4.0 -1111t~ue addstructural -ennmrc*deev pth dje t~o2 the esose-spectru (u) army engineer waterways...
TRANSCRIPT
t~UE ADDSTRUCTURAL -ENnMrC*dEEV Pth djE t~o2THE ESOSE-SPECTRU (U) ARMY ENGINEER WATERWAYSEXPEIMENT STATION VICKSBURG MS INFOR. P NIERSHA
UCASIFIED AUG 89 IES/TR ITL-89-6 F/G 0/2 2
mohmomm-som soou
4.0 ___-1111
Destroy this report when no longer needed. Do not returnit to the originator.
The rindings in this report are not to be construed as an otficialDepartment of the Army position unless so designated
by other authorized documents.
This program is furnished by the Government and is accepted and used
by the recipient with the express understanding that the United StatesGovernment makes no warranties, expressed or implied, concerning the
accuracy, completeness, reliability, usability, or suitability for any
particular purpose of the information and data contained in this pro-
gram or furnished in connection therewith, and the United States shall
be under no liability whatsoever to any person by reason of any use
made thereof. The program belongs to the Government. Therefore, the
recipient further agrees not to assertany proprietary rights therein ortorepresent this program to anyone as other than a Government program.
The contents of this report are not to be used foradvertising, publication, or promotional purposes.
Citation of trade names does not constitute anofficial endorsement or approval of the use of
such commercial products.
REPORT DOCUMENTATION PAGE M8 NO J 0 ,
0ES~ -Q C VAR. NC
Unclacssif ied:1 _iSS I.-. A - r,.R 7y 3 DI5,'PBON AAAe'w
Approved for public release; distribution. a ON DO'NGRADNG SC-.ED,.LE unlimited.
CRGANZA,ON REPORT NM3ERS; 5 "VONi-OR NGC N'3i
Technical Report
.aN-k O- "ERPORMING ORGAN!ZA- O,
6t 'J " CE S VBO. 7a NAME OF MON O'C" CGA ZA ONOif aoo'caole) USAEIWE S
See reverse. Information Technology Laboratory
Ec DDRESS Cry. Stae and ZIPCooe) 7D aDDRESS City. Stare an 7(P Code)
'' 'HI,. F, :v R,,I d PO Box 631
; o-. :5 ES9!q-o,19 Vicksburg, MS 39180-0631
SAE ,OF _%D'NG SPONSOR NG Sb O.CE 9 PROCjREMENr \SRLEN" DENT'.CAl ON% NvMBER
'iCA% ZAr
ON (it apo icatilt
See reverse. I
.XZCDEESS (Cry State a d Z!P Code) 0 SOQRCE OF F. N;G NV'ERS
PROGRAM PROEC7 AS AORK _%i'
Washington, Of "0314-1000 ELEMENT NO NO NO ACCESSON NO
. (Inciude Security C islsifcation)The Response-Spectrum Dynamic Analysis of Gravity Dams Using the Finite Element Method;
Pbse TI
ERSO%4,A'0RS(iersma, Paul
a E 31 aE0GRTb 'ME COVERED 14 DATE OF REPORT Year Month Day) 15 PAGE COUNT
Final report FROM___ August 1939 110.A6 5_PPLE%!E1'aPY NO-'TOPN
Available from National Technical Information Service, 5285 Port Royal Road, Springfield,VA 22161 .
COSAT, CODES 18 SBEC- EPvS Continue on reverse r necessar and :dentfy by block number)
_D O:ROUP SB-GROUP Finite element methsod (LC)
Gravity dams (LC)
Materials--Dynamic tLcst in, ( Ic)'9 ABsrRAC" Continue on reverse if necessary and idenify by block number)
This report is a part of a Corps-wide project to provide guidance for the use of fi-
nite element analyses. A continuation of the Phase Ta study dealing with a static finite
element analysis of a gravity dam and Phase Tb study dealing with foundation-structure
interaction, Phase II gives direction in performing a responso-<pectrtim dynamic finite
element analysis of a gravity dam. An example problem is presented to demonstrate how toverifv the results for a simplified structure before moving to the analysis of the real
structure. An analysis of a realistic gravity dam is then presented with results.
"9 9 S+ VP S <S A i ARLA t OF AS~-
~R 11 ABSTRACT SECA CA*ION',0<5 VIALM'D V D SAVE aS S [ YECSPS Unclassified
DM ',VE v ESPONS BE ND1 DDZ 22<o -ESaHOE lest/Fe Srea Code) 25< <S,<F(E (*\'RO
DL* ORM 1 , ,134VAR 4 . - o j a :, e .e 3a d - <e"
A.i ot,-, ec o_. aTe )o r ete Unc lassifiled
89 9 18 .g
Unclassified86CUINT CLA IIFIcAtIO OF THIS PAe
6a. NAME OF PERFORMING ORGANIZATION (Continued).
USAEWES Information Technology Laboratory and
CASE Task Group on Finite Element Analysis
8a. NAME OF FUNDING/SPONSORING ORGANIZATION (Continued).
Civil Works Directorate
Office, Chief of Engineers
Aooession For
NTIS GR.A&I VDTIG TAB F1Unaanounced 0Justifieatlon
By
Distribution/
Availability Codes
Avail ano/orDist Special
Unclasgnifi9dSECURITY CLASSIFICATION OF THIS PAGE
PREFACE
This report is aimed at providing guidance for the use of the finite
element method of analysis for the analysis of concrete gravity dams. This
Phase IT report will address only the dynamic analysis of the gravity dam and
the need for including a foundation. The Phase Ta report addressed the static
analysis of the gravity dam. Phase Ib will address the effect of the founda-
tion in the static analysis of concrete gravity dams. Other reports will ad-
dress guidance for other phases of finite element analysis. The work was
sponsored under funds provided to the US Army Engineer Waterways Experiment
Station (WES) by the Civil Works Directorate, Office, Chief of Engineers
(OCE), US Army, as part of the Computer-Aided Structural Engineering (CASE)
Project.
Input for the report was obtained from the CASE Task Group on Finite
Element Analysis. Members and others who directly contributed to the report
were:
Mr. David Raisanen, North Pacific Division (Chairman)Mr. Barry Fehl, St. Louis DistrictMr. Dick Huff, Kansas City District
Mr. Paul LaHoud, Huntsville DivisionMr. Jerry Foster, Federal Energy & Regulatory CommissionMr. Ed Alling, USDA - Soil Conservation ServiceMr. Paul Wiersma, Seattle DistrictMr. Terry West, Jacksonville District
Mr. Lucian Guthrie, OCEDr. N. Radhakrishnan, WESDr. Robert Hall, WESMr. H. Wayne Jones, WES
Dr. Kenneth Will, Georgia Institute of Technology
The report was compiled and written by Mr. Paul Wiersma.
Dr. Radhakrishnan, Acting Chief, Information Technology Laboratory (ITL), WES,
and CASE Project Manager, along with Dr. Robert Hall, Research Civil Engineer,
Structures Laboratory, WES, and Mr. H. Wayne Jones, Civil Engineer, ITL. moni-
tored the work. Ms. Gilda Miller, Information Products Division, ITL, edited
the report. Mr. Lucian Guthrie was the OCE Project Monitor.
COL Larry B. Fulton, EN, is the Commander and Director of WES.
Dr. Robert W. Whalin is the Technical Director.
CONTENTS
Page
PREFACE ................................................................ I
CONVERSIY FACTORS, NON-SI TO SI (METRIC)UNITS OF MEASUREMENT ...................................................... 3
PART I: INTRODUCTION ..................................................... 4
Background ........................................................... 4
Objective ............................................................ 4
PART II: FTWTTF ELEMENT DYNAMIC ANAlYSIS OF A SIMPLIFIED STRUCTURE... 9
Selection of a Simplified Structure .................................. 9Finite Element Models ............................................... 9
Discussion of Results ............................................... 17
PART III: GRAVITY DAM EXAMPLE PROBLEM ................................... 28
Description of the Problem .......................................... 28Finite Element Method Analysis ...................................... 29
Chopra Simplified Analysis .......................................... 33
PART IV: FOUNDATION EFFECTS ............................................. 50
Finite Element Yodel ................................................ 50
Stress Comparison ................................................... 51
PART V: SUMMARY AND RECOMMENDATIONS .................................... 65
Summary .............................................................. 65Recommendations ..................................................... 65
REFERENCES ................................................................. 66
APPENDIX A: MODELING OF HYDRODYNAMIC EFFECTS ............................ Al
APPENDIX B: INPUT FILE FOR MESH 2 MODEL ................................. B1
APPENDIX C: NATURAL FREQUENCY CALCULATIONS OF A CANTILEVER BEAM ....... Cl
Elementrary Beam Theory ............................................. C2Timoshenko's Theory ................................................. C3
APPENDIX D: GTSTRUDL INPUT AND OUTPUT FILES FOR GRAVITY DAMEXAMPLE PROBLEM ............................................. DI
APPENDIX E: COLE/CHEEK COMPUTER PROGRAM INPUT AND OUTPUT FILESFOR GRAVITY DAM EXAMPLE PROBLEM ............................. El
2
CONVERSION FACTORS, NON-SI TO SI (METRIC)UNITS OF MEASUREMENT
Non-SI units of measurement used in this report can be converted to SI
(metric) units as follows:
Multiply By To Obtain
feet 0.3048 metres
inches 2.54 centimetres
kips per foot 1355.818 newton-metres
kips per square foot 47.88026 kilopascals
pounds 4.448222 newtons
pounds per cubic 'not 16.01846 kilograms per
cubic metre
poundlo per foot 14.5939 newtons per metre
pounds per square foot 47.88026 pascals
pounds per square inch 6.894757 kilopascals
slugs 14.5939 kilograms
THE RESPONSE-SPECTRUM DYNAMIC ANALYSIS OF GRAVITY DAMS
USING THE FINITE ELEMENT METHOD
Phase TI
PART I: INTRODUCTION
Background
1. This report has been prepared as part of the ongoing effort by the
Computer-Aided Structural Engineering (CASE) Committee on finite element anal-
ysis. It is part of a Corps-wide project to provide guidance for the use of
f'inite element analysis.
Objective
2. The primary objective of this study is to give direction in perform-
ing a response-spectrum dynamic finite element analysis of a gravity dam using
a general-purpose computer program. This is one of several methods currently
being used within the Corps of Engineers (CE) for performing a dynamic analy-
sis on gravity dams based on guidance provided in ETL 1110-2-303 (Department
of the Army 1985). Other procedures and computer programs commonly used are
outlined below with the differences between them discussed. Table I contains
'\a summary of the procedures and computer programs.
a. The seismic coefficient method as presented in EM 1110-2-2200
(Department of the Army 1958) is in reality a static analysis
with static forces representing inertial and hydrodynamic loads.
The dam is analyzed using elementary beam theory and assumes a
fixed foundation and incompressible reservoir. The hydrodynamic
pressure distribution increases with depth. This procedure is
good for horizontal motions only. The procedure is currently
used only to evaluate overturning and sliding stability.
b. The "simplified" method for earthquake analysis of gravity dams
was developed by A. K. Chopra (1978). The analysis of the dam
is similar to that of the seismic coefficient method in that the
inertial and hydrodynamic loads are represented as static forces
and the dam Is analyzed using elementary beam theory. The sim-
plified method also assumed a rigid foundation. The difference
between the seismic coefficient method and the 1978 version of
the simplified method lies in the way that the inertial and
hydrodynamic loads are determined. In the simplified method
these loads are based on the fundamental mode of vibration of
4
-c4 'C cc (t
- -4 ( C. m m 0 c ( a
0- C 0 41 C -W0 0.C' NN N N N N W
r4 -~4) .,4 >) >~14
C 00 0 0 0
to ~ -4
4 )0 w-' m "a c co
a 0 W41 0' -0 02C Q- ) w ,q 4) 4
Z~- - C40.C I' 'CC 0 -~ 'CC 4 ) cu Q)' 4) cu QC )
04 14 *-4 ~ ' -
014.4 U) En02ci
CO cc co
0)4 "a "a C13 0 -4 'C IV
b1 m -) 0'C 0' CO c'C 0'C '-4 cc 4-hcu(D QS.,q v4 C 0 r-4 0 r-- 0 -4 co -C cu
-H 'Cr a -4 M.0'C-4 .0 0 "a- -0C -4 a0 .0 m2 cu0 0P24) * 4) ,-4 U) -44 1-4 ) - 4H M2
'- 2 > a) " Wi2 AJ 4.j -H 0'4U u 020. 24 " -4 2-4 14 Uucc 4-4 Er-1 -H T 0- U)- w (U Q
QO ) Ow0 -CL W0.J 4.j)u4) - 06 41 4 . A. Q c a)0 ) m 0 Aj(A F- a2 m0C 2-.(a. ca w k" 0. (t CL cO v-0 240 U, 3 "Q) 0 m 0J 0 4-J Ia. 44) " " 0 4--, .. 0. Qcc 0 co (
C 0 0 C 0
4) UO
c- 4-4 . W -4 w-
r.0 aO wO u )
is' '! 01 r-4 C>H r.. 1 C: .
4 -4 0. ")4 4)4 A)4 CO1 CO4- CO-- 1$4 .-. .4 cu. 4Jo 40 M) 4)a
'-44) 4 w -44 w'-4 4
L4 o0&-
>.00w >W u . W 0 0 2-.
