Moment = 32.048xl09 in-lbs; Shearing Force = 11.81xlO lbs; and Axial Compression = 2l.77xl04 lbs at the Base.
84)
Symmetry Line
/ I
I
I
I
I 1', . ~
I
7~
o
I
Gi
BASE
Symmetry Line
-"/--\
~ _\
~ \
\
\
\
\
~ ~ \
\
85)
Fig. 61. Contour Plot of the Axial Stress (ksi) in the Outer Surface of the Inner Shell Around One Flue Opening at the Most Critical Time (28.8 Seconds) with Bending Moment = 11 .28xl09 in-lbs; Shearing Force = 45.57xl05 1bs; and Axial Tension = 15.05xl04 lbs at the Base.
