‘Aawm1WW' -=- - «WIIWMII s | | V H w , _ !, ‘_— _, _,7 fi— _ _— — — l! gé M _| I U) j 5R3“ STUDY OF THE THEORY OF STRESSES IN NON-PKRALLEL CHORD TRL’SS AND TRUSS WITH SECONDARY WEB SYSTEM ”Ptesée for the chrcc of B. S. Siex'en Antonoff I927 EFT-{Est} 2 o . A ‘ fi-‘TL.’ § \r ”' ",";20_ .n‘ 2;.{; _' 3|. n ‘; g’ I': V 1 Nil ’3" y.- . .' ,q ‘ -. _ ’3 : 'tx‘ .13??? . 7 N flitu": u. 1‘...‘ 4 A. J Wait! , .3 J... hdlufllqlWJIJ'in 1.311.511 .0 5.. “WV P §.Mm,§s..fl.rl"fnn.|n \ .u .un’Jfinu ... . 0|. 0 It: 'e 'l."’. ‘l. ... . 3. . M, . . . . .......u_nL”..-.mcsnhvt”. .. A: .g. . ~o n - .. . . I .v.‘ (”fissnlflnNu VINYHYK . .. o . .. M.$..v....5:&. .fitlb A)" ; .l .JAlJ. In. '51 L“l VJ - r . . .- ,7; ‘1 ‘we ,.. 7L: wuurw ;.mn. w.. km? “1w“ .1”. . .5. i.» . '3 . . _‘ e..."|'.v' ‘- I y! I “II .‘,| I V." , ,1' -~ .1. aV-ti . .1" 5’5! I 2:; V” 'x vi a? .i- «A guy i I n...)- «5. ”av... r f ‘l ‘ fl. ’5 r. ;e . . o . . . . . ~ . ~ .I . e. .. . . . .;. . 1‘ . u . . . e . .. . .. . . . . .. t I I. .. .. . . . p . x. I.. .. J. . .. J. ..,. KI)” . . . .. ,. .... .. . . . .2 . . J 3.3% .fiflu bu J .. .r . . . ...... . .._ v \.. . . . 4 1 h , . .. .49. ‘ -- v .2? . .. a... ._ .QLVJ. an altrumfl .9 . 9.. .n . ._ .. .a; In?“ . rm... . . 3.. h. . w.“ M stw‘HWz. . ..mflm\ ,. J . m... .. n . .J J... .x 1.x . \J . J \ .. In. 2.. V . I I); .o . . 1.. J. a. .v .qu..V.«UJO. )0 . ”We... +......._. _ . .1 av ”4% r Brio! study of the Theory of Btrooaol 1n Non-Parallel chorGSTrull 3nd Trull with Secondtry Hob Syntan rho Tho-1| snhmittol to tha runnlty at the IIOnIGLN smtms COLLEGI ‘ of . elsrionltnro and.App11‘d Selena. by sum £2019” 03ndidnto to: tho Dosroo of B‘oholor of Scionoo. Juno 1927 {EH ESzc 00 F . 25, I dooiro to oxprooo my doop thanks to Proroooor R.A.Gonld for hio uoiotonoo. omootiono and ooflnotruouyo oritioiom of my work on thio thooio. Thonkofirojloo duo to Protoeooro 0.L.Allon and 0.11.03“ for augootiono. atovon Antonot! Richie“ auto collogo Juno 1981. 83787 n m CONTEHTS rho Pnrpooo lonnPorollol chord- Trnoo' Trnoo with aooondary Uob Byoton Gonolnoion summary of nook: need. I Iho purpooo of thio thooio in not a diooortation upon tho otrooooo in voriono momhoro or otool hridgoo; nor io it on attompm to onplify tho titoroturo on tho onbjoot. Ma y oxoollont booko and nogooino ortioloo howo olroody boon pnbliohod in whioh tho oanoot or otrooooo in bridgoo io oovorod thoroughly..fly only sorrow woo, oo 1, oooooflionoly, lookod ovor oomo at tho literaturo on thio onb- Joot. that o tour yoio oonroo in onsinooring, an prootiood by moot ooilogoa in tho country. 10 a vory ohort oonroo in which onitioi- ont tilo ond ottort oan ho dovotod to tho aubjoot of the theory or otrooo in otrnotnroo. this oubJoot io brood, vory important and intorootin3. It woo thon my ondoovor to find oomo way which would onablo.mo to otudy tho onbjoot o littlo doopor thon it io gonorolly poooiolo for on nndorsrodnoto otndont in hio rognlor oouroo. Ihio thooio thon io. oo for no ito prooont writing io oonoornod, an attempt to writo down thooo fundomontolo which i oould loarn in this ohort thoo that woo ollottod to no. It woo novortholoao, my intontion in doriving'rornnloo. to writo down ovory otop, in ordor to moko tho work oinplo ond oonprohonoiblo. I an Ithio with opooial nmphooio on tho froo body nothod. oooouoo tho froo body mothod prinoiploo oo thoy woro taught by tho mon in our dopartmont in courses of Moo- chonioo, atoongth o£.lotorialo ond Thoory or struoturoo, oro roolly tho oono prinoiploo upon whioh tho onbjoot of otrooooa in bridgoo io baood. It io in no woy boood on any highly intriooto or oonplox term”. A .(. D‘- II lhon tho lonsth of tho opon oxooodo oortoin dootanoo (obout 175 It. i it io thon oonoidorod odviooblo to uoo vorioblo hoisht of truoo in ordor to ooouro groator ooonony. Ourvod uppor chord in ouoh o oooo io gonorolly oollod upon to onowor tho purpooo. rho loot idool typo of truoo with ourvod nppor ohord oppooro to ho tho ono in whioh, for uniform looding tho ponol pointo would toll_on tho roopootivo pointo of tho nomont diogran ourvo for thot loodina. But bridgoo oro oloo oubjootod to partial looding. fhio condition produooo on ontiroly dittoront ottoot on tho otroo- ooo. rrmo ooro otrooooo in diogonol woo nonhoro and oquol horizon- tal oonpononto in jop‘ohord produood by uniform landing, tho port- iol looding oubjooto oll diosonolo to rovorool a: otrooo. Suoh don- dition oollo for oountoro in ovory'ponol. Tho prootioo, howovor. tol'ht onsinooro to uoo tlottor top ohordo ond thuo ovoid uoinz oountoro oxoopt tho oontor ponolo. iho uoo at tlottor top ohord not only oloninotoo oono oountoro but it oloo oontributoo oonoi- dorobly to tho ooothotiool otroot noting tho otruoturo, ot tho oolo tino, roooonoblo ooonomiool. rho otrooo in ony top ohord or non-porollol chord truoo. oo ohown-in diguro I. may be dotorninod by tho gonorol tormulo, I 1' .OOOOOOOOOOOOOOOOOOOOOOC I whoro 8 2 otrooo in top ohord ll 8 nonont ot o point oppooito thot ohord r 3 porpondioulor diotonoo tron.tho top chord to tho enomont oontor. r- O... O. O ligure I represento a common , oight panels, ourved tap ohord truoo. ( noe'iho ihoory of atruoturoo" by charloo.u. opofford page 184, figure ids.) P P P “3 P4 ‘U5 ‘0‘ / / \ \ rue ' £7 \ I // // \\ \\ 31 ?% I“ 1‘2 L“ e4 w the V" W Iigure.i consider tap chord UIU2. By general formula (I) tho otrooo inthia member io, .H r where ll 3 moment at panel point b2, and r : Lgb ( fig. 2 I per- pendicular distance from the moment center to the member, the le- ver arm. The value of r may be determined by prOportiona from ei- milar triangleo UrUza and Ung b, from which we have, r : l : : h : a. it must be noted that u‘i- angle ”102‘ is right angled tri- angle and the value of a ao obtained therefrom is, o 3 [h - hllz or 121* thuo maxing r, when this value of a is oubatituted in prose- eding equation equal to, 1.x.h :3 E'hi’z*1§jr ° The otrooo in any tap ohord member, of ourvod top ohord truoo. may also be determined by resolving it into its horiaontal and ver- tical components. In figure 2 lot 3h roprooent horizontal oomponont of otrooo 8, and 8., the vertical. Then by equation I and 8 33h 3.0 O ooooooooooooooooooooooooeo 3 where O denotoo the angle between ohord member and horiaontal. Both equation I and oqzation 8 give the oamo roaulto. h ‘ 8 and 8h 2 ( from equation 3) nee 0 solving these two equations simultaneously we obtain, 3 See 0 I h and : To determine the stress in any diagonal web member as UI guro 3, cut section 1-1, as shown in the figure, and consider the part to the left of the section as free body. By general formula the stress in Ule is .00....OOOOOOOOOOOOOOOOO...O‘ 3 2 r whoro.lo is the moment of the applied leads to the left of soot- ion I-I taken about point 0, and r is the perpendicular distance from cite the line of action of “132' The value of Nb may be ob- tained by considering,in figure 3, part to the left of section 1-1 as free body and taking moments of the applied loads about point C. ( clockwise moments are negative and counter-clockwise are po- sitive), thuo, RId - 3( d e l ) - I (d + l) - 3r 3 0 and solving for azufl‘d*1)r*w‘d*l);_E-Ii oooooooooooaeoa - 5. QCII..0..‘ an... ‘ .-- ._. -- “a- but Pfld +1) -l(dfl)"-Rxd2lo substituting this value in equation 6 we have equation 4. namely rho value of r may be calculated from similar triangles 031.2 and 011113. from which, by proportions, we obtain. 03 z 0.13 z : VIII: 0113. solving for OH . 5 - ‘ . 011' r :(l 81,111} . “I (“h- + 1) (th * 12): By reason of similar triangles the value of deal may likewise be determined. consider triagloo “103‘ and 0021.8 ( Pig. 8 ) in which 01.3 : U312 3: 111a : Ugo “I - - hi cig.e+si-tee A4 (11 - 1:1) Another method by which the stress in diagonal web member can be determined is in terms of its vertical component. has, resolving the stress 3 into its 3v an 8}; (vertical and hori- w‘sontal ) components and applting general equation the vertical component or the stress in 1111.: may be expressed by equation, 3's no ______ oeooooooooooooooooooooeos ‘I ,whero lo is the moment of all the loads applied to the left of section 1-! taken about point 0, and t is the distance from 0 to the panel point 1.8. ' - 5. «two ——.~ *.~_.u—-~—. .. - c O .— ‘vlflcm \ “C . .UIICO... ll 0 C O. .- t s I I \ ___.._.. .. L I I t - — .~ . I m... .—..- *0. w-t fl 0 U—a—‘gm- i S 8 3 8'3”. .0000.eOOOOOOOOOOOOOOOOOO" where O is the angle between diagonal and vertical. raking moments about point 0 the results may be obtained in the following form. moealcumqa) when) “319 l - PM - (I ‘5 Ml. But 3‘ ."+P) .V is the shear on section 1-1, and" 1- DH is the moment [Io about panel point Lo. Arranging the terms of the above stated equations the value of a. may be expressed by the following equation. no . Vd - “1.0000000eeooooooooeoeeoooeo 8 lets: it: further proof and criterion of this case see article 34. page 51 of " the Theory of structures" by Charles 1!. Bpof- ford. q ‘ to producer; maximum tensile stress in the diagonal web member VII; full panel loads must be applied to the right cm the sec-- tion 1-1 and no loads to the left of this section. considering the conventional method of leading, the value of no in equotion 0 must be positive and as great as possible. Any load to the left of the section 1-1 will produce negative moment am there- fore reduce the positive value of no. In present case lot n 8 number of panels . m = number of panels not loaded i’ : weight per panel on lower chord. (Upper chord loads will now be omitted in order facilitate the work and simplify the formulas. It is my belief that this stop ”7‘?! 56 fairer/7 - 7 - oeosnoc O 0 e' O I. . ‘ O assessed as k as. I Vt scoot eooseoe II H H H without any detrimental effect to this work. 150031180 the upper chord loads may easily be taken care of by the formulas for the lower chord loads and the combined effect thuo determined.) Returning back to the problem and taking moments abount point 0, figure a, we have, no a BIG - 3r eoeeeoooeeeeooooeeos 9 from equation 7 3 z 3'8ec9 and (111' 4» 12¢ BecO ' a a 111 Combining these three equations and solving fcr 8' we obtain. R 1111 .4_——-_- .OOOOOOOOOOOOQOOOOOO sv . I . Io rth’f + 13H: rho value of 31 can be obtained by taking moments Rz ( right reaction) of all the loads applied on the truss. Using notations stated above in this work the equation is 3 I t3—“(11'n‘i)(n-.’eeeeeeeeooeoeaoeII . 2nl - Substituting value of Eros obtained in equation II, in equat- ion 10. and remembering that th 1" ”‘4 - (h: + 1"” equation IO becomes 'd {n " m + I)‘n . l’eooelz u ‘ II I an assesses-00" .powsaOODeOO A. est‘.°"‘ esas\- ‘ 00.... so. i 0 ll the diagonal web members are subjected to compressive stress as well as to tensile. For maximum compression in dia- gonal Ulla the momentMo must be negative and as large as pos- sible. luch condtion can be obtained when the negative moment due to loads applied to the left of the section 1-1 is greater than the positive moment produced by left reaction hp, It is that placing loads on all panels to the left of the section [-1 and no loads to the right at that section will produce maximum momenteflo. As before, consider n : number (1 panels in whole truss m 8 number of panels loaded I 3 uniform load per lanel the maximum compressive stress by equation (I) is ( vertical com.) s, o 2:59. 4— t By application.of the same method of analysis as was used in case of tensile stress it is possible to express the vertical component of compressive stress in this diagonal by the fol- lowing equation. 3' '.:L_( m . I)(d * ln)............13 ant - Sinoe equations 12 and Is express the value of vertical components of stresses in diagonal members, it is than possible, by application of these tww formulas, to obtain stresses in any vertical as‘Uzlz. tor all loads to the right of panel point Lg” the vertical member 021.2 is in coupresstion and equation =3 [2 O O — .- —. .- a— h - .- . o ‘ fl 0 . o a - ‘ - _ ~ ‘ _- - ‘ i -- a D ' _. a . , I . ~ a— applies, while for all loads to the left of the panel point U2 the member is in tension and equation IS applies. Since,in these equations, the positive sign indicates tension and nega- tive compression the application of these equation to the ver- tical members involves interchange of signs in the right hand member of both equations. iII Stresses in Members of Truss Containing Secondary sob System. [or spans of considerable length the maximum economy is secured by means of subdividing panels and adding secondary diagonals and verticals.The method of determining stresses in truss of this type becomes somewhat complicated, especially when dealing with secondary system. The application of ordinary methods of Joints, moments and shear require a little modification. In some members however the stress can be obtained directly by one of these me- thods. The methods of determining stresses in various members of truss with secondary web system will now be given. 1.. Ll4|L2 I«511.4 1‘62) 1.5 1'7 L3 1' 9 ' ‘ ’ 3min ‘ I ) J ) ) A ) ) ) ) ) ) ) l ‘ ‘ 2 'I 2 we '4, as I5 I, 'e '9 gm ’18. ‘s Iiguro 4 represents onetype of truss with secondary web system. -10- The upper chord stresses in a truss similar to the one shown in ru. 4 may be found by using general equation I. consider the upper chord 31113. By general methods of moments the stress 8 is, l s s I“ h OOOOOOOOOOOOOOOCOOOCOI‘ where ‘14 is the moment at panel point Lg due to the applied loads and h is the height of the truss. The stresses in lower chord members can be found likewise by taking moments abount the upper chord points. Thus, for exaIple, the stress in L31“ is foung by passirg section 1-1 113. e, and taking moments about point 01. The resultirg equa- tion may be stated in following terms, sxnl-vxxievsxi-srh:o Solving for S, we have, 331.'I"z‘331) [VI -= soosooooo 15 h h ' C l 3 length of panel) Stress in 03115 is equal to the stress in U3U‘ and same method of analysis may be applied in case it is desired to determine this stress independent of the stress in 0311‘. The stresses in other chord members can be determined by w methods similar to the one already describe. The stresses in subvertioals as [11.1, 121.5 eto,. are equal to the loads applied at their respective panel points. tor example, the stress in "III is equal to II. The member a in tension. -11- oessssssw -.<- O Tho stresses in sub-diagonals as Leila may be obtained by resolving- to stress 8 into horisontal and vertical components and finding value of vertisal component by method of moments. b’ H /0 v '2 . . 1'1:- 5 In figure 5, consider the forces acting on Joint [2,. An equat- ion of moments about point 1‘4. gives, I211-8'13120 Solving for 3' we have, 3' 2 1/2'2sooooooeossesoeseoesss 16 Hating use of eqaution 3 previously derived we find 8 to be, S : S'Ssco s I/ssgsoce ( compression ) The stresses in lower ends of diagonal web nmmbors are equal to the product of shear in their respective panels and secant of angle between member and the vertical. Stress in H3136 may be obtained as follows. The shear on section 2-8 in Fig; eiis, - V = 31 . m. Osssssesasssssssssssoo. 17 where m : number of panels from left reaction to the left end .12- Osooe-oe. FOO$OOIOA sseo 11 00‘. a C U 0 D it out panel containing section 2-8, and RI and I as before. Solv- ing for BI , I 31 .7“ - I). (n : number of panels) Substituting this value of 31 in equation 17, we have, I v .T(n 'I' 2! " I) oeseeeeeosessseeess 18 Therefore the stress in 1151.6 is, i 3 ‘ T (n u an - I)fl.oa ooooeooeeeos 19 where O 8 angle between member 11315 and the vertical “335' Stresses in upper ends of diagonal web members can be de- termined by method of shears. (liagonals as 0,112, Salaam.) Consider diagonal 021:5. Pass section s-s tip, 4, and consider the portion of the structure to the left of this section as free body. The shear on this section may be given by equation varl.(s;esg+v3+v‘) This shear is distributed between Uglla and 121.3. It is evident the n, that vertical component of stress as in 02113 may be ex- pressed by equation, 8' 3 V O a; sessssoeoeooosoess 80 where V I shear on section 8-8, am Si ; vertical component of stress in sub-diagonal 131%. - Equation 80 makes possible to determine the stresses due to uniformly distributed dead load. for as explained before, 8 . S‘sece msaeeeoeesowsses ' .-e'. . ' