A New Method of Stress Analysis for Arch Dams Citation Noland, Thomas Jefferson, Jr. (1931) A New Method of Stress Analysis for Arch Dams. Master's thesis, California Institute of Technology. doi:10.7907/yc44-1j62. https://resolver.caltech.edu/CaltechTHESIS:09112024-185103473 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Civil Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Civil Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: May 1931 Record Number: CaltechTHESIS:09112024-185103473 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09112024-185103473 DOI: 10.7907/yc44-1j62 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16715 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 12 Sep 2024 18:47 Last Modified: 12 Sep 2024 18:59 Thesis Files PDF - Final Version See Usage Policy. 10MB Repository Staff Only: item control page

A NEW JJIETHOD OF STRESS ANALYSIS FOR ARCH DAMS by !i:homa.s Jii l\loland, Jr., IN PARTIAL FULFILLMEMT OF THE REQ.UIRElVi.El~TS FOR THE DEGREE OF NlASTER OF SCIENCE, CALIFORNIA INSTITUTE OF TECH1'l"OLOGY, 1931. TABLE OF CONTENTS Foreword •· ............• ~ ..•. e • • • • • • • • • • • • • • • • • • • • • • • • o.. 3 ..4 Explanation of Kalman method of analysis o•••••·•••·•••• 5-10 Mathematical Treatment: Step 1: Determination of moment at any :point in arch 11-15 Step 2: Ohoioe of additional load on arches b•ooeo 16 O••·······························••o Step 3: Elimination of deformations in bottom arch 16 Step 4.: Evaluation of Step, 3. applications of equations to bottom arch e••••~••o•••••••• 16-17 Step 5: ~etermination of stresses in arches and cantilevers ••••••••••o••o•o•••••••• •••••• 17-25 Step 6: Integration of expressions occuring in the integral of work • o • • • • • o. ·• •• • • • .,a o • • • • . , . . 26 ... 29 Step 7: Determination of the parameters that min imize the work of deformation o • • • • • • • . 30-32. Discussion •••••••••••••••••••••••••••••••e•••••••••~••• 33~34 Appendix" Results of numerical calculations •·•·•••·••o 35~39 J FOREWORD In the past most o! the arch dams have been designed according to the cylinder formula~ When this formula is used the stresses are figured in the same way as the stress in a pipe is figured. S= PR/~ where S denotes the stress in pounds per square inch, P denotes the pressure in the pipe in pounds per square inch, ii denotes the radius in inches, and T denotes the thickness in inches. It has been shown that this formula is entirely inadequate when applied to the design of arch dams. Fo Noetzlip Ao Nadai, and others have devised methods for computing the stresses in arch damse These methods which have been devised either involTe a great deal o! labor, or they are based on a trial and error solution. The method of analysis discussed in this thesis attempts to put the design of aroh dams on a more scientific basis. new and unique. The method is both Because the method is new, few men in the world understand it at the present time. The author claims little originality in the preparation of this thesis. Credit is due mainly to Dr. Eugenio Kalman, visiting professor at the California Institute of Technology. Dr~ Kalman devised the method of analysis; the author