C" co I. cli 0. cc CL 06
&-' 4.j J0 41 J.,4 w 1 41-1-4 Z;" tr-a U H 02, 4)U4H0C: 0J4 4.4 0-40
w~ 4-4) 0- S OG uG OH MC 04) U 1Q )w t2-.~~ 0 0' '-'- wC' 0--'4 0 0C0C 0
:3u2. )C a Qc 00 C 2 -0 o '-. 0 w vw) U C 4) w " >1 U C 4) 4)0 C 4) . cu .0 ) t&4 4 ) .00.W
com4- c c"a 5
-H- 0 . 2-4
4) '-4 41 0 -jO 0 cc 02-. 444 4) 14 W .- 4 24 2 -
'C co P-. to. 0.4) 0 C )' 4) -11 I-. w
U 4> 4> 0 . .0 'C G- O %0- C) rL -, -4
2-4 00 C 44' '4-40 u0 0. mr= A - I M.4'0cr Q) P..' 0.' 4)
E" r'- r'- -a CL CO4) '- 4 0 a)
cn Ln0 L2 U
the dam, with and without reservoir. In this method the reser-voir is assumed to be compressible. As with the seismic coeffi-
cient method, only horizontal motions are considered. In 1986,Fenves and Chopra revised the simplified method to include theeffects of an elastic foundation (by using a period lengtheningratio and added damping) and the effects of sediments in the
reservoir.
c. Tn 1986, Cole and Cheek (Technical Report SL-86-44, Departmentof the Army) combined the simplified method (1978 version) witha finite element model of the dam (in lieu of ,, ing elemcntarybeam theory). At present the procedure assumes a rigid founda-tion but plans are to include the effects of a linear elasticfoundation modeled by a finite element grid.
d. EADhi (a computer program for Earthquake Analysis of GravityDams Including Hydrodynamic Interaction) is a finite element
program that uses a time-history (but in the frequency domain)method of dynamic analysis. The program assumes a rigid founda-tion and models the reservoir as a comnressible fluid. Thehydrodynamic effect is represented by applying added mass to theface of the dam. The added mass is computed using methods simi-
lar to those used in the simplified method discussed earlier.
e. The EAGD-84 (Earthquake Analysis of Gravity Dams IncludingHydrodynamic and Foundation Interaction) program is very sinmilar
to the EADHI program except that the foundation is modeled as a
viscoelastic halfplane.
f. Several general-purpose programs are available that will performeither a 2-D or 3-D finite element time-history or response-spectrum dynamic analysis. Both dam and foundation can be mod-
eled by a linear elastic finite element grid or the dam can be
modeled on a fixed foundation. The influence of the reservoiron the dam is typically modeled using the cuncupt of added mass.Westergaard's added mass method is commonly used which assumes a
rigid dam and incompressible fluid.
3. This study is a continuation of the Phase Ta study which familiar-
ized the beginning finite element analyst with the steps necessary to perform
a static fir.ite element analysis 7f a gravity dam (Will 1987). Many of the
steps in performing this dynamic analysis could also apply to other Corps
structures. The beginning engineer shculd be able to develop an understanding
of the steps necessary to perform a dynamic analysis by carefully tollowing
this study and using supplemental material.
4. F-llowing is a list of the necessary steps in performing a dynamic
finite element analysis. The steps are very similar to those for a static
analysis as presented in the Phase Ta report (Will 1987).
a. Select a finite element computer program currently in use by theCorps or in widespread use by private engineering firms and sup-
ported by a vendor with the desired analysis capabilities.
6
b. .elect a simple problem as close as possible in overall geome-
try, material properties, boundary conditions, and loading con-ditions as the real structure to be analvzed. This structureshould have closed-form, experimental, cc other analytical so-lution results available.
c. Select the type of analysis (i.e. response-spectrum, time-h+istor v ) to be performed to obtain the desired results.
d. Select the finite element types to be used in the analysis fromthe library of elements available in the program chosen in
Step a.
e. Develop and .ralvze finite element models of the simplifiedstructure and compare results, such as deflections and stresses,with the closed-form results.
f. Develop modeling guidelines for both the grid and loadings fromthe results o! Step e, which may be extended to the real
structure.
g. Prepare a finite element model of the real structure and performan analysis.
h. Ask the following question: Is the solution accurate? If theanswer is no, refine and reanalvze until the answer is ves.
Before actually performing the analysis, further detailed discussion
Of the-e steps is warranted to understand their necessity:
P. Tn Step a, the key concept is thot the finite element programshould be currently used by the Corps or other engineering firmsand supported by a vendor. There are numerous finite elementprograms available today, therefore care must be taken in theselection process. Vhile factors such as ease of use, func-tional capabilities, and price are extremely important, an over-riding consideration is the use of the program within the Corpsor other engineering firms and support bv the vendor. An idealsituation is to find a program that is easy to use, has the nec-essary functional capabilities, is reasonably priced, and iscurrentlv being need by someone within the engineer's group and
is supported by the vendor.
b. The motivation for Steps a-e is to provide an opportunity forthe engineer to build confidence in the use of the program, fi-nite element modeling techniques, and to develop an understand-ing of the convergence criteria. Another important reason forthese FLeps is to provide the engineer with an understanding ofthe ty7pe, quantity, and quality of finite element results. Amuch too common occurrence is for the engineer" to devote anenormous amount of time iii developing the finite element modeland after results have been obtained, too little time is devotedto the interpretation of these results, i.e., the accuracy ofthe results, or their actual usage in the design process.
c. From the analysis performed in Steps b-e, the engineer must thenextrapolate the information gained from the modeling of the sim-ple :tructure to the modeling of the real structure. Guidelines
such as the number of subdivisions of the mesh in the horizontal
and vertical directions may be developed for use in the firstmodel of the real structure. Extreme care should also be usedwhen deciding how to handle the mass distribution of the struc-
ture and the necessity to include any additional mass, such asfoundation and adjacent water.
d. In Step g, the real structure is modeled, analyzed, and the re-
sults are interpreted. This leads to the crucial questions inthe analysis: Is the solution accurate within an error criteria
developed by the engineer? How much error is there? These arethe most difficult and crucial questions in the entire process.In many instances, the only correct way to answer these ques-
tions is to refine the model, reanalyze, and compare solutions.The following question should then be asked: Have the results
changed significantly due to the refinement? If not, an approx-imate solution has converged and the engineer must determine ifthe results make physical sense. If the results have changed
significantly, other models may be required and comparisons re-peated until convergenzc is satisfied. The engineer must remem-ber that the finite element method is an approximate solution
technique.
e. Also the questin of whether to perform a time-history or
response-spectrum analysis should be considered. In either casethe dynamic loadings and mass distribution should be examined to
ensure that they are appropriate.
6. In performing these steps for the analysis of a gravity dam, this
phase ot the ;tudv is limited to developing a method to analyze the deflec-
tions ird Lre-<cs ,'f the gravity concrete structure only. Interaction be-
tween t1'e qtrc turv and foundation is not considered at this time, however the
reservoir is considered. The program selected in Step a was GTSTRUDL* since
it i -pp ,rd V.; vendor, and currentlv widely used by the Corps. Also,
STR['lI '€ ;pr. t: t ive of a general-purpose finite element program.
- .PALI ' h is r-port presents an example of Steps b-e in prepara-
tion fnr t'i alvsi a gravity dam. An actual analysis of a nonoverflow
monolith similar to t!,(, Richard B. Russell Dam is presented in PART TIT. The
same eximple proI,Iem is then analyzed using Chopra's simplified method to
illustrate the different results produced by each method. Part IV studies the
impac:e of including a foundation with regard to combined static-dynamic
stresses within a gravity dam.
G-TSTRi'DTI is a generai-purpose finite element program owned and maintained
by the (;TICES Systems Laboratory, School of Civil Engineering, GeorgiaInstitute of Technology. Program runs used in this report were made on theControl Data Corporation, Cvbernet Computer System.
PART I: FINITE ELEMENT DYNAMIC ANALYSIS
OF A SIMPIFIED STRUCTURE
Selection of a Simplified Structure
8. The simplified cantilever structure representative of an idealized
gravity dam (Figure 1), finite element models, parameters, along with
80'
WATER
SURFACE -
0
Figure 1. Simplified structure
recommendations established in Phase Ia of this study were used here. Again,
the finite element runs were all made using GTSTRUDL. The program can be used
in the analysis of the static and dynamic response of linear two- and three-
dimensional (2- and 3-D) structural systems. The element used was the "IPQQ"
eight node isoparametric quadratic quadrilateral element.
Finite Element Models
iirilt element meshes
Three different models previously developed in the Phase Ta report
ts compare the convergence characteristics were used again. The various mod-
els are called the coarse, fine, and very fine meshes to indicate the r2lative
degree of refinement. They are also referred to as Meshes 1, 2, and 3 as
9
shown below. These meshes are illustrated in Figures 2, 3, and 4. The node
and elements are labeled in these figures. A summary of the meshes is pre-
sented below:
Number of Elements
Description Number of Nodes (All IPQQ's'
Mesh I Coarse 45 10
Mesh 2 Fine 149 40
Mesh 3 Very fine 537 160
Modeling procedure
10. The models were assumed to be completely restrained along their
bases and to be in a state of plane stress.
Material properties
11. The weight density of the material was assumed to be 150 pcf.* The
modulus of elasticity was 4,000,000 psi with a Poisson ratio of 0.20.
Dynamic structural properties
12. Dynamic analysis requires the same input to describe the structural
properties as does a static analysis. additional requirements are that the
inertia and damping of the structure must be specified.
13. GTSTRUDL will automatically compute member/element inertia contri-
butions by either the lumped or consistent approaches. The member/element
weight densities must be provided via the CONSTANTS command or the MATERIALS
command prior to a dynamic analysis if automatic computation is to take place.
The lumped mass approach is always more computationally efficient and is a
reasonable approximation for most problems.
14. Damping is specified in GTSTRUDL in one of two ways depending on
whether a modal superposition or direct integration transient analysis is to
be performed. In this study, a modal analysis will be performed, thus damping
ratios or percent damping would be specified. A 5 percent damping ratio was
assumed. Had the stiffness and mass matrices beer input via the MATRIX com-
mand, damping would have been specified by proportional damping constants.
15. The 5 percent damping ratio is appropriate for a mass concrete dam
interacting with a competent rock foundation if the calculated stress levels
A table of factors for converting non-SI units of measurement to SI
(metric) units is presented on page 3.
10
- U.4
41 C1
it du W. wt 0* 0 t
IN 3.h vs44 e b't f O 0
'A UN I mo it I. A I I V ft Ij
01 EU 0- IOU m n fw P^ - I
~ ~ ~ mr- IA: U w~ r~fu
: ~~~ ~ h 0*1u (AdP UU -I -V
1 ft m me me c ecc
14 z-L -
i Liy.
. - - .A U.- -. -- * ~ .. m.& m i- . &ft Iw CI im du in 40 - w uA I 2 f
fn Upa -8 *
-Us -r4
-VI m
cm-
at 'a
ImiuiuU IL±&Au 9w vd wd tr - - n - V
0 fu a 0 v .Al 'a1u1 40i 0 a-'
04 1, al -- -u tD LA o o V .
oa oP. Al -m r- wA 'A r, a
- Nm %o 4D__ f- w A rLc A 4D
(w f cra - - - - M- N
01 -p fu-l g fI cb w " mVa
he ;a ; ~ou gu du ru cua-
*Yl-** * q-w d '
ow u
-20
a
ta
a ~ ~ h 0 A a U V a - P
ab a f
Iv o- a- AlA aN 'aj~jrn m uI N ORU V V P
z
z
N P s a- __ v U i % '
IA 'I w w w l r% t
~ V,0
z
mm eol JA c o - fML v aiI.±..i.t !L A
-- 0
W N IV 10 C
9. b, to.11
#d el (9 x9 9 9 9 9 99 £ 9 £ fP.2 : 61 e A okV "cpF .in I" P4 du£9 £
995 f" P. 1 LA L 1 f.6 0 b Ob, 6, WY -Y oN v Vr f
UN 1! to s
13
are within the basic strength capacity of the materials. The damping ratio
would be increased if the stress levels go above the basic strength capacity
of the material.
Dynamic analysis
16. In performing a dynamic analysis, the GTSTRUDL program first com-
putes the mode shapes and frequencies of the structure. These results are
then used to perform a time history analysis or a response spectrum analysis,
depending on the choice of the program user.
17. The response spectrum analysis is perhaps the most common technique
used in many design offices. The computational effort required for a time-
history analysis is often prohibitive, since the response of each selected
time point must be computed and stored in order for the maximum response to be
identified. Due to this often substantial effort, the response spectrum anal-
ysis becomes an attractive alternative technique.
18. Response spectrum analysis is an approximate method of dynamic
analysis. It uses the known response of single degree of freedom systems with
the same natural frequency and percents of critical damping as the modes of
vibration of the structure being analyzed, when it is subjected to the same
transient loading.
19. Once the maximum response of each mode is obtained, the maximum
total response could be obtained by adding the maximum response of each mode
since. In general, however, different modes will attain their maximum values
at different times. Therefore, the superposition of the modal maximums will
be an upper bound on the actual total response and will significantly over-
estimate the response for many cases.
20. GTSTRUDL currently computes response spectra maximum responses by
combining the modal responses by seven different approaches. ETL 1110-2-303
(Department of the Army 1985) recommends the use of the Complete Quadratic
Combination (CQC) method (Der Kiureghian 1980). The CQC method degenerates to
the better known Square-Root-of-the-Sum-of-Squares (SRSS) method for simple,
2-D systems in which the frequencies are well separated. Combining modal max-
ima by the SRSS method can dramatically overestimate or significantly under-
estimate the dynamic response, especially for 3-D structures.
