Formulas, Tables, and Graphs for Trapezoidal Reinforced Concrete Beams Citation Howard, Charles Stetson (1923) Formulas, Tables, and Graphs for Trapezoidal Reinforced Concrete Beams. Bachelor's thesis, California Institute of Technology. doi:10.7907/RZB9-SP40. https://resolver.caltech.edu/CaltechTHESIS:06182015-150631884 Abstract No abstract. Item Type: Thesis (Bachelor'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: None, None Defense Date: 1 January 1923 Record Number: CaltechTHESIS:06182015-150631884 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06182015-150631884 DOI: 10.7907/RZB9-SP40 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9028 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 18 Jun 2015 22:19 Last Modified: 16 Aug 2023 22:05 Thesis Files Preview PDF - Final Version See Usage Policy. 7MB Repository Staff Only: item control page

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'f- 3 ~c ( Kd'J'(.bg ~3b) = c =ffcf?y c//}" /~ - ~[fcxPa. r ~K2(b-J:>2)Jd~ -;j kif/ (/(t/)2. ~ ( k~ Jo = -fc;rbg clK +- f;:K2. (b-b 12 )t:IA. fc X" K~ 2 k~~ b2Jk"' fcx ]kd(b-bz) + 2 kt?! () : fc kd ( P11. ; 3 J(~d,. 20 ) re (ktT/ { /?~ 7' 12.b.J 6 = Kd(b3 ~J.6) 2 ( b,z r-26.J 0 5 /?fl :: b/ r- ( 6-h/ }{1-,1cJ EJul' = br *fb/-h/ :: b -k(b-b,) = Kcl[br- k{b/-,b}~3b y J?~k(b/-6.J r--<.6 k d b r k.. ( b/ -b) ] :. Xa = J [-4 z 3 b 'I-/(_ { "/ -bl llv"hen b, = b (lf:pcf-gn911/t:rr Pt'e?m) yo : k d [-4 b 'I- /( ..t o ] = k ti _:J_ - 2 k / /'• 2.. p=. f1 cT;j a'J ::: Z f-1= 0 TAJ-yo 9bro . ,,.....,::: : IT b;b _/ £:r p { Jn•-i;J/} d 2. or C: T C::: f~kt/( 6-l r2b) = T:: a's Ts fc kd(b. ~ 6 -1- 26 1 = / f'sec/(b~6,J ~ . = ,·. l3ur b2 '7 "k{.6, -b.J f~A:d' (bi-k{6/-bJ) = f'sp~(P?-6,,.) or ~~kd [J b r* (b, -6)] = ~~ e'{(bf-~} Z3vf ~:: -h/1(/;~J , '. .k ::: -f=c n IS -1- h ii: . 1'1~/·e/ [ b- Q/'c:(~-b)] - • • b (J51'-hTc) J -61'H* - rs pd(br~} 2. 6 . ·. p (nf;/el)(Jb;'st-3bn'lf)-nb, -f'c -1-n b-h:. or 6(rs -r/7 rc )2. = 3 3 2. (.J'n b ~sTc ~-1- ~nfb - /1 ~b, = 3(.,CS -/-/1 rc} ~rs ( b 7'-b/) 11 Tc l.(3 brs 7'-""f'1 b /7 ~ nc) rc = rc re - ~75- ( 6 rHTc) 2 ( b-rh,J When b, =b (7f'pc/-~n9v/n/ bt"nm) 11/'f i.(!b;!sr- ~nbrc -nbrc} p ~ Jfs (~ ;1-H}f)~ { b ~/,.) 11 re~ 3(i:J = 6 .b rs ( 15- ~1?6 re) T:s r "'rrc} z /7 7f z. ,., .J;c 'L ,,/£ - - 26(61-nf'eJ 2(-fs~~nT.sfcJ Jd::: d-(lat'-ro)= a'-kd-1'-/;, ~ cl - I< p/ ~ tr.L [ '</ b i- .k ( b/ -6) 2 3.brk(b,-b} 1 = d-kd [1-..L[~bt--l<(h/-b) Z Jb r-~ (6/ -bJ .. • •• J [ / [ 4 b i- k. ( b, -b) ] = /-/(, 1-2 5/,J-1-f({p/-b) · OK. . J} t J When b, =b (leecrnn9ulu;-- P~dm) j::. /-I< (- ~ xy +I} = 1-f k. O.~. 7 .:J~/V//19 for /( rrom C = T fc Kd f:rbrk(b,-b}] = '1fc(l-k_J pd(b-rb,) 6 2 k.. 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JC(b,-b) 1-f(~{b)-pn(l-l<J(b-rk) =o 3 JV'/c = .ri:;jdz.[.;b ""* (b, -b) J /Vis = p6jd2.(b'l"b,) 2. 