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A WATER RESOURCES TECHNICAL PUBLICATIONENGINEERING MONOGRAPH No. 36

I---Guide for Preliminary Designof Arch Dams

UNITED STATES DEPARTMENTOF THE INTERIORBUREAU OF RECLAMATION


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A WATER RESOUR CES TECHNICAL PUBLICATIONEngineering Monograph No. 36

Guide for Preliminary Designof Arch Dams

By HOWARD L. BOGGSDivision of DesignOffice of Design and Construction,Engineering and Research Center,Denver, Colorado

United States Department of the Interior l

BUREAU OF RECLAMATION


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. . _ _ . . . . . _ _ .

As the Nations principal conservation agency, the Department of thelnterior has responsibility for most of our nationally owned publiclands and natural resources. This includes fostering the wisest use ofour land and water resources, protecting our fish and wildlife, preset-ping the environmental and cultural values of our national parks andhistorical places, and providing for the enjoyment of life through out-door recreation. The Department assesses our energy and mineralresources and works to assure that their development is in the bestinterests of all our people. The Department also has a major respon-sibility for American lndian reserva tion communities and for peoplewho live in Island Territories under U.S. administration .

ENGINEERING MONOGRAPHS are prepared and used by the technicalstaff of the Bureau of Reclamation. In the interest of dissemination ofresearch experience and knowledge, they are made available to other interestedtechnical circles in Government and private agencies and to the general publicby sale through the Superintendent of Documents, Government PrintingOffice, Washington, D.C.

First Printing: N ovember 1966Second Printing: August 1975

Third Printing: Janu ary 1977

U.S. GOVERNMENT PRINTING OFFICEWASHINGTON : 1977

For sale by the Superintendent of Documents, TJ.S. Government Printing Office, Wash-ington, D.C. 20402, or the Engineering and Research Center, Rureau of Reclamation,Attention 922, Denver Federal Center, Denver, Colo. 80225. Price SO wntn.Stock Number 024-00340097-3 Catalog Number I 27.34 :36/976

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P fe aceIN THEIR PLANNINQ of water resources develop-ment projects, Bureau of Reclamation engineersmay be required to prepare preliminary layoutsand estimates of arch dams. Such preliminarystudies aid the project planners in evaluatingthe feasibility of arch dams and their relationshipto other types of dams in the project plans interms of comparative economy, availability ofconstruction materials, and other technical con-siderations.

In the past, there has been no convenientmethod of preparing preliminary designs of archdams. This monograph provides a rapid methodto aid engineers in preparing such designs. Theengineers will f nd the empirical formulas andcharts developed here useful as a guide in pre-liminary layouts. They will also find that theformulas are useful in reanalyzing previouspreliminary designs to obtain more accurateestimates of dimensions and volumes of archdams being considered in project plans.

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I . X . . _ . ,


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SummaryT E EMPIRICAL FORMULAS developed in thismonograph for use by project planners or otherinterested persons will assist them in estimat-ing volumes and basic dimensions for arch damson future projects and in updating past estimates.The information contained here is to be usedonly as a guide for the preliminary layout ofconcrete arch dams.Topographic dimensions necessary to computethicknesses, projections, and volume a re structuralheight, H, and horizontal distances betweenabutments including estimated excavation tosound rock at crest elevation, L1, and 15 percentof H above base, L. Generally, results fromthe suggested formulas are conservative ascompared with values computed more precisely.

Empirical formulas derived from a statisticalanalysis of existing concrete arch dam data are:Tc=0.01[H+1.2L1];

T,.a=O.95Tg

AcknowledgmentsJohn R. Brizzolara and Kenneth G. Bell of and Mr. Bell developed the volumetric formulasthe Stress Analys is Unit, Concrete Dams Section,Dams Branch, Division of Design, were major and prepared the nomographs. The monographcontributors to this monograph. Mr. Brizzolara was prepared under the supervision of M. D.assembled major portions of the statistical data Copen, Head of the Stress Analys is Unit.

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ContentsPreface ____ ___________ -___- L_________ - ______________________Summary--.-----------------------------------------------------Acknowledgmenk------_----------------------------------------Definitions------------------------------- ______________________Introduction---------------------------- ________________________Statistical Data------------------------------------- ____________Analyses---- ____ ____ ---..-----------..----_---_-------_---_---Results--- _____ _-__-- _____ ___---_---_-_- _________ ____________CrownCantilever-Thicknesses--------- _________ ___- _______

Crown Cantilever-Projections-~----~---c________ -__-- ______ _Volume___---_---------------------------------------------LimitationsandAccuracy-__---------------------------------Examples_------------------------------------------------------Examplel_--__----_-__---_---------------------------------Exarnple2_________-___-------------------------------------Layout--_-___-------------------------------------------------

LIST OF FIGURES1. Damsit.e topography and required dimens ions for nomographs andlayoutofdam__--__-______-___------------------------------2. Nomograph for obtaining crest thickness and projections on crowncantilever__--____-____-___-_-_---_-------------------------3. Nomograph for obtaining base thickness and projections on crowncantilever__--__---____,___________-___---------------------4. Nomographforobtaining VI- _______ __ -_-__---__--_--- ________ _5. Nomographfor obtaining Vz---- ______ -_-_----__-__--- _______ _-6. Planforapreliminarydesign __________-__----__-_--_- ____-____7. Crown cantilever and lines of centers for a preliminary design- _____

page.a.111VVix135777101013131617

28911121415

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DefinitionsCONCRETE ARCH DAM TERMINOLOGY is not univer-sal and physical terms used in this gu ide may, insome instances, seem ambiguous. For this reason,words, phrases, and symbols used in this guidereferring to arch dams are defined.Double curvature---- Continuously curved in plan andelevation.Single curvature----- Curved in plan only.Depth of excavation- Depth from ground surface to

sound rock as determined fromavailable geological sources (geol-ogists, geologic maps, cores, etc.).Foundation-..-- ..---- Total mass of sound rock support-ing the dam.Abutment __________ The rock mass which supports thehorizontal elements, such as can-yon walls.Base ________-___.__ Bottom surface of vertical elementresting on foundation.

Extrados- __ _ _ __ _ __ _ Curved upstream surface of hori-zontal arch elements.Intrados- _ _ _ _ _ ___ __ Curved downstream surface of hori-zontal arch elements.Crest- _ __ _ ____ __ __ _ Top of dam.

Axis- _- _ _____ ____ __ Vertical reference surface, cylindri-cal in plan and coincident withthe extrados at crest elevation.Axis radius--_--_..-- Radius of axis equal to extradosradius at crest elevation.Central angle- _ _ _ _ _ _ Angle at extrados center formed bylines extended to arch abutments.

Crown cantilever- _ _ _ Verti cal section positioned aboutmidway between abutments andwhose base is genera lly the lowestelevation of the dam.Structura l height- - _ _ Vertical distance from crest of damto lowest point of foundation.Thickness- _ _ _ _ _ _ _ _ _ Horizontal distance between up-stream and downstream faces ofdam on line normal to extrados.Upstream projection- Horizontal distance from extradosto axis on line normal to extrados.Downstream projec- Horizontal distance from intradostion. to axis on line normal to extrados(thickness=upstream projection+ downstream projection) .

H =structural height.TC =thickness a6 crest.TB =thickness at base.TO.aH=thickness of crown cantilever at O.&Nabove base.USP =upstream projection.DSP =downstream projection,Ll =straight line distance at crest elevation be-tween abutments assumed excavated tosound rock.LZ =straight line distance, at 0.15Habove base,between abutments assumed excavated tosound rock.V =estimated volume of dam.

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- . . - . . _


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IntroductionTHIS MONOGRAPH is a guide which presents for- From statistical analyses of arch dam geometri-mulas and charts for preliminary design of arch cal properties, empirical formulas were developeddams. Included are the procedure and examples for t,hhe apid determination of initial physica lfor preparing a preliminary layout of a concrete dimensions for design and volume estimates. Thearch dam. Figure 1 shows the topographic map more complicated formulas are expressed inused in example 1. The figure indicates the re- nomographic charts, shown in figures 2, 3, 4, and5.quired measurements for determining the volume Figure

6 illustrates a plan and figure 7 illus-trates the crown cantilever and lines of centers forof the dam and dimensions of the crown cantilever. example 1.

1


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..--._- .._.

GUIDE FOR PRELIMINARY DESIGN OF ARCH DAMS

L>I I ,w -FIGURE 1 -Damsite topography and required dimensions for nomographs and layout of dam. For example 1: H= estimatedstructural height (890 feet); L,=chord length, crest elevation (660 feet); L,a=chord length, 0.16H above base (160 feet);and crest assumed at elevation 326.


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Statistical DataDATA USED in the statistical analysis were takenfrom a.nalytical studies currently on file in theStress Analysis Unit,. Data used in preparationof this guide include :

a. Structural height.b. Length, thickness, and central angle of arch

at crest of dam.c. Axis radius.

d. Length and central angle of lowest theoreticalarch in analysis.e. Thickness and upstream projection at base ofcrown cantilever.f. Volume of dam.g. Sustained modulus of elasticity of concreteand rock.h. Approximate loaded foundation area.i. Profile of dam developed along axis.

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. . _ . . .


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AnalysesSTATISTICAL ANALYSES of tabulated data werebased on observation, experience, and intuition.Combinations of height, thicknesses , projections,and distances were incorporated with constant orvariable coefficients in linear, nonlinear, or expo-nential equations to determine proper relation-ships for the basic dimensions and Volume.A foremost consideration in developing empiricalformulas for concrete arch dams is the shape of thecanyon. Initia.lly, the tabulated data were sepa-rated into two general canyon shapes, U and V.Several formulas were developed in part for eachshape, considering arc length at crest and struc-tural height. These observations disclosed tosome degree the relative importance of the selectedvariables. Including another variable, the canyon

width near the base, enable both canyon shapes tobe handled with a single formula. The most satis-factory elevation for measuring the lower canyonwidth is 15 percent of the structural height abovethe base.Difficulties experienced in arriving at simpleequations were due in part to the inclusion of datafor both single and double curvature dams.Although a greater number of single curvaturedams have been designed, data from the moreefficient doub le curvature concrete dams were usedto develop the final empirical formulas. As moredouble curvature arch dams are designed, moredata may become available for refinements in theformulas and nomographs.


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Results

T E ~0sT IMMEDIATE INFORMATION necessaryfor the design of a concrete arch dam for areconnaissance study is an estimate of thevolume and a general plan. As a result of thestatistical analyses , empirical formulas were de-veloped for computing the volume of concrete ina dam and for sufficient dimensions for a crowncantilever to produce an adequate shape. Di-mensions, in feet, required for solving the equa-tions are: H, the structural height (which is thevertical distance from the crest of the dam to thelowest assumed point of foundation) ; L1, thestraight line distance at crest elevation betweenabutments, assumed excavated to sound rock; andLz, the straight line distance at 0.15H betweenabutments, assumed excavated to sound rock.Crown Cantilever-ThicknessesThicknesses which are necessary for shapingthe crown cantilever are at the crest, T,; at thebase, TB; and at 0.45H above the base, T,,.u,.Formulas for computing each of the thicknessesare:a. Crest thickness, in feet,

T,=O.Ol [H+ 1.2 L,];b. Base thickness, in feet,

;c. Thickness at O.45H, in feet,

T,,,4m=0.95 TB.Nomographs for estimating the crest and basethicknesses are shown in figures 2 and 3. Thecrest thickness is found by intersecting the TO-scalewith a straight line from H to L, in figure 2.Thickness of the base is obtained from figure 3 bythe following procedure:

1. Mark on the S-scale the intersection ofa straight line between known values L1 andL-2.2. The base thickness is read from theT,-scale at the intersection of a straight linebetween S, and the known value H.Crown Cantilever-Projections

Upstream and downstream projections are hori-zontal distances used to locate the extrados andintrados relative to the reference surface, the axis,as shown in figure 2. Projections re lated to thepreceding three thicknesses are defined as follows:7


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.


- mo

--II00

- ,000

-000

- so0

-700

SW

so0

-.OO

M-

so -

70 -

so-

so -

40-

SO-

- so,

-,OO

- 400

-so00

pooo

- 1000

NOTESDlmonsions roquirod for using shotis

H SEstimotod structural height of dam from crest tobore including assumed depth of excavation.

L1 &might line dirtonce at crest elevation betweenobutmonts including l stimatod excavation to sound rock.

L2 =Stmight line distonco between abutments including l stimotmdexcovation to sound rock ot O.lSH above the base.

Procedur.Crest thickness, TC, on Figure 21. Intersact TC with o straight line from H to L1Bose thickness , lg. on Figure 3

1. lntersoct S1 with a stmight line from L1 to L22. Intersect fg with a straight line from Ii to S1

Upstream pro/*ctions:At crest, USP = 0.0At 0.45, USP = 0.951sAt baso, USP = 0.67Ts

Downstream pmioctions:At crest, DSP = TCAt 0.45H, DSP =O.OAt base, DSP s 0.331~

FIGURE 2.-Nomograph for obtaining crest thickness and projections on crown cantilever.


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RESULTS- I200

- 1100

-11ooo

-000

-000

- 700

-600

-300

--400

WOO

5000

4ooo

- woo

woo

-IO00-wo

-WO

1

Too600C--600400

/300

///- /I /

- j

/

1

900

L

-034-000

500

400

rwo

wo

I-- 100 1oo1 L I

ID00II001000

WO

Ii

Iwo -II00 -1000 -

WO-

wo-

roo-

300

400

3007

PO0 -

--- _---,--- -

too -:SOw

To

60

80

40

M--

to -o -

I II aS L 8

9I

FIQUMO 3 -Nomograph for obtaining ba.se thicknees and projection8 on crown cantilever. See note8 and drawing on figure 8.


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_ . . .

GUIDE FOR PRELIMINARY DESIGN OF ARCH DAMS0a. Crest USP=O.Oh. base DSP= T,

USP=O.67 TBDSP=O.33 TBc. At 0.45HUSP=maximum upstream projection=0.95 TBDSP=minimum downstream projection=O.O.

VolumeThe formula for computing the volume in cubicyards is:

where v=v,+v2,

v~=0.000002 H2 La [ H;y;1)2]1and v~=0.0004 H L,[H+l.l L,].

Formulas for VI and V2 are graphically repre-sented in figures 4 and 5. The procedure forestimating the volume from these nomographs is:To obtam VI,On figure 4-A,1. Obtain a value of U by intersectingthe U-scale with a straight line betweenknown values L1 and Lz.2. The value of W is obtained by inter-secting the W-scale with a straight linefrom L1 to known value H.On figure 4-B,

Upper and lower limits of the results are shownin the nomographs. The minimum thickness of3 feet in figure 2 is an arbitrary lower limit;factors other than stress become determiningconsiderations for very thin dams.Accuracy of the formulas for volumes and thick-nesses s within 10 percent of designed values for75 percent of the doub le curvature dams studied.However, formulas have been adjusted to assure,for the most part, conservative quantities anddimensions.3. Mark on the S-scale the intersection Numerical results from the formulas or nomo-of a straight line between H and L2. graphs are solely for preliminary design of concrete4. Indicate on the L&scale the inter- arch dams-that is, for estimating the quantitysection of a straight line between U and W of mass concrete and computing thicknesses for(values for U and W are the scaled values initial layouts or cost estimates. The final designfrom figure 4-A). must be prepared by specialists in the design and5. VI is now found by intersecting the VI- analysis of arch dams.

scale with a straight line between the markson the &- and &-scales.To obtain V2,On figure 5-A,1. Drawing a straight line between valuesH and L1 intersecting the X-scale.On figure 5-B,2. Mark on the &-scale its intersectionwith a straight line between Hand L1.3. VZ is found at the intersection of theVs-scale by a straight line between thevalues of X and S,. (The Xvalue is scaledfrom figure 5-A.)To obtain V,V, the total volume, is the sum of VI and Vs.

Limitations and AccuracyUpper and lower limits on the known data are:


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RESULTS

,-- _.__

HFIGURE 4-A

NOTES

L2 = Straight line distance betw.en obutmentrincluding estimated excavation to round rockat O.lSH above the base.

rocodur. for using chartson f ipure CA1. Intersect U with a straight line from L1 to L;

2. Intersect W with o straight line from Ii to L1On Ftgure 4.B

3. Inkr soct S1 with a straight Iin. from tl to L24. Intersect 52 with a straight lio. from U to W(Values for U and W are obtained from

Figure 4-A)5. Intersect V1 with o straight line fro m Sf to I:

m-to-lo-so-IOQ-

Loo-

x0-

600-7w-MO-mo-looo-

rm60W

40.000,occ

UFIGURE 4 -B

c

_-

FIWJRE 4.-Nomograph for obtaining VI. (V= VI + VZ).


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12 GUIDE FOR PRELIMINARY DEISGN OF ARCH DAMS

HFIGURE

x Ll5-A

NOTESwnrionr r.qu1r.d for using charts.N = Estlmoted structural height of dam from

crest to bore including assumed depthof l rcowtien.

L1 = Straight l ine distant. at crest votionbetween abutments including estim atedl mwation to sound rock.

L2 = Straight l ine distance bohreon abutmentincluding estima ted wwwtion to soundreck at 0.15N above the base.

dum for using chartsOn Figure 5-A

1. Intersect X with o straight l ine fromHto l .1

on F igun LB2. Intersect S3 with o straight l ine from

H to L13. Intorsoct V2 with a straight l in. from

53 to X (Value for X is obtained fromFigur. bA)

- -

i

c=; - - - - - _ _ - - - -- - - - +

f-- - - - - _ _

WOW E

FIGURE S-B

-iHW

-

J?x.cwa~ b.-Nomograph for obtaining V2. (V= VI + V,).


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ExamplesTwo EXAMPLES ARE PRESENTED to demonstratethe procedure for using the formulas. Knowndata are from dams designed and analyzed by theStress Analysis Unit. Actual values used in thedesign studied are indicated following the com-puted values.

A typical damsite is shown in figure 1 fromwhich were measured the dimensions Ll and Ls.The estimated structural height, H, together withL, and La are illustrated in the nomographs,figures 2, 3, 4, and 5. A layout for example 1utilizing procedures outlined in this guide, isshown in plan in figure 6 and in elevation infigure 7.Example 1

From the estimated structural height, H-290feet, and the measured chord lengths from figure 1,L, =550 feet and L,=160 feet, find thicknessesand projections on the crown cantilever andvolume.a. At crestThickness:

To=o.01[H+1.2L,]=0.01[290+(1.2)(550)]-9.5 feet- _ _ __ _ __ _ __ __ _ _ _ 10.0 feet.Upstream projection:USP,=O.O _-________--_-____ O.Ofoot.

Downstream projection:DSP,=9.6 feet _____________ 10.0 feet.b. At baseThickness:

,=~0.0012 H L1 Ii%($) g

=28.9 feet _-___--________ 27.8 feet,Upstream projection:USP,=0.67 TB=(0.67)(28.9)=19.4 feet ____________ 20.5 feet.Downstream projection:DSPB=O.33 TB= (0.33) (28.9)=9.6 feet ______________ 7.3 feet.o. At 0.45HThickness:To.(aa=o.95 TB= (0.95) (28.9)=27.5 feet- ___________ 25.7 feet,.

13


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. ..a .- . .I__~


I

EW


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El. 325;-,Normal W.S. --LB.El. 30 5-----

T, = 28.9.:

,.--Axis of dam

,-----Slope 1.25 : 1.00

MAXIMUM SECTION LINES OF CENTERS(ALONG PLANE OF CENTERS)FIGURE 7.-Cmwn cantilever and lines ojcenterejor a preliminary deeign.


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16 GUIDE FOR PRELIMINARY DESIGN OF ARCH DAMSUpstream projection:USPo.4aa=0.95 T,zj= (0.95) (28.9)=27.5 feet- __ - _ - - _ 30.0 feet.Downstream projection:

DSPo.GsH=O.Ooot- - - __-__ _ -4.3 feet.d. Volume v= VI + v2,wherev,=o.ooooo2HaL (HL+yyE 1 2 1

=0.000002(290)2(160) 2go&(0~;~]C=36,800 cubic yards,andV,=O.OOO4HL,[H+1.1L,]=0.0004(290)(550)[290+ (1.1)(550)]= 57,100 cubic yards.Then,V=36,800+57 ,100=93,900 cubic yards- __ __88,300 cubic yards.

Example PGiven: El=736 feet, L1=1,350 feet, andL=400 feet.Find: T, USP, and DSP at crest, base, and0.45H and volume.a. ThicknessesAt crest:T,=O.ol[H+l.2L,l=0.01[736+(1.2)(1,350)]

=23.6 feet------- ____ __-- 25.0 feet.At base: I FTTB= d3 0.0012 H L, b(-$)

isi736

3J 0.0012(736)(1,350)(400)(~) a=113.6 feet------------- 110.0 feet.

At 0.45H:To.yH=o.95 TB=(0.95)(113.6)=107.9 feet- - - _______ 100.0 feet,b. Upstream projections

At crest:USP,=O.O- - - - - - - - - - - - - - - - - _ ___ 0.0.At base:USP,=O.67 TB= (0.67)(113.6)=76.1 feet- __-_- ____ - 65.0 feet.c. Downstream projectionsAt crest:DSP, = T,=23.6 feet- _ __- _______ 25.0 feet.At base:DSPB=O.33 TB= (0.33)(113.6)

=37.5 feet------------ 45.0 feet.At 0.45H:DSP,,.rsH=O.O ___ - ____ ___-__--___- 0.0.d. Volume

v= v, + v2,wherevI=o.ooooo2H2L (HL+o*yII 1- 2 1

=0.000002(736)(400) { 736+(0.8)(1,350)}1,350-400 1= 1,504,OOO ubic yards,

andV2=o.0004HL1[H+1.1L,J=0.0004(736)(1,350)[736+(1.1)(1,350)]=883,000 cubic yards.

Then,V=1,504,000+883,000=2,387,000 cubic yards-__-__--_---_-_-- 2,240,OOOcubic yards.


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LayoutBECAUSE IDEAL DAMSITES are virtually nonexist-ent, a final design for an arch dam is the result,of judicious evaluation and selection of physicalpropertie s which b est satisfy site conditions,stress requiremen ts, and design cyiteria. Thisfinal des ign is arrived at by several cycles oflayout, analysis, evaluation, and improvement.The initial layout in the series is based on resultsof formulas in this guide and the judgment ofthe designer . A proce dure for making the initiallayout is as follows:a. From the structural height, H, and chord

lengths, Ll and Lz, describe the crown cantileverusing equations or nomographs in this guide.

b. Compute the axis radius: RAxIB=0.6L 1.c. From all available geological information

on the damsite, estimate the depth of excavationto sound rock.

d. Draw on vellum overlaying a topographicmap of the site a circular arc, with the axisradius connecting abutments at the crest eleva-tion. This arc should be so oriented thatthe angle of incidence t#o each abutment isapproximately equal.e. On the axis, locate a point about midwaybetween abutments and in the riverbed (crowncantilever). A line on the drawing connecting

1 The RAXIS should be longthcned if either the arc fails to make contnctwith the abutment at the estimated depth of excavation, or if the subtendedcentral angle exceeds 120.

this point and the axis center may be usedfor the plane of centers.. On this verticalplane, extrados and intrados centers are locatedfor drawing circular arcs which represent con-tour lines on the faces of the dam. The systemof centers for each face must form smooth andcontinuous curves to produce a satisfactoryline of centers.

f. The extrados and intrados centers at eachselected elevation should be spaced on the planeof centers to produce a variation in the ratioof abutment thickness to crown thickness of1.0 : 1.0 at the crest to about 1.5 : 1.0 at mid-height and 1.1: 1.0 at the riverbed.

g. Contour lines on the dam pass throughthe faces of the crown cantilever and terminate,at the abutments. For reasons of expediency,the contours should be selected at convenient,elevations on both faces, equally spaced wher-ever possible, and at intervals not greater than100 feet nor less than 20 feet in elevation.

h. Abutments are drawn radial from theextrados center. The perimetrical contact ofdam and foundation should be smooth andcontinuous.

i. A tangent, to each contour on the down-stream face at t,he abutment line should makean angle not less than 30 with a line generallyparallel to the canyon wall at that elevation.

17

. : .U S GOVERNMENT PRlNTlNG OFFICE 1995-941 .783


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