Design earthquake
21. Professor H. B. Seed's design earthquake response spectrum (Fig-
ure 5) (Seed, Ugus, and Lysmer 1974), scaled to 0.25 g peak ground acceleration
14
0.75
5% DAMPING
RESPONSE SPECTRUMSCALED TO 0.25gGROUND ACCELERATION
ROCK SITE
0.50
0 25
0 0.5 1.0 1 5
PERIOD, SEC
Figure 5. Response spectrum (after Seed, Ugus,
and Lysmer 1974)
with 5 percent damping, was used to represent the earthquake for this example.
22. In an actual analysis, the analyst would have to do a geological
and seismological investigation of the dam site. The objective of the inves-
tigation would be to establish the controlling earthquake, and the correspond-
ing ground motions to be used in the study.
Hydrodynamic effects
23. It has long been understood that the inertial resistance of the
water in a reservoir has an important influence on the earthquake response of
concrete dams. This study considered a reservoir, or hydrodynamic head of
170 ft on the upstream face of the dam. Except in the case of the EADHI
(Chopra and Chakrabarti 1974) and EAGD-84 (Fenves and Chopra 1984) codes,
finite element models usually use the concept of an "added mass" or "virtual
mass" of water moving with the dam to represent the hydrodynamic interaction
effect. There are several methods of approximating this effect. The one cho-
sen for this study is an extension of the Westergaard method that was
15
originally developed for gravity dams (Department of the Army 1958,
Westergaard 1933). Westergaard reasoned that the added mass would produce the
same net effect on lateral loads as the parabolic hydrodynamic pressure dis-
tribution. The effect is approximated by determining and attaching added
masses to the fa-e of the dam. This increased mass results in increased iner-
tial resistance to the motion of the structure when an earthquake is applied
and is intended to simulate the actual resistance to the motion of the struc-
ture caused bv the water mass. Calculation of the added mass mav be found in
kppendix A.
24. Although research has shown the Westergaard added mass formulation
provides a convenient, simple means for representing reservoir interaction in
the dynamic analysis of gravity dams, there are limitations that should be
noted. The underlying assumptions of Westergaard's work are that the dam is
rigid and the water is incompressible. Chopra (1970) has shown that flexibil-
ity of the dam and compressibility of water are very important considerationsin the dynazic response, hence the development of the FAPI nc EAGD-84 codes
(Chopra 1970). The main drawback to these programs is that the only dynamic
input applicable is an acceleration time-history record, lie has therefore
developed a simplified response spectrum analysis procedure (Chopra 1978) for
use in the analysis of nonoverflow concrete gravity dams and uncontrolled (un-
gated) spillway monoliths which can be modeled for 2-D analysis. The princi-
ples involved are those basic to structural analysis by response spectrum
methods. The method represents the hydrodynamic interaction effects by an
added mass of water which moves with the dam. However, unlike Westergaard's
added mass, this mass is dependent "on the frequency and shape of the funda-
mental mode of vibration of the dam, and the effects of interaction between
the flexible dam and water, considering itp compressibility, on the fundamen-
tal frequency of the dam."
Tnput
25. To illustrate the use of GTSTRUDL to solve a problem such as this
simplified structure, the input file used for the Mesh 2 model may be found in
Appendix B.
16
Discussion of Results
Mode shapes and frequencies
26. The center-line deflection of the first four mode shapes of the
rectangular beam are presented in Figure 6. The actual deflected shapes are
shown in Figures 7 to 10, respectively. GTSTRUDL results for the frequencies
were compared to Timoshenko beam theory and elementary beam theory results
(Table 2). The equations used to generate the results for the closed-form
solutions are presented in Appendix C. Timoshenko beam theory included the
effects of transverse shear and rotary inertia which is significant due to the
relatively short and deep characteristics of the structure.
LEGEND
1ST MODE
A 2ND MODE3RD MODE (AXIALI
E 4TH MODE
0J
Figure 6. Center-line deflection of mode shapesfcr simplified structure
17
~g."1'*
2
b.* .
1-4
(.J- - - -
5-4
* *2
EI~- - - - - - - a)
k
- - - -- - - -
0
- - - - - - - - a)
U,
- - a a)0E
U,1-4
r~.a)'-4I
II..
18
Q)
-~ C)
01 E
0
0
tl I ci
C)
U1
oz.-V.o
U -r'.
U0
,. .-
r~20
• ' ' l l q P '
a rz
S.-..gS$.4
-'33
ar.4
- - 21
Table 2
Comparison of Frequencies for Simplified Structure
Frequency, cpsGTSTRUDL (Mesh 2) Ircludes Added Mass
Fixed Elementary Timoshenko (fixed base) GTSTRITDLtMode Base Pin and Roller* Beam** Beam** Mesh I Mesh 2 Mesh 3
1 3.72 3.63 4.65 4.11 3.34 3.34 3.34
2 15.04 13.52 29.15 16.90 12.90 13.04 13.09
3tt 15.37 15.00 .... 15.03 15.04 15.04
4 32.67 28.19 81.59 35.90 25.82 26.11 26.25
* Although a fixed base should be used in an analysis, a pin and roller base
was also run for comparison.
** Appendix C for these results.
t Frequencies of dam including effects of stored water. Other frequencies
do not include any stored water effects.
tt This is predominately an axial mode, Figure 6.
27. The second part of lable 2 indicates the deviation in frequencies
due to the various model refinements.
Comparison of models
for the dynamic analysis
28. A comparison of the deflection results along the height of the
rectangular beam is presented in Table 3. The difference in the maximum dis-
placement is approximately 3 percent which indicates a reasonable agreement
between the three models.
Table 3
Comparison of Transverse Deflections Along the Simplified Structure
Distance from Deflection, in.Base, ft Mesh I Mesh 2 Mesh 3
18.5 0.021 0.022 0.022
37.0 0.064 0.066 0.066
55.5 0.122 0.[?s 0.126
74.0 0.193 0.196 0.197
92.5 0.273 0.276 0.277
111.0 0.358 0.362 0.363
129.5 0.446 0.451 0.452
148.0 0.536 0.541 0.543
166.5 0.625 0.630 0.632
1851) 0.71L 0.716 0.718
22
29. A comparison of the SXY (shear) and SYY (normal) stresses was made
at two locations along the height of the simplified structure. The results
for these stresses for the various meshes at a height of 37.0 ft above the
base are presented in Table 4, while those at a height of 111.0 ft are in
Table 5.
30. In theory, the finer the mesh the more correct the solution.
Therefore, Mesh 3 should be the most correct solution. The results indicate
that Meshes I and 2 are converging to this solution. For Mesh 1, tf,- differ-
ence from Mesh 3 in the maximum stresses is 16 percent for shear and 3 percent
for the normal stress. Mesh 2 is within I percent in both cases of the re-
sults of Mesh 3. Contour plots were obtained for the SXY an SYY stress compo-
nents for Mesh 2 and are shown in Figures 11 and 12.
31. The analyst, in addition to deciding on the degree of mesh refine-
ment so as to achieve sufficient accuracy, may need to consider cost. The
relative costs for the computer runs for Meshes 1, 2, and 3 was $0.81, $3.31,
and $23.25, respectively. (These costs are from runs of GTSTRUDL on the Con-
trol Data Cooperation Cybernet Computer Service and should be used only as a
relative measure.) While Mesh 3 should produce a more accurate solution, the
additional cost does not appear to be justified. Mesh 2 provides an accept-
able solution, balancing both cost and accuracy.
Model truncation effects
32. An investigation was made to deturmine the effects of using various
numbers of structural vibration modes. Analyses were made using the fundamen-
tal mode and subsequently increasing the number of modes until the final solu-
tion showed convergence. Results in Table 6 show no difference between modes
three and four, thus indicating that only the first three modes are necessary
to establish complete convergence. Results also show that there is only a
minimal difference in the response using one mode and the computed combined
response Laximum as represented by the four-mode analysis. This indicates
that the first, or fundamental, mode of vibrations participation is predomi-
nate in obtaining the response-spectra maximum in this particular dynamic
analysis.
23
Table 4
Comparison of Dynamic Stresses along the
Simplified Structure at Height of 37 Ft
SYY, psi SXY, psix, ft Mesh I Mesh 2 Mesh 3 Mesh I Mesh 2 Mesh 3
-40.0 639 612 618 20 7 5
-30.0 470 468 60 62
-20.0 304 313 313 97 111 107
-10.0 157 156 127 129
0.0 8 4 4 178 141 137
10.0 157 157 127 129
20.0 305 314 314 96 111 105
30.0 470 468 60 62
40.0 638 611 618 20 5 1
Table 5
Comparison of Dynamic Stresses along the
Simplified Structure at Height of 111 Ft
SYY, psi SXY, psi
x, ft Mesh I Mesh 2 Mesh 3 Mesh 1 Mesh 2 Mesh 3
-40.0 185 184 184 19 8 7
-30.0 145 145 39 41
-20.0 102 101 101 65 72 69
-10.0 52 52 83 85
0.0 3 3 3 108 93 90
10.0 52 52 82 83
20.0 102 101 101 64 72 70
30.0 147 147 37 39
40.0 188 187 187 18 4 1
24
.- 4-4
-r-
z
4; x
- 25
.1 14
*4 4
*00
0J 0>4
26r
Table 6
Effect of Number of Modes Used in Analysis on
Dynamic Stresses (Mesh 2) at Height of 37 Ft
SYY, psi SXY, psi
x, ft 1 Mode 2 Modes 3 Modes 4 Modes I Mode 2 Modes 3 Modes 4 Modes
-40.0 611 612 612 612 5 6 7 7
-30.0 470 470 470 470 58 60 60 60
-20.0 313 313 313 313 107 ill 111 il
-10.0 157 157 157 157 123 127 127 127
0.0 0 4 4 4 136 141 141 141
10.0 157 157 157 157 122 127 127 127
20.0 313 314 314 314 107 ill il ill
30.0 470 470 470 470 58 60 60 60
40.0 611 611 611 611 5 5 5 5
27
PART III: GRAVITY DAM EXAMPLE PROBLEM
Description of the Problem
33. An actual earthquake analysis using both the dynamic finite element
response spectrum method as outlined in Part II and Chopra's simplified re-
sponse spectrum method is presented to demonstrate the procedures and illus-
trate the different results produced by each method.
34. The structure to be analyzed is a nonoverflow monolith similar to
those of the Richard B. Russell Dam. The dam is 185 ft high with a reservoir
depth of 170 ft. Seed's design response spectrum, scaled to a peak horizontal
ground acceleration of 0.25 g with 5 percent damping (Figure 5) (Seed, Ugus,
and Lysmer 1974), will be used in both analysis.
35. The geometry of the nonoverflow section is defined in Figure 13.
17'EL 495'
NORMALPOOL
EL 480'EL 470'
EL 453'
12L8
EL 356'
12
1 10EL 310'
143.25'
Figure 13. Geometry of dam monolith
28
Finite Element Method Analysis
36. A listing of the input for the GTSTRUDL finite element analysis of
the gravity dam may be found in Appendix D, pages D-2, 3, and 4.
37. Previous results indicated that a mesh with four elements across
the base, Figure 14, was a reasonable compromise between accuracy and cost.
The mesh contained 36 elements and 135 nodes. The monolith was assumed to be
completely restrained along the base.
38. The strurture was loaded by hydrostatic and hydrodynamic loadings
starting at 170 ft above the base and a self-weight of the concrete of
150 pcf. The hydrostatic pressures were input as uniform edge loads on the
upstream elements. The hydrodynamic effect was approximated by attaching
Westergaard's (1933) "added masses" to the upstream face nodes, Table 7.
Analysis
39. The analysis is performed in two parts. The static (stiffness)
analysis and dynamic analysis are performed separately. These results are
then combined to give the final results. The static analysis consisted of two
load cases: (1) hydrostatic pressure on the upstream face of the dam, and
(2) self-weight (dead load) of concrete.
Results of analysis
40. Results of the independent load cases were obtained. It should be
noted that elements incident on a common node will have different stresses at
the same node. This is due to the fact that continuity of stresses is not en-
forced or required for the finite elements in GTSTRUDL, as is true in all
other major finite element programs. To obtain a more useful representation
of the stresses, one can use the CALCULATE AVERAGE command. To compute the
weighted average, GTSTRUDL sums the stresses for all elements incident on a
given node, and then divides the sum by the number of elements which are inci-
dent on the node.
41. Using the COMBINE command, it is then possible to combine the in-
dependent loading conditions to obtain a final result. In this example, one
would add the two static loading cases (loads I and 2) to obtain a total
static loading response (loading combination 5). The static loading condition
than can be combined with the dynamic loading.
42. It is first necessary to operate on the dynamic loading (load 3) to
transform the results into the form of a static loading condition. The result
29
'j)
-LJw
-Ii
0 cc
U-) C., 0
0 0
CI co c
0l
0f 0)
0 I000
NOI-VA bo
30) -r
Table 7
Structure Loading for Gravity Dam Example
Hydrostatic AddedY Y y Pressure Mass
Node Elevation ft ft ft psf slugs/ft
127 495.00 185.00
122
113 480.00 170.00 0.00 0 552.50
108 5.00 312 231
7.50Q9 470.00 160.00 10.00 624 463
14.25
94 18.50 1,154 76122.75
85 453.00 143.00 27.00 1,685 1,136
33.0780 39.13 2,442 1,580
45.1971 428.75 118.75 51.25 3,198 1,811
57.3266 63.38 3,955 2,012
69.4457 404.50 94.50 75.50 4,711 2,199
81.5752 87.63 5,468 2,367
93.6943 380.25 70.25 99.75 6,224 2,527
105.8238 111.88 6,981 2,675
117.9429 356.00 46.00 124.00 7,738 2,742
129.7524 135.50 8,455 2,793
141 .2515 333.00 23.00 147.00 9,173 2,909
152.7510 158.50 9,890 3,021
164.251 310.00 0.00 170.00 10,608 1,551
Notes: Y is measured from base of dam (el 310).*Y is depth below water surface (el 480).*
y is depth below water surface to midpoint between nodes.Ce and Mi are defined in Appendix A.
Ce = 5 51.54
0.72 ( 170 2
1,000 - 1.5o
= 51.54 x 1700.5 - Y = 13.91 ( - Y1. = added mass
All elevations (el) cited herein are in feet referred to the National Geodetic
Vertical Datum (NGVD) of 1929.
31
is called a pseudostatic loading. The command performs this function by
copying the results of the specified modal combination (CQC) of the response
spectrum loading condition into a static loading condition (loading 4). This
can then be added or subtracted with the static loading condition to obtain
the maximum or minimum stresses in the dam.
43. A listing of the output can be found in Appendix D. Page D-5 lists
the dams first four frequencies. Pages D-6 through D-21 give the stresses for
the loading conditions; load 4 - pseudostatic loading, load 5 - static load-
ing, and load 6 - static plus dynamic loading. Extracted from the output and
presented in Table 8 are the vertical (SYY) stresses for the monoliths
Table 8
Stresses in Dam
SYY, psiStatic plus
Node Static Dynamic Dynamic
1 53 313 366
9 -41 44 3
15 -23 275 253
23 -60 88 29
29 -59 260 202
37 -89 167 79
43 -56 261 205
51 -78 192 114
57 -51 260 208
65 -56 197 141
71 -49 257 208
79 -33 199 166
85 -49 273 225
93 -13 194 181
99 -28 188 160
107 -29 285 256
113 -15 79 64
121 -14 54 40
Note: Positive = TensionNegative = Compression
32
upstream and downstream face nodes. Column 2 presents the weighted average
stress at -f- rarfou- -1- due to the static loeding. In Column 3, the
dynamic stresses are given. The final column presents the combined stresses,
the addition of the static and dynamic stresses. This would represent the
maximum tension in the monolith due to the prescribed earthquake with an up-
stream pool at el 480 ft.
44. The upstream and downstream face node SYY stress component along
with the contour plots for the various loading conditions are shown in Fig-
ures 15, 16, and 17.
45. As a sidelight to the above example problem, one additional analy-
sis was made. As previously mentioned, the inertial resistance of the water
in a reservoir has an influence on the earthquake response of a dam. But, how
great is the influence? To answer this, an additional computer run was made
without the hydrodynamic effects (attached "added masses"). Figure 18 shows
the resulting face node SYY stress components. Comparing this with Figure 16
one can see the higher stresses in the dam due to dam/reservoir interaction.
Chopra Simplified Analysis
46. The more rigorous dynamic analysis of gravity dams is obtained by
using finite element computer programs. Due to the capability of these
programs to model the horizontal and vertical structural deformations of the
dam, to model the exterior and interior concrete, and to include the response
of the higher modes of vibration, the interaction effect of the foundation and
any surrounding soil, and the horizontal and vertical components of the ground
motion, some amount of specialized training is required to use them effec-
tively. Chopra's simplified analysis procedure is a compromise on that
complexity.
47. Using a set of standard curves for the fundamental mode shape, the
ratio of fundamental period of the dam with and without stored water, and the
variation of hydrodynamic pressure over the depth of the water, one can calcu-
late a set of equivalent static lateral loads. These forces are considered to
act separately in the upstream and downstream directions, and their effects
added to the effects of all other design loads.
33
0
LO
In
(N
z 04
z Li)0 cc
0, -0 i
niLw o
Inn
LPn
ci)
"-4
I
00
8)l
34
0
C0
z (n0 IC)
It). C.
Z~Z W uiH
wlol00,
Z w0 I-
Ul 0
00(3))
00 LO LO 0) -n
C14 -- (N( (N (N N C14 m
H.Ld3G
U')-
35
CW
-4
co
LO C)
LOC
00
0 un
zwN
(n~~I ODLL
z n n 5;C)T n 4
0 0(0
+ CL
a Ia
C1-4
-v-4
m 0 0Y) C-' '4C4a 0 0) (T
C4 0" 0C--
In Inca
0 0
In
IoC)
NOI.LVA3
36
z
00rA
Ul ui
Z 4U cz) LW
Lu L0
z0
C)CLn)
(N 0-I
U)
m) U) 0
Cl)
M C
Cl) OD -
0) LP
IT 1U
N0liA3(N
(N Cl ~ 37
48. The computation of the earthquake forces is carried out as follows:
a. Compute T , the fundamental natural period of vibration of-- S
the dam, in seconds, without the influence of stored water,from
1.4 x HTS =s (1)
where
H = height of dam, feets
E = modulus of elasticity of concrete, psi
b. Compute T , the fundamental natural period of vibration of-- S
the dam, in seconds, with stored water
T = RIT (2)s I s
where R1 = period ratio determined from Figure 19.
c. Compute R2 , the ratio of the fundamental resonant period for
the impulsive hydrodynamic pressure to T froms
R 4.0 x H/C(3
2 Ts
where
C = velocity of sound in water = 4,720 ft/sec
H = depth of water (feet) in Figure 19
d. Compute f (y) , the lateral earthquake forces over the height
of the dam, including hydrodynamic effects, from
A I x S a(T s )f5 (y) = a [Ws(Y)(y) + gp1 (y)] (4)
where
A, = scaling constant, with assumed value of 4
S a(T s) = a spectral acceleration at period of vibration
Ts, from the design response spectrums
g = acceleration due to gravity
38
w (y) = weight per unit height of the dams
I(y) = fundamental mode shape factor from Figure 20
gp1 (y) = pressure distribution factor from Figure 21 cor-responding to R2 and multiplied by the quantity
(H/H ) 2s
e. Compute f(y) , the lateral earthquake forces without hydro-dynamic effects from:
Sa (Ts)
f(y) = A I 1 Ws(Y )y (5)
where
A =3
S (T) = spectral acceleration at period of vibration T sa s s
1.7
1.6 E 7 x 106 PSI
1.6
I-
t a 1.4 E Sx 106
. E=4x 1060 0 I
0
2 2 1.3
W W
Z 14 E=2x 106
00 /< 1.2 E Ix 106
11.0
0. 1.2. 07 08 . .
TOTAL DEPTH OF WATER, H IH, - 0.92HEIGHT OF DAM, HS
Figure 19. Illustration of how to determine standardvalues for Rl , the ratio of fundamental vibration
periods of the dam with and without water, for agiven H/H ratio and modulus of elasticity (after
Chopra, 1978)
39
1 0
09
0.8 -
0.7
0.6
Figure 20. Illustration of how todetermine the standard fundamental
o= 0., period and mode shape for a given< ~HS HEIGHTOF DAMN FT
y/H ratio (after Chopra 1978)S
01 0.32L' I I/, , 10 0.2 04 0.6 0.8 1.0
MODE SHAPE.0
1'0
4.Ox HIC
0.9 R2 TS
= 0.90
x 0.8
Figure 21. Illustration of 0.7
how to determine the stan- 0
dard variation of gpl over o
the depth of water for H/H 0= 1 and various values of s =0o.9
R where W is the unit 0 0.40.9z _0
weight of water (after \Chopra 1978) .39 0.97
0 01 0.3 04 05 0.6 07 08 09
40
49. The equations for distributed lateral force (Equations 4 and 5),
can be integrated between appropriate limits to yield concentrated forces
which are then applied statically to the dam. Stress computations are then
carried out as with other design loads.
Computation of earthquake forces
50. For this analysis, the nonoverflow monolith of the gravity dam is
divided into nine sections (or horizontal slices). The elevation of each sec-
tion is equal to the corner node elevations of the elements used in the pre-
vious FEM analysis. Thus, section I is at an elevation equal to that for
nodes 113 through 121 (see Figure 14), section 2 is at nodes 99 through 107,
section 3 at nodes 85 through 93, etc.
51. Steps in the computation of the earthquake forces are as follows:
a. For E = 4.0 x 106 psi and dam height H = 185 ft, fromEquation 1,
1.4 x 185T = 1.4 x 18 = 0.13 sec
4.0 x 16
b. From Figure 18,
R 1 = 1.23 for E =4.0 x 106 psi
and
H 170. . . . 0.92H 185s
From Equation 2,
T= 1.23(0.13) = 0.16 secs
c. From Equation 3,
4.0 x (170/4,720)R2 = 0.16 = 0.90
d. Equation 4 was evaluated along each level throughout the height
of the dam, by substituting = 4 , and [Sa(TIs) / = 0.63
(from Figure 5) for T = 0.16 sec, by computing the weight ofs
41
the dam per unit height, w (y) , from the section dimensions' S
(Figure 13) and the unit weight of concrete, and by substitut-ing for (y) from Figure 19 and gp1 (y) , for Figure 21.
The distributed lateral (earthquake) forces f (y) and equiv-
alent static load are listed in Table 9 and pictorially shownin Figures 22 and 23.
Computation of stresses
52. At this point, simple beam theory
Mc PI - A
could be used to calculate the stresses at each horizontal section. As was
done with the earthquake forces, the distributed gravity and hydrostatic
forces are replaced by concentrated loads. The direct and bending stresses
are then computed based on section properties (see Table 10) at each horizon-
tal slice.
53. Tables 11 and 12 show the results of the stress calculations. The
earthquake loads have been applied both upstream and downstream and the re-
sults combined with the static loads. Figure 24 shows the static stresses
(SYY) while Figure 25 shows the maximum tensile stresses (SYY), a result of
combining the static and dynamic stresses.
54. An alternative to using simple beam theory to calculate stresses
would be to perform a static finite element analysis. This approach has been
implemented by Cole and Cheek (Technical Report SL-86-44, Department of the
Army) in a new user-friendly computer program. The program was developed
using the finite element methods of analysis to determine the dam's inertial
response along with Chopra's simplified procedure for estimating the hydro-
dynamic loading. The program is menu driven, allowing for ease of use by the
novice. The only input required is the dam's geometry and appropriate re-
sponse spectrum. Output consists of principal surface stress values at
selected elevations for both the upstream and downstream faces of the dam. A
run using this program is included in Appendix E. The results are tabulated
in Appendix E, page E4 and shown in Figure 26.
Comparison of Procedures
55. Corresponding calculated stresses are plotted for each method of
42
r-- '0 r D 0 0 LC) '0 0 M4)-~- oz r- r- 00 C a, U-. r- %C
-- .0 N. '0 ' LI1 r- oc -: IT -c U u - - -
> LIH -LI) -4 (7 m 00 -040&j- 00 () (N MN 'T nr- - U
c c C/ C4 I
'D c', - V 0 c4 - 0- zr->1 cl 00 - all (4 00 00 rN-
's U-7 z. ~ C 0 a, 00 'D l '4
0
'- C. 0'! C C' c! '0) 0& r~ ~ U-~ .4 C 4 0 f
to2 4.4x Dt
. . 0 0 0 00 Lt) - - -un U'N ul( U) r, C0 c') U) 1.0 oo 4-
3 c-4
wE
U 0 L) 0 N 0 ' - c) 04C ) - C
0; c ; 1 ; ; C ;G
u0
- 0 01 00 r- C. ) m0 '0 - C(Nc
:I4 low- -. Ia a
(42I -c -- 00 (NN rN (N (
E 4 -4
U2 -47 0 4I c,7 4 '~
(a I v) W a a 0a
Co r- L N IT Le) N-0
04 0' '0) - 4 Zr 00 00~ -i -..2-4-C) 0 0 0 c' co( N ( C) (
IV IlI 0 0 C0 C 0 C. 0 Q
u ~ - 0 - N -4 cc A- -0'-4 - * *r4- V!CC!I
4.) 4 0 0 0 0 0 0 0 0 II 0
'4 0ON- (N4 4-4W 0D 0 0 U- 0 LPn 0 0 0) -~ C4
-'. 0 ' 0 0T N- %Di m 0C. w 4j 42(V U-I 0 0 0- ('1 - -a, 0 ' 0 m2144 -
-1 - - - - - (44 2
000>0 (C C 0 0 4-4 Q.4-- 40 CD 0 0 N-) (C (N 0 0 0 442W4.)
to2 (C 0 0 00 IT 0 '0 m~ 0 02 to 0w> all 00 N- (C' (N 0 00 (C' m~ -440 H W
14 u) P, c44 -
434
500
G0 --
450
50
200
10 40
350
150
SCALE
2 1 0 . 10~ LB/FT
Figure 22. Dynamic force (inertia + water) on gravity dam
Soo
G0 -
450
50
z0
~400
100
350
150
310 1701
SCALE
3 2 1 0 x 105 LBS
Figure 23. Dynamic load (inertia + water) on gravity dam
44
Table 10
Section Properties of Gravity Dam
Base
Base Width W A iSSection Elevation ft lb in. in.
1 480.00 17.00 38,250 2,448 83,232
2 470.00 17.00 63,750 2,448 83,232
3 453.00 28.33 121,546 4,080 231,254
4 428.75 46.52 257,679 6,699 623,264
5 404.50 64.71 465,435 9,318 1,205,967
6 380.25 82.89 733,883 11,936 1.978,777
7 356.00 101.08 1,068,478 14,556 2,942,544
8 333.00 122.52 1,454,188 17,643 4,323,211
9 310.00 143.25 1,912,641 20,628 5,909,922
analysis; finite element analysis using Westergaard's "added mass" (FEM),
Chopra's simplified method using simple beam theory (CSM/SBT), and Chopra's
simplified method using finite element analysis (CSM/FEM). Comparisons are
presented for both the upstream (Figure 27) and the downstream (Figure 28)
faces.
45
Table 11
Vertical Stresses in Gravity Dam with Earthquake in
Downstream Direction Using Simplified Method
Static Stresses+
Static Stresses Dynamic Stresses Dynamic Stressespsi psi psi
Sertion Flevation Uostream Downstream Ustrear Downstream Upstream Downstream
1 480.00 -16 -16 93 -93 77 -1092 470.00 -24 -28 257 -257 233 -285
3 453.00 -6 -54 275 -275 269 -3294 428.75 -30 -46 283 -283 253 -3295 404.50 -44 -56 299 -299 255 -3556 380.25 -50 -72 311 -311 261 -3837 356.00 -52 -94 318 -318 266 -4128 333.00 -62 -102 299 -299 237 -4019 310.00 -70 -116 287 -2R7 217 -403
Note: Positive = Tension.Negative = Compression.
Table 12
Vertical Stresses in Gravity Dam with Earthquakes in
Upstream Direction Using Simplified Method
Static Stresses+
Static Stresses Dynamic Stresses Dynamic Stressespsi psi psi
Section Elevation Upstream Downstream Upstream Downstream Upstream Downstream
1 480.00 -16 -16 - 93 93 -109 772 470.00 -24 -28 -257 257 -281 2293 453.00 -6 -54 -275 275 -281 2214 428.75 -30 -46 -283 283 -313 2375 404.50 -44 -56 -299 299 -343 2436 380.25 -50 -72 -311 311 -261 2397 356.00 -52 -94 -318 318 -370 2248 333.00 -62 -102 -299 299 -361 1979 310.00 -70 -116 -287 287 -357 171
Note: Positive - Tension.Negative - Compression.
46
500
G0 __-16 -16 psi
-24 -- -28
450 -6 -54-
50 -30 -- - -46
z LEGENDo + TENSION
< 400 -COMPRESSION ~ - - - -5
100 -50 -- - - - - -72
-52--------------------94
150 -62-------------------- -102
310 170 .- 70 -1
Figure 24. Static stresses (SYY) in gravity dam from simplified method
500
0 77 77 PSI
231 229
450 269 -- 221
LEGEND50 + TENSION 28323
-COMPRESSION
0
255--------------243> 400
100 261 239
350 266--------------------224
150 237-------------------- 197
310 170 217 171
Figure 25. Maximum tensile stresses (SYY) in gravity dam fromsimplified method
47
49536 19 PSI
0 - 122 165
2b8 295
450328 270
50
329 2550
~400LU a. 333 237
Uj1 0344 208
350I 379 163
150
Fig:L L6 Maxim principal surface stresses 14
48
SINGWIESTEGAARDOS
1A/SETS~ CHOP.A SIMPCYIED ANALYSIS
:sFM SOPA SIMPLIFIED ANALYSIS
__,SING FIIT ELEMENT
CSS OSBT
0 '00 .00 10040 0S-RESSES -PSI
Figure 27. Upstream stresses
0 I
03
02
0, CSMI E-1
STRESSES PSI
Figure 28. Downstream stresses
49
PART IV: FOUNDATION EFFECTS
Finite Element Model
56. The purpose of this portion of the qtAdy in to determine the neces-
sity of including a foundation in the finite element model used for the dy-
namic analvsis of a gravity dar. The same gravity dam geometry and mesh size
as described in the Part III FEM analysis were used for this study with the
A dition of a foundations block (Figure 29). Limits on the width and deT:h of
a foundation block were evaluated in the Phase lb report (in preparation).
The model used in this study included foundation material in both the
upstream and downstream directions equal to the base width of the dam
and a depth equal to 1.5 times the base width. The resulting mesh contains
68 elements and 247 nodes.
Y
239 24765 68
61 225 233 x211 219
57197 205
53
183 191
49169 177
45155 163
41141 149
37127 135
122 33123 124 125 26105, 109 Il 1" 113; ,1s 1 7 12125 110 112 114 116
79 - 95
17
53 -69
9
28 2927 -43
18k9203 21 22 5 23 i24 25
18 1 19 2 3 4 5 6 7 8 26
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Figure 29. Mesh for dam monolith
50
57. The foundation boundary was assumed fixed at the corners (nodes 1
and 17) and free to slide along its base and sides. Negligible mass was given
to the foundation to avoid wave propagation effects at this level.
Stress Comparison
58. In order to assess the effect of the foundation, the dam was ana-
lyzed five times varying only the modulus of elasticity of the foundation
material. The various Er/Ec ratios (the ratio of the modules of elasticity of
the foundation (Er) to the dam (Ec)) were: 0.05, 0.25, 1.00, 1.75, and 3.00.
An analysis assuming an Er/Ec ratio of infinite (-) is equivalent to the
analysis, as performed in Part III, assuming the dam is completely restrained
along its base.
59. Comparison of the first mode frequency and period for the various
Er/Ec ratios is presented below:
Frequency PeriodEr/Ec (cyc/sec) (sec/cyc)
Infinite 6.042 0.165
3.00 5.543 0.180
1.75 5.256 0.190
1.00 4.836 0.207
0.25 3.353 0.298
0.05 1.731 0.578
The Pddition of a foundation to a model, regardless of its size, will decrease
its frequency and increase its period. This substantiates that in addition,
as the foundation's modules of elasticity decrease, it will further reduce the
frequency and increase the period.
60. The upstream and downstream dam face node SYY stress components
along with the contour plots for the static, dynamic, and combined static and
dynamic analysis are obtained for the various Er/Ec ratios and ioT-.m in Fig-
ures 30 through 44. Previously presented in Figures 15, 16, and 17 were the
stress components and -ontour plots for the analysis assuming a completely
restrained base (Er/Ec = infinite). Combined results for the SYY stress com-
ponents are presented in Figure 45 through 50.
61. Figures 45 through 46 show that as the foundation's modulus of
51
495LEGEND
0 -~ - 15 -1 TENSION
-28 -29 -=COMPRESSION
-9-13 -25450
50 -48 -31
z
4 T -48 -55> 400 i
-50100 -61 -101 -75
-109 -176 -75-0
35 100 150 _200
150 -232 -253 2- ~ ~ )\250
-289 -25d~)2
310 170 - 144 -162 __
-55 -71 -83
Figure 30. Static stresses (SYY) for Er/Ec ratio of 0.05
495
0 - 31 19
77 117 LEGEND
+=TENSION
450 12 90- COMPRESSION
50128 10150
Cr4137 108
> 400
00 139 12210
131 153 100 50350
150 178 \11194 150
236 200o 203L 10151 210
24 12 36
Figure 31. Dynamic stresses (SYY) for Er/Ec ratio of 0.05
52
495
0 SZ 16 5 LEGEND
89 88TENSION- COMPF1:SSION
73 77450
50 79 70 50
z0
89 53< 400
100 79 21 50
22 -23
350
10 5-55 50
- 5~0, -50\-53 -52-5
310 - 170 L75 5-31 -60 -47
Figure 32. Maximum tensile (static plus dynamic) stresses (SYY)
for Er/Ec ratio of 0.05
LEGEND
0;7 -15 -14 + TENSION
-28 -29 -25 - =COMPRESSION
-49 -13 -25450
50 -48 -32
0H-49 -55
400 _
IU)-50-
S 0 -51 -90
- 100-85 -135 -75
147 -15 -150
1 5 0 7 9 2 -1 5 0 - 0-171 -92 -75
310 170 -87 .---- . ... - - - - -115 -- --
-71 -84 -94
Figure 33. Static stresses (SYY) for Er/Ec ratio of 0.25
53
0 61 37
147 222 LEGEND
TENSION228 166
450 - COMPRESSION
50 231 182
C245 192
> 400
350 50100 0150
00 252 209 750200
100
245 227
35 - 4\ 203
0 250 ' x
V8 300 OMRS3O
15 339 224 3500
1550
49 26 83 140
Figure 34. Dynamic stresses (SYY) for Er/Ec ratio of 0.25
495
0 46 23119 193 LEGEND
150 TENSION
189 153 -1COMPRESIN450
50 16F3 150
z0
4 I196 138
100 195 120 1002\r /E rao 050
160 92 1 W6053501
150
10192 59 200
276 250 '
310 170 200-0
-22 -27 -11 25
Figure 35. Maximum tensile (static plus dynamic) stresses (SYY)for Er/Ec ratio 0.25
54
495
0 7 -15 -14-- 25-28 -29 LEGEND-2
TENSION
-49 -13 25COMPRESSION450
50 -48 -33
0
400< -50 -554 -0-50
100 -85 -3
-70 -110 -75350 -75
.'9 -109150
1 -50 -1000 1310 170 -31 • -__ __
-89 -99 -104 -75
Figure 36. Static stresses (SYY) for Er/Ec ratio of 1.00
495
Z781 5208 5 LEGEND
192 2879 2+ - TENSION
286 203 - COMPRESSION450
50 274 211
0
, 280 213> 400 \
100 281 218
150
277 210 25 00 50 0 0350 \200 1 50 0
300
150 348 , 159 350
310 170 275 15 0 _.. .
67 43 107 95
Figure 37. Dynamic stresses (SYY) for Er/Ec ratio of 1.00
55
495
0S- 66 38
165 299LEGEND
200 TENSION
23R 190 - COMPRESSION450
50 226 176
z
230 158 150>400
100 226 135 100
100
207 100 150 50 0 05
150 209 50 250
361 350
30L 1023-22 -56 3 20
Figure 38. Maximum tensile (static plus dynamic) stresses (SYY)for Er/Ec ratio of 1.00
495
0 -7 -15 -14
-6 -29 -25 LEGEND-2 COME SIN
-48 -13 -25COPES N
450
50 -48 -33
z
05
1 00 -55 -81
-67 -104350 -75
10 -- 61 -93
1500
-95 -105 -108
Figure 39. Static stresses (SYY) for Er/Ec ratio of 1.75
56
K7 83 55
197 295 LEGEND* TENSION
290 205 - = COMPRESSION450
50 274 211
z0
277 211
100
100 278 212 150 2
20015
100274 198
350 50
331 136
150 350
395
310 170 252 --- 76
74 49 112
Figura 40. Dynamic stresses (SYY) for Er/Ec ratio of 1.75
495
0 %7 68 40
169 267 LEGEND+ TENSION
241 193 - COMPRESSION450
50 226 178
150
- 227 156400
50 100
100 222 131 100
150 50
350 208 94 0 0
270 43 250 0 )
366 \ 350
,10 L 170 239 2156 415--21 -56 4
Figure 41. Maximum tensile (static plus dynamic) stresses (SYY)for Er/Fc ratio of 1.75
57
495
0 - -15 -14-LEGEND-28 -29 -25 + TENSION
-49 -13 - - COMPRESSION450
50 -49 -33
S4 '-51 -55400"a 0. -50
100 -55 -80 -75
X_ -75
-64 -99
-48IS-1 -50 -0
31.1L -25 1075-100 -111 -111 -51
Figure 42. Static stresses (SYY) for Er/Ec ratio of 3.00
495
0 \7 84 56
198 297 LEGEND
+ TENSION450289 209 = COMPRESSION
50 271 208
2
273 207 5040
100 273 206 200
200150
271 188350
250 100
150 314 119 300
364 35
310 170 232 --- - - *
81 54 115 62
Figure 43. Dynamic stresses (SYY) for Er/Ec ratio of 3.00
58
495
69 42 LEGEND
170 268 + z TENSION
- COMPRESSION240 192
450
50 223 176 150
222 152> 400
U 100U 0
0 03550
207 .,9100350 150
266 37 20
346 -0
310 1- 170 233-19 -58 4 11
Figure 44. Maximum tensile (static plus dynamic) stresses (SYY)for Er/Ec ratio of 3.00
59
~ _______ErfEc RATIO
300 --75--
05..
00- -- o3
34 - - ___STRE __ES -PSI__
02 r 5 p e f Stti
30075-- ~025 o0
06 00
,
00 -100 -200 -300
STRESSES -PSI
Figure 45. Upstream face: Staticstresses (SYY)
E RATIO
3 00
O'S 1 75
0., lO-.- -
0.7
.0
CI6
05
0 3
02 ) Il300 ' ) \2~
00 00 -200 300 -400 -500
Figure 46. Downstream face: Staticstresses (SYY)
60
I Er/'Ec RATIO
- - too -
08 025 - -
.00
'06
13 J / 0.25(044 3.00
7.75
03 1.00
0 05
01_
Or_ _ _ _ _
0 50 100 ISO 200 250 300 350 400
STRESSES - PSI
Figure 47. Upstream face: Dynamic tensile stresses (SYY)
10 1_____ _____ ________
Er/Ec RATIO
09 30 --
0.05 ----
0 7
0.25005
30009 - 1.75
13 7.00
04 \ kH
\.I.
02 --.-
0I
0 50 100 150 200 250 300 350 400
STRESSES -PSI
Figure 48. Downstream face: Dynamic tensile stresses (SYY)
61
E,/Ec RATIO
-~ 1300-
o S _______- - ______-025-
0.05-
0 7 /
06
0 5
050
041.0
0.2
02
0
S 90 100 150 200 250 300 350 A0IM
STRE SSES _ PSI
Figure 49. Upstream face: Static + dynamic tensile stresses (SYY)
1 0
/ Er/Ec RATIO
0 300-
I1.5- -- -1 00-
0( - -- -025 - -
4
o 0.05--05
-- 0.5 --- 7
0.0
02
100 1O 10 200 350 300 350 4007
SIRFSSES PSI
Figure 50. Downstream face: Static + dynamic tensile stresses (SYY)
62
elasticity decreases, the static stresses (SYY) below midheight of the dam
increase. This was confirmed in the Phase Ib report (in preparation).
62. Figures 47 and 48, showing the dynamic stresses, are nit quite
clear cut. To understand the results, one must first look at the response
spectrum. Principally, the first mode of vibration results in the largest
participation in the response spectra maximum. Therefore, the model whose
fundamental mode shapes period results in the largest corresponding maximum
response values will generally have the highest dynamic stresses.
63. Seed's smoothed response spectrum (Figure 5) was used in this anal-
ysis. Figure 51 shows the actual digitized response spectrum used for the
computer programs input. The first mode periods associated with the various
Er/Ec ratios are plotted on this response curve along with the corresponding
maximum response values. It so happens, in this example, that the model with
an Er/Ec of infinite results in the paak maximum response of 0.63g. As the
Er/Ec ratio decreases, so does the response acceleration. This would indicate
0,65
063062 0
061 10.55
052 0
045 -
0.35
0 25 Er/Ec RATIO
3.00
1 751.00
0 0,25005
0 __ _ __ - -- _ I ,
0 01 02 03 04 05 05 07
PERIOD FIC
Figure 51. Digitized response spectrum
63
that the analysis using the Er/Ec ratio of infinite would result in the high-
est dynamic stresses, and stresses would decrease as the Er/Ec decreased.
64. In this particular case, results shown in Figures 47 and 48 do not
quite bear this out. The maximum stresses resulted from an Er/Ec ratio of
1.00. Er/Ec ratios of 1.75, 3.00, and infinite resulted in slightly decreas-
ing stresses. This is opposite of what would be expected and must be attrib-
uted to foundation-structure interaction. Stresses for the four highest Er/Ec
ratios were, however, generally within 5 percent of each other.
65. Er/Ec ratios of 0.25 and 0.05 did result in lower stresses, as
expected, except in the lower portion of the dam.
66. Figures 49 and 50 show the combined static and dynamic stresses.
Comparison of the various plots show that combined stresses for Er/Ec ratios
above 0.25 are within 25 percent of each other except at the toe of the dam.
Tiis indicates, for this particular case, the finite element grid need not
include a foundation until the foundation materials modulus of elasticity is
less than 25 percent of that in the dam.
64
PART V: SUMARY AND RECOMMENDATIONS
Summary
67. The primary objective of this study was to illustrate an approach
for performing a finite element response-spectrum dynamic analysis of a grav-
itv dam. The illustrations should serve the beginning finite element analyst
with a better understanding of the behavior of a gravity dam which is subject
to dvnamic loading.
68. The analyqis of a monolith similar to those of the Richard B.
Russell Dam determined that a mesh with four elements across the base was a
reasonable compromise between accuracy and cost in the dynamic analysis. The
finite element grid need not include a foundation block as long as the founda-
tion materials modulus of elasticity is at least 2L percent of that in the
dam.
69. Results of the comparison between a FEM analysis using
Westergaard's (1933) "added mass" and Chopra's (1978) "simplified response
spectrum" method showed the simplified method to be conservative, but not
excessively -,o. Results indicate the simplified method could be used to make
a first-cut estimate of the surface stresses.
Recommendations
70. Conclusions reached in this report were based on studies using a
single monolith size. Varying the overall dimensions of the dams may signifi-
cantly alter the results. This report should illustrate to the engineer the
importance of making verification studies to ensure the use of a proper mesh
size and the necessity of including a foundation from which usable results can
be obtained. The analyst should develop and analyze finite element models of
simplified structures, then extrapolate the information gained to the modeling
of the real structure.
REFERENCES
Chopra, A. K. 1970 (Aug). "Earthquake Response of Concrete Gravity Dams,"Journal, Engineering Mechanics Division, American Society of Civil Engineers,
Vol 96, No. EM4, pp 443-454.
. 1978 (Jun). "Earthquake Resistant Design of Concrete GravityDams," Journal, Structural Division, American Society of Civil Engineers,Vol 104, No. ST6, pp 953-971.
Chopra, A. K., and Chakrabarti, P. 1974 (May). "EADHI, A Computer Programfor Earthquake Analysis of Gravity Dams Including Hydrodynamic Interaction,"
No. EERC 73-7.
Der Kitireghian, A. 1980 (Sep). "Probalistic Model Combination for EarthquakeLoading," Proceedings, 7th World Conference on Earthquake Engineering,Istanbul, Turkey, available from Turkish National Committee on EarthquakeEngineering, Deprem Arastinma Enstitusa, Yuksel Caddesi 7-B, Ankara, Turkey.
Fenves, G., and Chopra, A. K. 1984 (Aug). "EAGD-84, A Computer Program forEarthquake Analysis of Concrete Gravity Dams," No. EERC 84-11, EarthquakeEngineering Research Center, University of California at Berklev.
Foster, Jerry. "The Static Analysis of Gravity Dams Including FoundationsEffects Using the Finite Element Method, Phase Ib," (in preparation), US ArmyEngineer Waterways Experiment Station, Vicksburg, MS.
_ "Seismic Analysis of Gravity Dams," Technical Report SL-86-44, USArmy Engineer Waterways Experiment Station, Vicksburg, MS.
Headquarters, Department of the Army, Office of the Chief of Engineers. 1958(Sep). "Gravity Dam Design," Engineer Manual 1110-2-2200, Washington, DC.
. 1985 (Aug). "Earthquake Analysis and Design of Concrete GravityDams," Engineer Technical Letter 1110-2-303, Washington, DC.
Krusqewski, E. T. 1949 (Jul). "Effect of Transverse Shear and Rotary Inertiaon the Natural Frequency of a Uniform Beam," Technical Note 1909, NationalAdvisory Committee for Aeronautics [Functions transferred to NationalAeronautics and Space Administration, 19581.
Seed, H. B., Ugus, C., and Lysmer, John. 1974. "Site Dependent Spectra forEarthquake-Resistance Design," No. EERC 74-12, Earthquake Engineering Research
Center, University of California at Berkley.
Warburton, G. B. 1964. The Dynamic Behavior of Structures, Pergamon Press.
Westergaard, H. M. 1933. "Water Pressures on Dams During Earthquakes,"
Transactions, American Society of Civil Engineers, Vol 98, pp. 418-433.
Will, Kenneth M. 1987 (Dec). "The Static Analysis of Gravity Dams Using theFinite Element Method, Phase Ia," Technical Report ITL-87-8, US Army EngineerWaterwavs Experiment Station, Vicksburg, MS.
66
APPENDIX A: MODELING OF HYDRODYNAMIC EFFECTS
1. The "added mass" applied to a structure to simulate the hydrodynamic
effects can be computed using Westergaard's formula, EM 1110-2-2200 (Depart-
ment of the Army 1958),* which gives the hydrodynamic pressure at a depth y
below the water surface as:
p = C e ahy lb/ft 2 (1)e
P =2 Ca yihy (2)3 eg
where
p = hydrodynamic pressure at depth y below water surface, pounds per
square foot
P = total pressire to depth y from surface using the parabolic,
pounds per square foot approximation
h = total depth of water, feet
a = ratio of earthquake acceleration, a to g
a = acceleration due to the earthquake, feet per second squared
g = gravitation acceleration, 32.2 ft/sec
2
C = a factor depending principally on height of dam and the earthquakevibration period, t , sece
2. Westergaard's approximate equation for C , which is sufficientlye
accurate for all usual conditions, in pounds per cubic foot is:
C = 51 lb/ft3 (3)
1 - 0.72 (1,00t e
Period of vibration, t , is usually assumed as I sec. The mass per unite
area to be added to the face of the dam is then calculated by dividing the
pressure by the acceleration, a . This gives:
C2 3m - e hy lb-sec 2/ft3 (4)g
* References cited in this appendix are included in the references at the end
of the main text.
A3
M = 2 y/hy lb-sec 2 /ft2 (5)
where
m = mass per unit area to be added to the face of the dam
M = total mass to depth y
The total mass to be added to a particular area on the face of the dam is then
found by integrating this quantity over the area under consideration. These
added masses are lumped at the node points of the finite element grid on the
face of the dam. This gives:
2 Ce 0.5 (yl.5 1.5)
Mi =3g - Y ) slugs/ft (6)
where
M= added mass to be applied at node ii
Y2 = pool depth to the midpoint between node i and the node directlybelow
Yl = pool depth to the midpoint between node i and the node directlyabove
A 4
APPENDIX B: INPUT FILE FOR MESH 2 MODEL
STRUDL 'MESH 2' 'FINE MESH'$ 4 BY 10 MESHUNITS KIPS FEETGEN 9 JOI ID 1 1 X 0.0 10.0 Y 0.0
MOD 10 ID 14 Y 18.53EN 5 JO ID 10 1 X 0.0 20.0 Y 9.25MOD 9 ID 14 Y 18.5TYPE PLANE STRESSGEN 10 ELE 11 1 4 FROM 1,14 TO 3 TO 17 TO 15 TO 2 TO 11 TO 1G TO 10GEN 10 ELE ID 2 4 FROM 3,14 TO 5 TO 19 TO 17 TO 4 TO 12 TO 18 TO 11GEN 10 ELE ID 3 4 FROM 5,14 TO 7 TO 21 TO 19 TO 6 TO 13 TO 20 TO 12GEN 10 ELE ID 4 4 FROM 7,14 TO 9 TO 23 TO 21 TO 8 TO 14 TO 22 TO 13STAT SUPPORT 1 TO 9CONSTANTSE 576000. ALLPOISSON 0.20 ALLDEN 0.150 ALL,ELEM PROP1 TO 40 TYPE 'IPOQ' THICK 1.0DAMPING 0.05 4$
UNITS KIPS FEET SECONDS CYCLES
STORE RESPONSE SPECTRA ACCELERATION LIN VS PERIOD LIN 'SEED' DUMP
$ ACCELERATION (FT/SECAA2) VS PERIOD (SEC)DAMPING 0.05 FACTOR 0.2531.78 .0050 33.42 .026051.23 .0745 59.70 .080578.99 .1260 81.69 .163572.61 .2680 68.68 .289057.06 .3755 53.19 .400540.22 .5055 37.22 .541527.05 .6980 23.92 .795517.87 1.0905 16.00 1.19459.98 1.6610 8.82 1.8030
END OF RESPONSE SPECTRUM$
UNITS KIPS FEET SECONDSINERTIA OF JOI LUMPEDINERTIA OF JOINTS MASS$ HYDRODYNAMIC 'ADDED MASS'1 TRANS X 1.4110 TRANS X 2.7615 TRANS X 2.6824 TRANS X 2.6029 TRANS X 2.51
B3
29 TRANS X 2.51
38 TRANS X 2.4243 TRANS X 2.3352 TRANS X 2.2357 TRANS X 2.1366 TRANS X 2.0371 TRANS X 1.9280 TRANS X 1.8085 TRANS X 1.6794 TRANS X 1.5499 TRANS X 1.39108 TRANS X 1.22113 TRANS X 1.02
122 TRANS X 0.77127 TRANS X 0.36$
UNITS LBS INCHESDYN LOAD 1 'SEED RESPONSE SPECTRUM'
SUPPORT ACCTRANSLATION X FILE 'SEED'END o'F DYN LOAD$$
EIGENPROBLEM PARAMETERSSOLVE USING SUBSPACE ITERATIONNUMBER OF MODES 4PERFORM NO STRUM SEQUENCE CHECK ORTHOGONALITY CHECKTOLERANCE EIGENVAL I.E-4END$
DYN ANAL MODALCOMPUTE DYNAMIC DISPLACEMENTS FORCES STRESSES MODAL COMB ALLPRINT DYN DATAOUTPUT DECIMAL 4OUTPUT BY LOADINGOUTPUT FIELD ELIST DYNAMIC EIGENVECREATE PSEUDO STATIC LOADING 2 'COC OF LOADING 1' AS COC OF LOADING 1DELETIONS ; LOAD ILIST DISPL STRESSESCALCULATE AVERAGE STRESSESSAVE DIRECT 'MESSAVi'FINISHEND OF FILE7B
134
APPENDIX C: NATURAL FREQUENCY CALCULATIONS OF A CANTILEVER BEAM
Elementary Beam Theory
1. This discussion concerns natural frequency of a cantilever beam
based only on the elementary engineering theory of beam bending with no sec-
ondary effects (Warburton 1964):*
L
0 . X
M, 3.516/ 1/2MODE 11 . .. ;o = 2 Aj
MODE 2' - L .... Iwo 22.032 Eg)1/
M O D E2 6 1.70 /E l_ " 1/2
0 L2 k2A/I o.504L 0.... _ I
0. 132L
Figure Cl. First three modes
and frequencies of a uniform
cantilever beam based on ele-mentary beam theory (after
Warburton 1964)
where
w = natural frequency of beam excluding secondary effects0
L = length of cantilever beam (185 ft)
E = modulus of elasticity (576,000 ksf)
I = moment of inertia (42,667 ft4 )
P = weight density of material (0.150 k/ft )
A = effective total cross-sectional area (80 ft )
Mode I
3.516 (576,000 x 42,667) 1/2= 2 01 0 ) = 4.649 cyc/seco 185 2 0.15 x 80
* References cited in this appendix are included in the references at the end
of the main text.
C3
Mode 2
22.03 (576,000 x 42,667 1/2= 2 2 0.15 x 80 = 29.13o 1852
Mode 3
61.70 t576,000 x 42,667) 1/2 = 81590 185 2 k0 .15 x 80
Timoshenko's Theory
2. This discussion concerns the natural frequency of a cantilever beam
including the effect of transverse shear and rotary inertia (Kruszewski 1949).
The effect of shear lag and shear deformation of the web is to increase the
flexibility of the beam because of the additional deflection that is intro-
duced. The effect of rotary inertia is to increase the dynamic loading on the
beam because of the additional inertia loading due to the rotational acceler-
ation of the differential elements of the beam. Considerable lowering of the
frequency due to the secondary effects is obtained for the higher-mode num-
bers, as shown in Figure C2 (Kruszewski 1949)
W = kB E-14 ; ks E7s
o B - ; k- =o B L s
where
w = natural frequency of beam
kB = frequency coefficient where shear and rotary inertia are neglected
0m = mass of beam per unit length (I1E
k = coefficient of shear rigidity (L EA = 0.21
A = shear area ( AT =66.7)
V = Poisson ratio (0.20)
G = shear modulus 2 (i-+ ) = 240,000 ksf
kRI coefficient of rotary inertia F 0.12)
A = effective total cross-sectional area (80 ft
C4
1.00 1AA 1NACA
kRi
00.10
0.150.80 0.20-
0.70
Wo R
0.62 0.60 00.050.10
0.50 -- 0.15 kR i
0.47 __R0.20
k8 = 3.52 ko 22.0 00.40 L L k =6.17 L 0.05
0.10
o.o I I II I I I
0.30- % . ~ ~N0.20
0 0.08 0.16 0.24 0 0.08 0.16 0.24 0 0.08 0.16 0.24
0.19 0.19 0.19ks ks ks
(a) FIRST MODE (b) SECOND MODE (c) THIRD MODE
Figure C2. Illustration of how to determine the ratio and natural
frequencies of a cantilever beam with and without considering shearand rotary inertia (From Kruszewski 1949)
Mode 1
W - 0.885.". w = 0.885(4.649) = 4.11 cyc/sec
0
Mode 2
- - 0.58.*.w = 0.58(29.13) = 16.90
0
Mode 3
- 0.44 ..w = 0.44(81.59) = 35.90W
0
C5
APPENDIX D: GTSTRUDL INPUT AND OUTPUT FILES FORGRAVITY DAM EXAMPLE PROBLEM
3TRUDL "RBRDATI' 'RICHARD B. RUSSELL DAM NON-OVERFLOW MONOLITH'
UNITS FEET KIPS$
$ FIRST GENERATE ALL JOINTS AS HAVING ZERO COORDINATES AND THEN
$ GENERATE THE CORNER NODE COORDINATES FOR ALL ELEMENTS IN THE$ CHANGES MODE. GTSTRUDL WILL ASSUME THAT THE MIDSIDE NODES
$ HAVE COORDINATES OF ZERO.$
GENE 135 JOI ID 1 1 X 0 0CHANGESJOINT COORDINATES19 143.2529 3.84 46.
37 104.92 4G.85 11.92 143.93 40.25 143.99 11.92 160.107 28.92 IGO.113 11.92 170.121 28.92 170.127 11.92 185.135 28.92 185.$
GEN B 1 9 37 29XD 4 P EQYD 2 P EQGEN B 29 37 93 85XD 4 P EQYD 4 P EQGEN B 85 93 107 99
XD 4 P EQYD I P EOGEN B 99 107 121 113XD 4 P EQYD I P EQGEN B 113 121 135 127XD 4 P EQYD 1 P EQ$
ADD IT IONS$
TYPE PLANE STRESSGENERATE 4 ELEMENTS ID 1 1 F 1 2 T 3 2 T 17 2 T 15 2 T 2 _2 T 11 1 T 16 2 T 10 1
REPEAT 8 ID 4 F 14STAT SUvPORT I TO 9CONSTANTSE 576000. ALLPOISSON 0.20 ALLLIEN 0.150 ALLELEM PROPI TO 3G TYPE ' IP00' THICK 1.0[AMPING 0.05 41INITS I(IPS FEET SECONDS CYCLES
D3
STORE RESPONSE SPECTRA ACCELERATION LIN VS PERIOD LIN 'SEED' [jUMP
$ ACCELERATION (FT/SECA*2) VS PERIOD (SEC)DAMPING 0.05 FACTOR 0.2531.78 .0050 33.42 .02605i.23 .0745 59.70 .0805
78.99 .1260 81.69 .163572.61 .2680 68.68 .2899
57.06 .3755 53.19 .400540.22 .5055 37.22 .5415
27.05 .6980 23.92 .795517.87 1.0905 16.00 1.1945
9.98 1.6610 8.82 1.8030END OF RESPONSE SPECTRUM$UNITS KIPS FEET SECONDSINERTIA OF JOI LUMPEDINERTIA OF JOINTS MASS$ HYDRODYNAMIC 'ADDED MASS'I TRANS X 1.55
10 TRANS X 3.0215 TRANS X 2.9124 TRANS X 2.7929 TRANS X 2.7438 TIANS X 2.6843 TRANS X 2.5352 TRANS X 2.3757 TRANS X 2.206 TRANS X 2.01/1 TRANS X 1.81
80 TRANS X 1.5885 TRANS X 1.1494 TRANS X .7G99 TRANS X .46
108 TRANS X .23113 TRANS X .06$
LOAD 1 'HYDROSTATIC PRESSURES'ELEMENT LOADSI EDGE FOR EDG 4 GLO VAR VX 9.17 9.89 10.61
5 EDGE FOR EDO 4 GLO VAR VX 7.74 8.46 9.179 EDGE FOR EDO 4 GLO VAR VX 6.22 6.98 7.7413 EDGE FOR EDG 4 GLO VAR VX 4.71 5.47 6.2217 EDGE FOR EDO 4 GLO VAR VX 3.20 3.96 4.7121 EDGE FOR EDG 4 GLO VAR VX I.69 2.44 3.225 EDGE FOR EDO 4 GLO VAR VX .62 1.15 1.69
29 EDGE FOR EDO 4 GLO VAR VX 0.0 .31 .62LOAD 2 'DEAD LOAD'ELEMENT LOADSI TO 36 BODY FORCES GLOBAL BY 0.150
UNITS LBS INCHESSTIFFNESS ANALYSISDYN LOAD 3 'SEED RESPONSE SPECTRUM'
qUFPORT MCCTRANSLATION X FILE 'SEED'
END OF DYN LOAD
END OF DYN LOAD
EIGENPROBLEM PARAMETERSSOLVE USING SUBSPACE ITERATIONNUMBER OF MODES 4PERFORM NO STURM SEOUENCE CHECK ORTHOGONALITY CHECKTOLERANCE EIGENVAL I.E-4END$
DYN ANAL MODALCOMPUTE DYNAMIC STRESSES MODAL COMB ALLPRINT DYN DATAOUTPUT DECIMAL 4
OUTPUT BY LOADINGOUTPUT FIELD ECREATE PSEUDO STATIC LOADING 4 'CQC OF LOADING 3' AS COC Of LOADING 3DELETIONS ; LOAD 3 ; ADDITIONSLOADING COMBINATION 5 'STATIC LOAD = DEAD LOAD + HYDROSTATIC'COMBINE 5 1 1.0 2 1.0LOADING COMBINATION 6 'STATIC + DYNAMIC'COMBINE 6 5 1.0 4 1.0LIST STRESSESCALCULATE AVERAGE STRESSESSAVE DIRECT 'RBRSAV1'FIN IS HEND OF FILE
A EIGEN-SOLUTION CHECKS AAAAAAAAAAAAAAAAA AAA
0
MODE ------ E IGENVALUE -------- REQUENCY ------- FREQUENCY -------- PER IOD ------((RAD/SEC)AA2) (RAD/SEC) (CYC/SEC) (SEC/CYC)
I 1.441404D+03 3. 796583D+01 6.042449D+00 1.654958D-012 7.008182D+03 1.371488D+01 1.332364D+01 7.505458D-02j 1 .7458661+04 1.321312D+02 2. 102934D+01 4.755262D-024 1 . 975820D+04 1 . 405639D+02 2.237143D+0 1 4.46998GD-02
1)3
+ 4 * +4 . *lx 0 - * M . m 1
*u InI
0 0 0
40 IV
t. Ire
cu m Su f u c u t
* B P* 6 K
* * EU
* SEU EU EU EU EU EU U U U U U U E E E 4~ 4
* S 0 0 0 0 . 0 0 0 0 0 0 0 : 0* 4 4 4 4 4 4 4 . 4 4 tA 4*~~~~ x i h i h ~ i hi i i h ii h i* .X 49 @ 40 P- qt~ 4 .6go1U 94* 4 0 EU U w 40 - '- P I U 94 .6 y f.6MUW .
*I w-j a
* II
a ~ ~ ~ ~ ~ ~ ~ 0 m 494 in- EUa4s449 ~ E 9 04 4 0 E
A
M In
10 fla
ce u
w Z w w w w
AS n *v
0i m~ a ft wl a v- iv - U S * vA USe
*~~~ ~ WI Sb U v n S 4 A S Uc @ @ IL K aS a4 ain a a t 4..
* A I S AP EU A A A A U U E A A A A A U A AD7U
w w w w w w w w w w w Uw a 0 r t- ( M a w 91 o
fn v r- * W , 0 1LVA : All a. 100 *: ;; It 4
W ok fu # m i C- Ift W tID 0 o f" M i
a Q * 0 m - 14f- :ft w a ! t m l S
. . . . . .. . .6
M 0 WV 60 W f" 0 A 4. A bf"m 41 .4 W p o"4 6 ( . 0 I 4 A :: a1 m C 0 'Ih
am a 0 m * amLOU A m A I f In W ft a 4' ai X
p~m .4.4(a 4644 40 a A 4' v~ .4 mm 464
. . .4 .5 . .- p .fai . . p. .34 . 5 5 . .1 . 6.. .
* a *m IV. a a -W am p. u a m .a ft vi aIa i ~
p. W4 S Id .a m beU wI .4 A V a s. W 3 v 4 .
14 at A4 a3 . A . 4 . 4
* 00 4 4 * 4 4 64 64 4 4 4 4o 4 . . . . . . .a~ to * mo p. c am am in a. v 9 aCa a aw
~ *45 p. 4~ * . p. 4 VI '4D8
le m V
4a I-16 0 e 4WDa WWIWe m
l A m* ~ ~ ~ ~ ~~~1 WAf ~ lA lA 9lA lA A lA lf lA 1 Au
AD a C @ @ * @ @ C C S e O* 4. . 4. 4. 4 4. . 4. 4. 4 4. . 4. 4. 4 4. 4. 4 4 4. . 04 . S
NO hi h i h i h h i0 i h i hi h i h i h h 0W ~ h i h+ +~ + s r - I l . - I ' U - '
Pm m Sh t Al Al b 4 4' S ~ e 4 P 5*6 Idf~~~- CU U U A e . 0P A US S . ~ 4 U
9 U . t l In r. q4 fib- U A l. A 4 ~ ft U Se m AS PS S 4' Al A S I t S I S U Al q~ Y U l 0P
US ~ V IN .4.4 Alm 4 . 4 0 S l - S . l A . . 4 f
Al A PS ft P PS ft A PS PS 'S P ft Al~l Al t ft I'S ft t .4 ft t ft ft t.Ax c@ Se @ * * K~~. K w~~. c S
* . . .. 4. 4. 4. 4. 4 4 4 4 4 . . . . . 4. 4. 4. 4. 4 4 4D9.4
a w wft mT ! 4 ! ! T! .! T . + T
mm
Oh~ ~~ fu fu E U . U E u E 4 U U E U E u E 4 . u u E 94 . E E P
V- Is -V qn a f lb C 0 a 4 b E ~ t 9 t 4' r4 a* ft .-4. CC- to a m4 a mb S t 0 EU U n m mU fu r. a- vw v- U f%0r- 0 4
o ' n 0 4 . ' l ' U C f ' 0 0 i ' l ' u 0 S f A . '
* in ma 4'- - E EU @4. 6.4 4 Eulb - iO IA ft5 60T4' ~ i 4ai - S U C 40 WU C u f u 4 ~ E 4 4 C - .
UEUCICICIEU wE C I U ; 2 0. Eu ' .4 4000n0
v v0 s3 ft do ft lb v- a vi at vb a. at aI a6 4 at a
4'i -W. W, Wi N UCU 6t NS UilN N 1 1 Nu Nu lb1 I N' Iu lb p W4 C- .,
*~ an 1Eu 4 z4 Uc cb i t 4 U i 4 iU i -C
4 -l - to in at. W t f u f u C S 4 4 5 h f u S
U Eu W . C- .4 i ~ C 5 EU U EU 4 4 lb t EU .4. W ft 0I'
................................................................
* 4+ +
1 m
*a a
a I 0. @ 5.1it iliiiiha "- i )i
Id we w
;1 T m a1 v v a
T . P. o 'a M a
to, m 08 t, o a .a I Y a . 4e , j911d m 0 1 unm u, m-. -sia m 8- •. oI~ 1 o t n 1o t o 6
* I
fu N It|* 0
* I_
T "a vi fi 4 4 4 4 4 4. 4 . 9 4 4 4
ha ai -i h i I SW l ia ~ w so ni li i ha
1- a 01 in t S t m 9 MSm -0 'am '0 00 01 oft as6 om gab am am Is a
I aX ro m * a snI I
* .45
Eq
0
a-
ia-
UN faw w w la 6a fl1 uu na fw w 4 w iw
CU ~ ~ ~ ~ a in LA to toE 4 U 4* A L .fu .4 I, wU fl a i* V) *
No C A Nb La F- NI a Im *1 1~ .6 to LAC F-LUS ~ ~ ~ ~ ~ o R11- W, MO fl @ ~ Lf q1 u S AI
0 Sp a a w41 rd P1 40 t- 0. VS s * ~
I I I I
*t 0.. . . mO. . . to 0 I*
FA
go 'a. h h Ni h i i fIh
w f4E U v 1* fl a. ft a- am Eq a r- w a q a
I 61 g 1 W I N M M Ai W. !% I ; I I I -W.
*% Oda Na a . n a0 f o*~~~C co "C@ @ ww w ; T . .
D*-
f" r- In t w w fu I vi a-00 MI MVIm e
-t-
to
4 T
-, ,, v,,"rm nu.. ..IV a.. f% .v- f-- v' ~~~ pi ' gnii a'I,-,
p..
D13
W W 1 Iw wa u f " o r b n m v W
a -ia9 - 0 uI
e- ou V
00F-r- I
P. 0e * .0? ? ?a p .IV r rn x 9 V - N
* ~cu 00 aO fl:l 9 0 . S U4 9 6 5 V
U8 a 4 a9 a I* aU N w aY vW
I b. 11 W. M9 -, !I.4 4 !IA I. :
* 4 4 4 4 4 4 ow d 4 4 4 .4 4 4 e ow 4 4 4 4ci c hi hi hi hi ci hi hi hi i i i ih Si ai a
v C- 10 Y . O C 4 - 4 3 4 l . N
r* p1 z9 4 - A b9 W w N 9 - i F N
lb19N JNW 0 NWW1 D 1 E
Ws
4 F ,0NOW W W W W W IAW W aW
v 6 W 0 n r-
: o
Z SX
cu v t a40 u1,1- 0 "
1 0 .44 U) C 00 Me
40 o 0 of me 0m e t sa~v . 0 -s e n
* 3*v ~ -a- U a.5M 3 v s a asnm-e~ fv m el'.4+
on 2 2 0 2 agob a a 0 I I
04 - -4.4 .4 we 4 .4
* D15
CA cb M-0 a a1,, A
8 1 1a T0OW aW " a- .
b at
W., Wa adb
0. a a"M Io o . i a 0 O
z z
* a a4 4 Aric ppItI .
g a Wa
*," A a ax-. C4.4 am~ 10 ~6W m n m a
it mm so re it it a a a a a- a a
em r w C4 ga r- m ia a au's mrrap ~ .4 shBAA S * S . .4 US a a 'a i* Of- 684Ais
P. 44 a U B a a S .4O S C I54
4 P. U P P. 4 0 6 a AAD16e
A
cA
WW W W w w w w w w
0
w 0501010 0 w La 01 0a Ad w 010 01 W b91 1 1 1 1 A w 11111
9 m or 0 to a m!r 0 9 l 14 ; 1. a 0 fu 49 00 m- vS lw ela lu11- 0 M1. 01 t- t- .4 01a in .4 la to a a lb Ao 0 0 0
r)S01 0 0 V A o -0 1 2 "A K1 m 1 01 so f- 0 A Io- Ml*r #A 010 V 61 m r- t. M a4 Ag.. 1 U A f
01-' A e 00 01 01 10 0 A 0 0110 A * AM M 14 104 4 +0 0 4@ + * OT OT 0* + +0w 4 4a 4s 4 4 d I" 4 w 4 w id 4 4 4 4 w a 4 k 4d m 4 Ad l ad
r.h i hi h i h hi h i a hi hi hi hi hi V hi Le hi hai h i h i-to 01 - 1 1 01 r 1 * 4 0 . 4 S r- 5
r- 4 1 1 01 5 . 1 1 1 - ft .o a - * m a a s mr
I~~ ~ A
01 . .40 010 01 0 m.4 4U m 101. 010 010 0101 - 0010
@00000
0D11
L -1A 347 -S VTW-AIDn-D STUCTIJRAL -9kG ER K-CSTHEEREESPONSE-SPECTRI(U) ARMY ENGINEER WATERWARYS
I XEIMENTSTATION, VICKSBUR6 MS INFOR. P WIERSMA
LNLSIFIED AUG 89 WES TR-ITL- 9-6 F/G 13/2 M
1-______:7 -I2*8~
1.0 :E 14.0 .
111111.25 '. ____
0 0
I.
r-U
v~~4 a a , I. U u '- DEa E a a 1 48 E1 o IV t a O a 1 0 1 a I a w
m ar- a aa a u -m a r a- t
- a ma n 48 u - * -~ e r * ma m e.g Y
ma ~ ~ ~ ~ o 6"a a a- a a a . u. -m -A A U O AS
S 0 1, a .
D18
19
0 0 0 00 0 0
* 6 + + +
100
a.40505 919W 05 E EU W~ 0 EU 919 (U 0 W EU 190 Eu 66 m m 191
*~~~~~ a@ 0 0w0 SO 0 0 0 0 0 @ 40
4.~ ~ ~ ~ ~ ~~~~~~~~~~~o *.4 .44 .4 ~ 44 .4 . .4 .4 .44 .4 .4
-4 ej 4r v vi aV a a ft vi a a
19 a s r-w IA su .~ e qD19
Z1
4, U, t
4 00
fla
011
* . 4 4 is T 4 4 4 4 4 4
A is ma in a C a 4014 a A m i w y S in @1 4 A 0@ s 1 C 8 - 0 4 isS sSiA AU
63 am W, -- d W .#, .8.0 - W U s i B * Ca a un -r i ma a- a On 00 dab-~ ~ 6 6~~~~. " - A S U 1i i S i - A 4 4 4 1 1
-~ ~ ~ i EUc .4 - . f4 46 U 41 A U60 4 1
i g 01 301 A A 010 5 . MA A A US 4 D20m
.+ OAIA
t, NU t N a 'Sb
0 0 nf
* U3
*~~~1 .. . . 2 .*~~ ~ fu to44' C- A -' E
w U S - fI IA f .4. S 194 I U 9
~ ni C- w u~ 'm S lS- w E C
~ ~ . ~ - Al IA (~~ 0
* U 4 .4 N .4 .4 4 94 .4 . .4 4 .W.m C m It 0 C m mm
4. 4 4. 4. . 4 4. 4. . 4 4.4. . 4hi i h hi hi hi i h hi hi hi ^so h
* ~ ~ ~ ~ ~ b .4 C I - E -SC 45~t 9 P 0 Fl 4. ma It a- 4* S 4OU A FlY Al.4 98 ECFl F U 1 FlY 4' A P1 l * 0 9
140
D21.
APPENDIX E: COLE/CHEEK COMPUTER PROGRAM INPUT ANDOUTPUT FILES FOR GRAVITY DAM EXAMPLE PROBLEM
tt)LD G2DGII/SDFDA1,R
DO YOU WISH TO SEE INFORMATION FILE ?YES (Y) OF NO (CR)
-h
HCOL, BO (-NROU IF SM -0.), S ?-- 4,9,0INPUT DAM GEOMETERY IN FT AND
ELEVATIONS RELATIVE TO ANY DATUM
ELEY. OF BASE ?-0.S
SLOPE : RUN TO RISE RATIOUPSTREAM SLOPE ?-0.0833
BREAK ELEV. OF UPSTREAM SLOPE ?
-143.DAM CREST ELEV. ?-185.CREST WIDTH ?n17.DOWNSTREAM SLOPE-0.67253BREAK ELEV. OF DOWNSTREAM SLOPE*16.RADIUS OF TRANSITION
U,.
SUFPLY NROU*1 VALUES FOR YHw9. 23. 46. 70.2S 94.S 118.75 143. 160. 170. 185.
E3
INITIAL PHASE - PROGRESS IrDICRTUR
...1/5 0
.................... >3/5.... ....... ... . ........ . . ** >4/5
MODULUS OF ELASTICITY s E (MILLION PSI) OR F'C (PSI)w4.
UNIT WEIGHT OF CONCRETE: GAMC (LBF/CF)-150.RESERVOIR ELEVATION I HUATER (FT)-17S.
NATURAL PERIOD OF DAM - 0.12950 SECONDS
NATURAL PERIOD OF DAM + WATER - 0.15963 SECONDS 6 6.26 HTZ)
DO YOU WISH TO USE SEED'S (MEAN)DESIGN SPECTRUM (SX) FOR ROCK SITES ? (CR)
SUPPLY YOUR OUN SPECTRUM FILE ? (1)SUPPLY SPECTRAL ACCELERATION VALUE ? (2)(ZERO VALUE FOR STATIC STRESSES ONLY)U
PEAK GROUND ACCELERATrON ?0.625
SOLUTION PHASE - PROGRESS INDICATOR** . * * .. *, j/5 4
.... *0 •e * ..... ................ ) 3/5
E4
HROU * 9 NCOL • 4 MEG * 90 MBAND - 14 NBLOCK I
........................................ >4/5
1509 HRS., 14 MAY 1987
DAM HEIGHT (FT) t 185.0POOL HEIGHT (FT) t 170.0
MODULUS (PSI * 10=Z6) : 4.00SPECTRAL ACCELERATION (G'S) t 0.61
PRINCIPAL STRESSES AS A FRACTION F'C- 4353.WHERE E - 33. S (GAMC St 1.5) * SQRT(F'C)
STRESSES INCLUDE GRAVITY AND HYDROSTATIC LOADS
UPSTREAM PRINCIPAL DAM HEIGHT DOUMSTREAM PRINCIPALSTRESSES (%F'C) (FT) STRESSES (%F'C)
36. ( 1%) 177.5 19. 0 O%)122. C 3%) 165.0 165. ( 4X)268. C 6%) 151.5 295. ( 7%)328. C 8%) 130.9 270. C 6%)329. C 8%) 106.6 255. C 6%)333. C 8%) 82.4 237. 5 %X)344. ( 8%) 58.1 208. S % )379. C 9%) 34.5 163. C 4%)
> 494. C11%) 11.5 114. ( 3%)
E5
LARGEST PRINCIPLE STRESSES (PSI) AT ELEMENT CENTROID
DOWNSTREAM 333.1 UPSTREAM 224.t
TIME SUMMARIES$PHASE 1: INITIAL FORMULATION 2.6PHASE 2: LOAD GENERATION 1.1PHASE 3: SOLUTION 0.5PHASE 4t POST PROCESSING 0.2
TOTAL TIME 4.4
DO YOU WISH TO TRY NEW DAM PROPERITIES ?YES (CR) OR NO (N)
aN
(CR) TO STOP, ANY LETTER TO CONTINUE
Eop
E6
f Br '~; 7. , c :'Fsrs iC A )
-- ~~~ t ';s' a ( p{ter P'aqrarn t,)r Analy;'s 'if i'- '<* '
'9 'i :iit Norhnear Supp-,rts EEAM(
'I 'J''' % 'i- C rpii~rProqrcirnj frin 9t armi ",picy A"',,'
-)f 0 h9JF n F,> a dat;ons, rCBFAR,
ICant It 1(
WATERWAYS EXPERIMENT STATION REPORTSPUBLISHED UNDER THE COMPUTER-AIDED
STRUCTURAL ENGINEERING (CASE) PROJECT
(Continued)
Title Date
.. -4 Frnie Element Studies of a Horizontally Framed Miter Gate Aug 1987Report 5 Alternate Configuratlon Miter Gate Finite Element
Studies-Additional Closed SectionsReport 6 Elastic Buckling of Girders in Horizontally Framed
Miter Gate-Report 7 Apolication and Summary
_7U-r - Users Guide UTEXAS2 Slope-Stability Package: Volume I. Aug 1987Users Manual
T: 5 Si(,cng Stability of Concrete Structures (CSLIDE) Oct 1987
. .,u ITL-87-8 Crteria Specifications for and Validation of a Computer Program Dec 1987for the Design or Investigation of Horizontally Framed MiterGates (CMITER)
.TL-87 - Procedure for Static Analysis of Gravity Dams Using the Finite Jan 1988
Element Method - Phase la
- , , -:,rt ITL-88-1 Users Guide Computer Program for Analysis of Planar Grid Feb 1988Structures (CGRID)
. - Pe urt 17 L-88-1 Development of Design Formulas for Ribbed Mat Foundations Apr 1988on Expansive Soils
-_. Rp rt ITL-88-2 User's Guide Pile Group Graphics Display (CPGG) Post- Apr 1988processor to CPGA Program
1.,tr" Rc-port ITL-88-2 User's Guide for Design and Investigation of Horizontally Framed Jun 1988Miter Gates (CMITER)
Report ITL-88-4 User's Guide for Revised Computer Program to Calculate Shear, Sep 1988Moment, and Thrust (CSMT)
P-,"port GL-87-1 Users Guidt. UTEXAS2 Slope-Stability Package, Volume II. Feb 1989Tneory
T,,:rr-ic a, Run Dori ITL-89-3 Users Guide Pile Group Analysis (CPGA) Computer Group Jul 1989
T, ch",§4, Report ITL-89-4 CBASIN--Structural Design of Saint Anthony Falls Stilling Basins Aug 1989According to Corps of Engineers Criteria for HydraulicStructures, Computer Program X0098
T chr,( a Re rDrt ITL-89-5 CCHAN--Structural Design of Rectangular Channels According Aug 1989Accordi'g to Corps of Engineers Criteria for HydraulicStructures Computer Program X0097
Tc¢hnical Report ITL-89-6 The Response-Spectrum Dynamic Analysis of Gravity Dams Using Aug 1989the Finite Element Method: Phase II
DISCLAIMER NOTICE
THIS DOCUMENT IS BEST
QUALITY AVAILABLE. THE COPY
FURNISHED TO DTIC CONTAINED
A SIGNIFICANT NUMBER OF
PA '.S WHICH DO NOT
REPRODUCE LEGIBLY.
=OEM
DTIC