9 .:50/v/119 for values of p b, =o.2b 17=15 nfc2.(3b'/s t--4xl5brc - /5)( o.zbfc J'= Yfs(6-rnrc)Z(b-ro.2bJ = /,,fc2.{3brs f-.7/Tc) .:Yfs (f"sT--/frc)Z,-r/.2b _ 1f"fc2.{fi.r/9Tc) - r., (Ts f- / .rfc) 2~ /. .< 61 = o. ~b 1fhZ-frs.;-17hJ P = rs('h-r/.rrcJ~xJ.6 ;.r/c~(4t-16~) b,-= 0 .6 b f> = f.s ( f; rl.J7i:)2.x /.8 b, =0 . ab ;3-/;2(fsr-!6rc) /> = & r fs -r1.rrc)<x 1.8 For Vt1/ves .5ee ,Fo//ow1119 Table. TR8LEI fc fs fl- rh l§flJk r1 B c D I 12Xlt> 3 2 11 " 3 100 16 II 18 ,, 'f 20 ti .1' 6 12" 1-f,, 7 8 ,;oo 16 9 /8" II /(J 2~" II V-? ,, ; . , . 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(b, -h) rK2.b - pn{1-k.J ( b +-kJ = o 3 b, -=0.2 b 3 -3.7.Yl<:i.- 67.3p/\. -1-67.3p=:o 3 - 5/(2. -/O~,t:J/( r/0.1"p =o /( b, =0. tf.b /( b/=0.6b K 3 - 7.f' I<.~ -t3opk. r /3op =o t>, = 0.8b /( 3 -1.s:o 1< 2 - .r;orpk.. t-48l!Jp =o Volves of K J % Q'S ~o /~'1r.S. e;rA4' y 1n IP6/t? ZL w-~,,...,.... To«nd .all of the ;;qua tions v1 r~ o f the fo nn (k 3 :-xk2 - yk + y == 0 J, t hsre b e int: but four diff' ::n ...;n t values of x ( iil.11 gr <J:ater than 1) fo r the who l o set of prob l .J!ns , :lnd th·3 v ;;. l u::s of k being known to r an::;-:1 b ct-_·;,'.!on .270 The e quat ion was r~writt en i n tho for m k 2 (k-::c) = y(lc - 1 ), and .550. o r since both x and 1 •.•1ere great~ r w!;en c c dividing t hru by 1-k , y by t h~ s mc c siv: v 2 l uc s of k co::nput~d wer·3 next and th:rn k, in th~ fonn lC' (.·-kj - y(l- k) , = 1i;: 2 (x- k )/ (1- k ). .270 , tabul at-:~d . .280, .2;10, and so on t o k "" .550 computat io,1 Vi~ s eas il y pr ~ for:ned 'l.'h on a 16 place ;.1o nroc by sett inz x-k up on or.. tn.e rizht , a nd buildin:; 1-k up to li;: wh~nc0 i:h8 values of y g iv en 2 th~ left o f the keyboard , 1 -k in the middl'~ c!f tho lowe r J.i a l, y app-:::ar::J ~t t .ht: bft of tile lower di -.1. For exampl e, for k = .43 (x - 1':) = 3 .32, 1-k"' .57, and k 2 = . 184:i'(found i n a t ~-, ble of squa res), k ·' .)yboard 3320000057 and b y forward tur11s of the crank build 57 up to 184 ':1(023 ) i n ' t th~ ,nid O. l ~ of th~ low e r dial , when y:;:: l.076 'il{ 74~) a p9ears the 1 ~ft of t he lower dia 1. Go py y = l . O77 , c 10a.!· dia 1 s, and change (k-k =3~31 , 1-k = . 56, and Build 56 up to l ';.1 35('376 ) in the miclll.l ~ of the lowe r di a l keyboard s e tup to 3310000056 for k = .44 k 2 l · 3·· = .:J oJ. 1i1nd when y = 1.1 440 (00 1) a pp::2 rs a t t !1e left. Valu·:?s of ;v fo r inter.:iedi a te 3 pla c e v :alue s of k w~re t h en computed by proportiona l y = 1 . 0 ?7 for k = .430 :md y = 1.144 for k = .4-40 , di 2 l board 67 (1144 - 10 77 give s y i nte rpol at ion, th:ii.t is, for example , having fou.nd. obt a i m~•l :r1e nta. l ly ) , on~ f o 1~,.1ard 10770 and. key- tur n of t hc "crank = 1.084 f ork = .431, anothGr turn e;iv e s y = l. 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