has merely made the computations under his guidance and direction. In making the computations the author has tried to attain accuracy. The work has been checked. Complicated terms to be integrated have been checked by the proof of integration~ viz~~ differentiating the integrated result, and showing that it is equivalent to the expression under the integral signo The author wishes to express his thanks to the civil engineering members of the California Institute faculty for helpful criticism and adviceo ?asadena, California, May 19310 s A NEW .METHOD OF STRESS ANALYSIS OF ARCH D.AlvlS The purpose of this thesis is to present the mathematical treatment of the Kalman method of stress analysis of arch dams. Before presenting the computations, the salient points of the Kalman method will be discussed. For a more comprehensive discussion of the method the reader is referred to the ":Proceedings of the American ~ociety of Civil b'ngineers for .i111arch 193lp pages 440 to 460." When this method is applied to the analysis of arch dams, the dam is divided into cantilever elements by vertical, radial plan.ea, and it is divided into a r ch elements by horizontal planes~ The intersection s of these hori~ontal and vertical planes form parallelepiped elementsD We shall presently consider the forces acting on these parallelepiped elements. The water pressure against a dam varies, of course, with the depth of the waterw is constant. The pressure at any given depth bince all points of an arch element are at the same depth, one might think that the arch elements are under uniform loado However, such is not the case. The arch elements would be under uniform load if it were not for the action of the cantilever elements and the shearing forces due to adjacent archeso The cantilever elements put bending moments in the arches, and complicate the loading considerablyo The question arises 11At any given point 9 how much of the load is carried by the arch, and how mttch of the l oad is carried by the cantilever?" ·•1, .... ... ·,:: -"'!" ·. : ·•:, Fig. 1. 3 2 2 w :> ~~ 3 4- Fig. 4. w Fig. 2. .!!'ig. 3. All arch dam analyses are concP,rned wi th this probleVi of the division of load between the arches and cantileve:is. Figure 1 shows an element forr'ied b~r the int~ rsecti on of an arch and a cantilever. ~·ignre 2 is an end view of 7 the element. Figure 3 shows a plan view of the element ., and Figure 4 shows a side view. There are four sets of forces that act on the element. They are: lo The forces (1), Figure 3 ~ These are shearing forces between the element and adjacent cantilevers. 2. ~he compressive forces (2 ) , Figure 3o ~hese forces a.re caused by the adjacent cantilevers pushing against the element. 3. The forces (3), Figure 2. These are shearing forces between the top and bottom of the element and the adjacent archeso The forces are also shown in Fig o 4. 4o The load (4) due to the water pressure e This load is proportional to the distance below the surface of the water. Since the element is in equilibrium, the resultant o:f the sets of forces acting upon it must be zeroQ ~hat is: From the figures on the preoeeding page we see that the forces which act on the cantilever are the forces ( l ) , ( 2) I) and ( 4 L, Let us call f(x) the resultant cantilever force, then: (b} .. or (1) f- (2) :: f(x) - (.Q,) 0 • • • • • • • • • • • • • • • • • • • (c} J'rom the figures on the preceeding page we also see that the forces acting on the arch are the forces (3 ) and (4). But from equation t a) we obtain: (3)-t(4) = -{1)-(2 ) f)GGOO•• . . · • . . . . O,,. (d ) ~ubstituting the value of (1) + (2) from equation (o) in equation (d), we obtain: ,i) - f(x) for the load on the archo But (4} is the force due to the water pressure and is equal to la:, where xis the distance below the surface of the watero Therefore the load on the arch is: kx- f(x) o We see that the load on the arch consists of two parts. lQ The load due to the water pressure, and, 2o The correction load f(x) due to the cantilever aotiono The correction load f{x} may be expressed by a Fouri er series of the form: f(x):::. (a +bx +-ox2}cos1 ~ + (a+ Bx+ ox2)cost?[ ~ t - - - d-'f'O "-CJIO where a,b,op and A, B~C, are parameters to be det ermined q ip .. ttri9te rYte~svreJ .fro WI ce,,,,te;r, (/) 0 = all'l,le -f r o m c e n te r to e:< lc vt l3ment In evaluating the paramaters two things are considered: l. The deformation in the bottom arch is zero, and 2o The para.maters minimi ze the work of deformationo THE CHIEF DIFFERENO.c; BlllTWEEN THE ~UJAl'i 11E THOD Ol!~ ANALYSIS AND O'l1HER .LVLETHODS I S T:HA T THE CORRECTING LOAD I N THE KALMAN METHOD IS EX:PRESSED BY .A FOURIER SERIE S 0 other methods of analysis In o the distribution of load between q the arches and cantilevers is determined by approximations involving the solution of simultaneous equationso at any point in the dam, the deflection of the cantilever is equal to the deflection of the archo each point that is taken. There is an equation for The more points that are taken, the better the approximationa By the Aalman method a correcting load is chosen containing a few parametersQ The stresses in the arches and cantilevers due to the load thus chosen are then computedo The integral of work is expressed in terms of the stresses acting on the arch and cantilever sliceso ~he equations for the parameters are derived from the condiuion that the work of deformation is a minimum» and finaliy the linear equations for the parameters are solvedo in presenting the computations for the Kalman analysis, some trivial steps and transformations have been omittedo However,the author has tried to make the computations sufficiently complete so that they may be followedo In order to codify the mass of computat ions the author has arranged then in seven steps as follows: ~TE? 1: The determination of the moment at any point in an arch" Three conditions of loading are considered: ( a) :p ( ¢ ) == Constant. (b) :p ( ' ) :: COS (c} p( ~ ) ::::. cos 1T J:f /P .3:f" qJ cl- 'Ill) In each of these three cases the h.alman formulas J 0 have been appliedo STEP 2: (.l?roceedings AoS.,C .E, .ili1arch, 1931, p ,., 43'7 ) Choice of an additional load. on the- arches f (x;, ¢ ) ,. (The additional load on the arch is the load on the cantilever. ) Eliminati on of deformations in bottom arch since the bottom arch undergoes no deformationo STE:t' 4: .illvaluation of Step 3o application of the moment equations to the bottom archo STEP 5: Determination of the stresses in the arches and cantilevers for the assumed additional load. l::>TE:P 6: Integration of the expressions occuring in the integral of worko STEP '7,. Determinat ion of the parameters a,b 9 c, and AgB,G, which minimize the work of defornie..tiono After the parameters have been obtained the analysis may be applied to a particular dam by substituting numerical values in the formulaso The author attempted to compare results obtained by using different central angles~ and different ratios of thic:knesa to raduus of damG The central angles of 120 degreesi 140 degrees and 180 degrees ( ¢0 :. 6d~ 7 o: ?ot} ) were ohosen; and t /r ratios of 1/5, 1/10, and 1/20 were chosen. Time did not permit the making of as many nu.rnerical oalcu.lations as the author would like to have mad.eo Numerical calculations which have been made will be founrl in the appendix., A discussion of the kal:man method will be found after the computations, just preceding the appendixo l n this step the moment at any point in the arch is determinedo xhree cases are considered: (a) .i::( 4> ) -== Constant { bJ .d (p) =- Cos (cJ P( cf; ) 3!_[ ~ "'~ Cos If. ~6 I n each of the three cases. the "-8.lman formulas are applied. ( ~ee ~roceedings of Ao~oCo~o for Earch 1931 Po 457) p \ I \ I \ / \I Choice of additional load on arches, ft~~). (The additional load on the arch is the lead on the cantilever ) . . We may let a .rrourier series represent the additional load on an arch. Thus: 1/+ -f ( X; ~)=(a+ H c -'f ~ ~ ~ +8J +- C" '") ,6,.. ~ -f (?() q~ i .s J oad ~ ~ canttl e v~6y; ~'f Q f a_J b ., c, and A, B.11 u are parameters that are to be determined later. STEF 3: ~limination of deformation in the bottom arch(since the bottom arch undergoes no deformationoJ If~ denotes the vertical moment in the bottom arch, then_ there exists a horizontal moment of magnitude 4 where~is joisson 1 s ratio_, f i.s moolvlv-!I of ela~t,'a lfJ, - - - (eJ arid t1 +- ~ t1 == 0 E f1 1.s t h e 11 o wz ~ 11 t w ·h I ch Cl c.- f.s a f1 t he a r- c '1 a 11 ~ .,AA,-jJ. --t h ~ r11 a W1-e Y1 t t h ~ t ~ ct.s o "1 T h .e E ca J1t1 lever. STEJ.) I : rn this step we evaluate step 3o f he moment equations for a 1 , M2 and M3 which we derived in step l are applied to the bottom arch and the results are substituted in equation (e } of step 3o The C an-file ve r m o n,1 e J1 t ,:S ·, H Mt:H ==ff x:: o (7i✓ 4) (H .-1) t/2< =- ~ 7[.. ~ , H(.ir H+~ H ~ ;l.'fo °\ 0f.) - ~,_-~ - £H__f t- ~~ q H(,1-H+ §I!'+ c H"') ~ . ~ q c:l..¢0 .;l.. 3 3 . .5TEP 4(co1'TO) 17 3 H4 -AJL - ~ -S-- . "1,.,- ~ .o\ flJ --( f- ~ *,._ +-- h HJ+- E:JL141L t /44 tf : ~ +c;Ji ") ~3JE Al I J.. j, .1.~b \ ~ f d-- C:, ..l~o ~ -r ~ e I o 4 ol o n -th t11 4 r- c. h 1'.s •. J:>(1) - (w1 H- (a +-1,H +-cHj~?L ~ -(A-+-BH-t-CH~4J7{~. J-~b t'o In this step we shall compute the stresses in the arches and cantilevers for the assumed additional arch loado .uetermination of 0-i'J v";; 1: (XJ r); t (<OJ rJ. {Notation is the same as in the article by Kalman in Proceedings of Ao~ o C oEo for ~arch 1931, Po 446. The notation is used by a number of authors.} H ~J ' f =-I< ➔ d'X+ 1 /i 1-:::.. la.. ~ILA)~ t A~~ ,• o1.'f D ~ 4) 0 -1-(b 4n. d-'fO IL"' 'r,h + B ~ ~ c:p) ~~ ~ +-(c Lt,,_ f~o ~ +- C -4n. TC tlJ • ar f =- I< + oX t ,z_~ ~ ti-- ff(!J(:x-.f) ds f =- o ~10 . JfK+ts + rz 5~ ('x-sJ J. 1 -::c ~ =- 0 1 , S (coni~) r .STEPG I~ /vlf ::: {- <'ti- A e] t -x(lwf-.J.J'-8£)-+ ,<L(-cJ'-Cc.) /'Ji~~ == (-at- A£.)'-+x'(!wf-A-f-8t) -z. + x'f (- cJ'-ce:.) +- ;. (- a/' -Ac) ( fw [ -k/' - .Be.) 1+ J (-ct-[£)· dx + J (-~t-A e.) (-c I'- Cc)~ -z3 (tt)I f -kt - B H ~' ,'.ff tit rdf /J.1 = ~ ~ o~-!" J r LH {- ~t- At) _, J 7J J / r -i. .i1l +-If- ld' w S - bI- 13 c:. +-!ts (-_]lct- c ~ .,__ + ~Jf- (-a..f- ~ (tw5=,tf-:ge: -1--!f (-cf - C t:} {!vv J - bf- B c) f ¥.a (-af- At:'(- cl - C e)] jf ,)... -:[ (-a.f -A'i:J (j):r t:-t-z-f~+- ~ ~ l-+fe + A i. ce2- '\--= j !J.. '-/ ";) ~ .,_ tJ> f[:c:. .,_+e ~ "JI.. cp +- A,,__~ -z- f ~~ -+- ~z r ~ h} ✓ + ~ r A,~ f ~ -rr ~ 'f- .1 o I ::::. [,. :::::. ~ IP o +- - · A ;Cos ~ ~ +- :J ..::... A ~ r ..q t- 4 b 2 ( q. i- ~ :i. it. L -+- ?Tf4o ~ Po r A ci-rr-i.--- J.;-~b ~ 8 - - - f- _ . - 11cP. 'if - O '+- r L\' ~rrcp ~ ¢o _ ~ 1f , ___- I;- ~ • 1--- - +- L; ..::.. 4 /4..,'...... ¢ (]] ::: a, ,_ [ c} 0 1.. -=:- .. +- (7 2 +- A ......+- 4:J ~tPo d tR ~ 2 r-1I + g ~ & (J" Tr +-A-_ j , -=. LL ~ ~ ¢(/ STEP S (CONT 'D) ®a Jf (d'w J -,,?3 - B t:) = /.., <-S ,._t-1:i',.J',. t-- J3 1--e""- j l'w b S J' 'Z.. (i)a (])~ ()a ©u:©cr f- }..bEJ£-~ctw f3S t:.. (!)Jr r ;t,., .,_ S -z..d 4... J.,@'L..rt/Jo[~ ?-~ - J. ~ ¢. ~ ~. 'I. -~o ~o r ~ .,__4o 7,d ,1, = if,,,/ (i)L ~. t-£ ., .,__,_;__ a¢• ,,:J r rj)() - 1/: ~ ·~o ~ (/Jo -f- 1. ~ ~¢Jo , c/) 47, cpl),_ 0 = if'w -z.@,. [} b +- ± ~ ~ ¢, - ~ ~:,- ~ oJ @.n: r ""-I d1 ~a .,.@.c~ - 'Z.+-1' +-4 .,_ +-f-= /7 J 4 t,, r 4 + /, =,4 ~ + IA._'"-~ ;i. p, + ?lr4o 'Tr z...._ If 'Vo J, "C;_ 7 b ,J.,J :2-- ~ [ E , _ c .,_ .,/ ~ ~ B-z. {Jo (;, Jl .,_ f- rf .,_+-- Z :_ +- J lt'i ~ 1- . i ~~ .,__¢ - - 4~ -~ t, = r- Lr .IL " , J--- J_ ..J( z) J A 4o 1- ~-lz1T ~o~tP~l 1T ,.__ if 47 .1-- J fr-=: ,ik fo t (I /4...!! e,,..~ ~ ,,,a_,;__ ~-o - 4'o ~ ~ 'l d q ~o - ~o - .,_;, o>v fs t:. _, b Ow ® + A4 _ 7 ~b 7.-- r ~ ~. . ~~ ~o ;2. <!Jo :J 1i1 r ; ~ 0 + 4 rcrr ~() ~~o L.. '7/1.-- 't ~o-. - - _,.. a V w l!;I +4 (~. t-'t~ .t~d - ~~ ~ ~ +P' ~~"' 6rr S,eP S (cont'd.) ~b BJJ'c: d, ~ = HB [~o ( :i.:=.J7. + ~II +d- ~ ¢0 (.f2. A +::: +? -t-rr~ Z ) rf) + r <f it. iHo'1T ~Pi;o 'L_ -c..- '; + 4 J{~ot½~:i.¢oJ- /J4 £.'if9o ~ ¢6 1rr~ Lt Po '1- ©n: - '-d'-w BfJ f dt= -d-//,1113 @fL.J?. &il.Qt-/~c~ +£ ~ ¥I ~ ~ ¢ - J2 ~ <P-2 - .0 • \of'O ~ !lo A-,;.., 1h, ~~ cpb l.f - z. ~~o ~ ~ (J d ~. ~ ifw 13 ~ &- J2 ~ ¢D +,,P {~o +l ~ tJ === - - ;;i_ / :)_ Z. 'fT 1° ,&.,_ ¢6 _ :;_ .fl ~ ¢ -ti A~'-rp 0 C/,P- t,. ¢a..,__ • cPo 1 +- -~37T £ ~ ¢0' '_j, CD 6) © .IIJ (-cJ- ce.J"---==-c\f ~~cCJ'e +- c'-e ... aJm c .,_Jt 1:c,/ P-=-c ""[jpo (; _ ~ . _ +- r .,__+ 4 .,_ t-F fl t-£. ~ ol ¢ +- 81T#o ~ ¢0 /14 ~ 0 CfT 2--_ Lt ¢0 ~ + if -s: A ~ ¢b J @.ra ~ c cf!c. dQ =-?- c.. C [p. 2 -R ~o*~~ 11~. - -f ~~ ¢0 (S/4+ ~.PJ+ rJ . !+'ff 4>, b¾Lp1l ~ 1=. ft. tf o.__ STEP s (coNT'o) ,-i -+- 4 J {~o +-~ ~ :l. ~ J-~ -=-z. 0 ' 311 ~o -/~ 42'ff ~o ~¢u q711-_'r- 16 'L- c--dQ = C. [?. J7. !..~ 0+,f1{(jJ O+~ ~ '1. tN +- ;z "- tPo + LJ- SL rf~ ¢ +- ~ J z 1f ~4~ (3) JII C •1-J 'L- ~Pt) - R c Ai '7 3 <fj L '-f'O_J 1T""'- l.f-¢c 1..... I (D ~ f f (-A !-A '0W'w-kf- 8 aj = -A t..,fJ +A hJ ,__ - • q (!# Cl) +ABft, -A{wf..5tAkcf£·+-ABC2- @a -Atw ]ff d ~ -= A d'w ~ f~ ~1+1¼\~t,,,A + LI /4M-. .,___ 4' - -===: ~ 4°. ~ r ~ ~•,Cea. 7L ~ - 4 ~ ~o - ~ - Ji]tfdo ,I\ Affi,._ 't' lf'' 0 -J- ~ /1 fi'(fo ~ ¢6 1T 7-- /..;. ¢r; 1-- ;p. ~•-:::: -A J-W @L?- 2 ,.a....:,, ~o -}- 4 { ~o +-"i_ A-&.,.. ~ ~oJ - ~~ ~ ~ c/Jo - 'f /1 ~ ~o .:.. ~ 4 ~ '-cfe>J '1f ~J]l: t--J_ ~b Ab f J''-dtP =Ali l4o ('."2. "=. 'L-+- r .... -1-4 - rj -j- ~ ~ ;.. ¢~ +- 81flo ~~. /1 4 -::- · 1T --. 2- t- / l f ~ A ~ ¢0 1f-i.-,__ L;- ¢() "l-- I (3) .rr:. A (bt-B J!1 i. d ~ -= A (btJ3) [~. ( :i- ~ Jc +- '-t.:!J - ~ z_ ) +:i..~¢o(-11L1 +~A ) fJ 3 IJ = ' + !:__J__ •4 7/10 ,C,s_ ¢ I + 4 J' (.1i O +i ~ ~ 40) °IJ 1-_ l.f jo z.. - _I~ A.z Po Mn-!b] q17 :. 'I- f O -z_ f @.rt - Ad';.; Ef J ¢ = -/2 d-'w [A (¢0 +-j ~ :i ¢0) - I J£'ff ~s- ~ ¢0 _ ::l qrr1.,,- 1'-f()'- rf ~ "¢0 +- ir ~ ~ <$0] ~~ -3/ A s{c'l.-d 4=A- B &o (J .fl.'/..+ rf '-.3; JLz.} + rf i-: _l_ ~ Q_ ¢ f- Lj fL ,..f ~ ¢ + ~,,f :z"TTcf,_ 4..¢0 ©il . , _ + - Z 0 D ~~'t ~l> 'Z-- ;i.. (- ~ -S:)[~J_:_J-_!_ ! e) =- C /~cf f cE,t.~f 1-+- Bc et - CIvt1 E' f +#CE!'+- BCet..- y C D : u : [ts 4°' -C [1t.~!_!; -.tr r~ ~- - 4 ~ --~oJ - c l w o f d ' w t , J P p + A ( ~ . t ± _ _ . ; _ _ _ ~ j_ f '-J ,f = b @l.u: b C- J' -r ~~ ~ ¢ D c: {}, ( ~ -=:. .,_+ (I f'- .L1 '-1-J 2 t- 1? 'JI f D -6,.,. ¢b 1 / 1-:_ {' {' J A +· 4 -==- A~ ¢01 Lf '() 1---- G:Ji: (8 c. +-.b C) f [ J' =-{j3c. +.b C) {cJ a:& JL t- 4: .fl. r!_ O ~ ( 11 --&- ~ ·z ") T ,:;, ~ ~, ( ..fl.. AT --= rf)-F r ~_: 477 t-e,._,t 1/ 1/'l-- Lj $b -z.- + A rf ( ~at t~ :lc/c.)- Ji 4£1[ 4 AJJ>.z.¢~ 7 0 ' q1r-i...-L;¢0 ~ ~ STE? 6; l n this step we integrate the quantities occuring in the equation for the work of deformationo ~~ w "" /ff [rrx ...t ~,. - ~ (~-r~) ~~(-ct; + t 4;-)J - -a, . '2-, fff Ox,. =f f/f 11; f '-_ H +-¢0 f / f 11x ...,,/ rJ ~X lo o - ~o fjj~,. =f/f 11/·d;:rdf. - qo H j f Jo-;~ , t'!j1, t!Jr-1 -}p jf11~ 11,. e=. JJJ'x ~... , _ ft f j V,, >d ti dJi. (D ( ' _,,I i // t1 -i-d rd~ -:. rt~ rz~, go oo Hj'/a-z--+ AZ--)+,s;ss'-L: -i J, ~t, 3 t,. tJao o , , H ' (ah +;4 L3) + 1..s-, 1/- '). 'D.J- H 7 ( a c. +-Al )-;-.i,-, 14 :;.1 H1(1:i 7-+JJ-z..) f 1t,n) 00 /-f P( b L-/-{3 l) f H q c~ 7--+- {_ '-) J. STEP 7: In this step we determine the paramaters in the 0 ]'ourier series which minimize the work oJ deformation. we may find the moments in the bottom arch by sub st i tut ing x-= li • in the formulas for .lili 1 i M2 , and » 3 in Step lo •i• the moment in the bottom arch due to constant load,may be written: 11, .»11 2 , "% ,~(R,+.J, ~4) A the moment in the bottom arch due to the cos 1[ ;;..tpo"-f · load, may be written: Ml --= t' '"{ R,._ +.J 2. 49. (j +1-;. MQ.f~c ~) .w1.3, the moment in the bottom arch due to the cos -!1f' q ~cp,, load, may be written: fh = ri..(!<-3 t ✓J3 ~¢ + w~ ~~D q, The load really applied to the bottom arch is: P{~J -: d'w H- {a.+,H+~Hj~~ 0 -(A+BH+cH7..)~ fi,~' The total vertical moment in the bottom arch is: f1tot 4 f ~ K H· r1.{f.. 1 tJ,~¢)-(~+J:>H+cH~(1?1-+.J1-m~ + T1- An. ij.;) r , ._- (A+ 8 If+ CH j { "'-3 +.JJ ~ 4 + w~ ~ ge P) r -z:i-fo eE 11 where ~ ,:.S p tJ 1· s ~ o h '-5 ra t, ·o , Th ~ com lo 1'yi e d h or i -z. o l'1 ta I a YI d r's r h c:q I mo w-, e "1 ts ,·~ : 11+ - f /1 = K HI~, - (11+bH +cH '-J /~1- -t4"+6 H t-c H~ R__j + 1< 1-UI - (a. f..b H+ c Ht-Js .. -(A+B. H+cw-) .5.3 £e<t ip 3F - Co. t b ff t-C Hj Tz_ A'w. ff,; q- l4 t BH /- C. H ~ 11/s ,e.,,.,,_ ~fo ~ + ~( ~ H,. ,_t-_bJ±! f- u±J") ~~ ~ +~ (~-1-JL!I' t-£!±.'tf~i-[~ r,_,.J=. ?-. b I~ r1.-p ~ JJ.. J~ ,0 0 'l'D STEP 7 (Conte/,) '.3>1 :r -f : ,-i t 1< ~ ~ 1- r,4,g_ 'IC Qt- c: ~ g ¢ -t Me11 A-=- J< --= t ==-- £ -=- a , F )"' b m t h ; .s Y" 6 IQ t i'o .:L(p o :J.4'0 //1 -= o - w ~ o b ta / 11 : u 1< H R1- ( ci +b H t c Ht..) R1.. -0 +s H+c H ,_) R,.3 ==- a - - - 1 K 1-f s,- (a. + b H f- c Hj (Si.) - '4 +13 f-f +c H ~ .5j -:: 6 - - - (2J 1 - (~t-bH t-cH-z..)T.. + /W (a.H ~ bH::!J + c H 'ti -=- 6 - rz..e ~ 6 ..1.. I -t4 BI-/ -1-cH j 10, + ~e. ·(A.Ji,._+ IJH:!.+f:. H 'ti ~ I -t • J:.-f & P=__e r~eT~ ~YJJ 9 -=- ~ r~£~ - =O - - - (.3) - - (If) I~ then,' a(~'--') t-b(eJf!-H)+ c(-Eji/- H"-) - o - -- (3~ 3 od A( £11-"-t) t-13 (lf -H) +- C(§f1/- H1..) =- 0 - - - ( 'f1 F v-o m I (/)1 ~ ) (3 ~ h1 f c, r tcu1 t d @ w e o I, -t-a / H t- h ~ e 9vat/ o .5 -r o ,,_.. th e. f 4 yq m t1 f-e r ..5, Cf vi YI lli,J?ORT.ANT EQUATIONS t't+otC= fw H.x R, ~ s s -- rw H R, S:,3 - R.!i S,L = I< R'2.. ~ Si .Sl ~~ R1 A+ Bt-C = dMI H 5~ R,...S.3 - R~S-a. s, == tw H R"L..S,- R, St.._- KCcap1't 1) ~ R."L. I\~ s '2.. s~ f<.,._,S~ - R~ S1... f Pr/ ~-/l.H-a..-,PH ....tl,_1 ., 6 = l<.H~{_PH 'L--1~)-~ P'H ~- I~ H,_- ~ P H.J+ I~ H. 8 =- KH'I-( qH=J .J.}-A f9 H ,,.-I ~ H.,___6 g H,_~ I 1 l Cy H 'I- - I ~ I+ ..._ ~ 5' Jt3 +- I ~ HI DISCUSSION A first glance at the computations of the Kalman method of analysis may give one the impression that the method is considerably involved. A great deal of time and patience ia required for one to follow through the computations and thoroughly understand each ste:p 9 However~ the method is not nearly as complicated as it appears to be at first sighte while long, are quite simpleo ~ost of the computations; Other methods of analysis also involve a great deal of labor. Some methods require for good approximations the solving of sixty to a hundred simultaneous equationsa 3olving this many simultaneous equations, even if trial a..'l'ld error methods are adapted, is no small tasko When the Kalman method is usedp and the general equations for the parameters are derivedg the parameters for any particular dam may be obtained by merely substituting numerical values in the formulas on page 32. The general equations do not have to be derived for each damg The author believes that the Kalman method has real merito The .Kalman method expresses in one equation the cantilever loading at anu point in the dam~ The equation for the cantilever load is: f(x,<P )==-(a+bx+-cx2)cosl£tP f(A+Bx+-cx2) Any point x and cos¾1fo/-,- - - in the dam is located by its coordinates "x" is the distance of the point below the surface of the water, and~ is the angle to the point from the center. When these tvlO quanti•liies are known the cantilever loading may be obtained from the above equation. The load on the arch element is simply the load due to the water pressure minus this correction load f(x, 1 ). ,lust how accurate the method is, the author does not know, but he believes that the method will give good results. It would be interesting to analyze a particular dam by _several methods, and compare the results. Such a research would involve a great deal of labor, but the results would be valuable. The problem of determining the stress distribution in arch dams is an interesting one~ and one which offers considerable opportunity for future research. APPENDIX Results of numerical calculations. NUMERICAL CALCULATIONS y ...L s- J //) {o L .... ---r"'" (I+ %(,1.) t (1+ 1~,.) -'-{I-:!...= t. ~r,. ) ,b'f /, tJ I 333 ,S06hf::, , 41ht7 "{) 0007 , 0 / I, 063 33 .56 /t,fo , l;-.9C/l0 ,l}O/ /,!>DOl3 ,SOO'f/ , 1.;- q9 79 ,b00!.;16, -t~ i: 'bb w- bi I" ._ 00 7 \ I ag -t /Dj -½_ ( 1- -;;pi. ~ ~ 0 0c-7.j-O 1. 70 66 1 ..L o, 00/ q, 70 0 l)CfS" 1. 6 J-0 0 , 0 0 0 3 {, o.3. _L /(J .J.._ T" J. h0° . , D..S S;).. 7o J.. JO 1.. 0 R 7, J 'J. -0- i f 7 1',6t/i7176 6, 6 I J 7 3f _Q 2... 0 SJ:J- 6 ,tJ;t7tf 3J- q 71 "- ::::... . 0.J;)- . 0 2 76 3 q, ,9q3~t,tJ oF D ::. I .s~' , OJ. 7 t I/ 0' :::.. , I O 7 ~- . {)!J-3 7 ~" ? t- I 000J,,ldl~ I' 1- 0000 ::. I rr~ 7, 8:::i.1~?7 {6tt006- Os t 2. 0 I~ Q -;i..+-,ooo/ J. .!)- 3f y /(,)e~ .1. Jo .L I oq. '"~ t: ,._, +i. ( 1-t sr.._, r ¢0 -t 'I,.. ) O () 0 0 \_ ':: I - ~I :l.311, :).. IP, j t57 , O!::J-3 3 / q :2..'-0, t, {)j-3.!)- · D 0 3 ~I I I fl, l~t ' ' . If o -t-, oo I'- ::::. Po~ ~ I fl o f 00 .3· I '8 ;)_ ~ I IJ I 1 09DG -- t- , ooo / - I o 1Io, o q o.!J- Io, I o3 /, 0 s- RESULTS Olr NU.MERI C.ti.L SUBS?ITUTIONS--STEP 1. ' TERM (!) • t .; '0 () -D "" .:: 70° E. 80 0 - ¢0 ;:; ~ (0 © ® (:i) SJ.bqq .so O 00 - ,J.oql. ,l,/,JfpO ,SDC/30 . iooo ~~o .71,qJb ,(p.571/i ;J..q 7 'f /, / 77 8 , q1',83 1,$3/3~~6 D I 70S.3l. , 8lb3S -, J;./7 9 a, ,s-.r.3.3 ')..,3C/b 7 -3 I 7 6',- 7 ~~ I (J) ~ (f) - I (S) ' (!) (@ @ ¢/) :: b~d ,.,l./70 ,lf831PP ,O~'J..q~ . o/1 o l. , ~ .r, qs,t,,JJ, ,b?-.f{)!J ° L;, 3 8C/ 0 . qqs1;-~ , os.:nr l~ ,r :,. q ,01~o!i·~ ," IO 1-11 6- gc/ I/. lLj-Sq ~, ' ' 8f :.. 76 ¢0 :. I ~ ~ Q) I .1. a @ ,0/,78~ I()/ I O:J..b'f~ ,01 b'1-'f4~0 - ,D~lft/ , 0 '7-.~ Olf ,O'tb8f f , oh 81l, RB~SUI,11 S OJl l'rUlviERlCAL SUBSTITUTIONS -- STEP 7. tPo ~ j_ s fJ; 0 0 J LO ...1.. +-, 0 )..3'17 -,oj,.Btf! '?~t; _, J.., 3/ 0 ~ +, 8 0 0 0 i~d i -,1,~13t, I I i, L~o. +. o I /;.3 "-J ,.. -.o/Bt,'L 4j,/;t,.d-9 - t~.Jo7J. t-/ ,j- 31.3 - ,Oj,$77"7 J..., ~3 3S-/ LJ.B,Jt8~ - , 'f-, o 84- I II 4, 8, l 8O!:,- 'J ..J_ tD +-.o~6'i1 ;2 0 ---;0360/'f } , U3L;. 87~ -, 0 ~"I 'fJ-1 '77,/0U j_ /0 - '1 3, J11-fI ~37,~LJ.7 IJ013't/7 - ,() I 3 J., 'f.J t, ,, 7 J. 7 B -r a, ~011. I ,DO l33 8 -, 0().J 3 /..!) v~ , ~ 4· - b'1,b~111- - 17, 867b s" U'.. · l_Lf-'f 7 2- -,JBf'"'-, r, o.SI qJ- IO , l l ~// -, 3 /J-7 / II .J_ 2..0 , ~ 7 i I/ -;3 )_j CJ 7 1) 3- '/J"]-3 7 - , I 8 ')...JJ +-,tJ]JoJ .J... 2.,0 , It:, 747 - ,1'17!J -, , 7 o, f -,-;J..";i..01,.. .1.. ~-- ,11.-;1,o --,0777,1, 7j tJ z..?.. 6.J j_ /ti .l. J_ JO I ..L. z.O - I, (;.,~3 4-__s· :..- .l Jr 7 00 S7, o i.1k J z_o '0 0 ,-,_ so, 7tJ't-~ -6/,1d-70 j_ 4a S2- ~,bOS11~ - ~ D 07J'-f'3 _L. ~oo R,_ ~o J,- 70° R, 51 -J,s7,3 /,/1tJ6 I 13 711 -, b Cf' 13 7 /3 7J .2. -, ~ 1I.J -:.5- - II II II I) - --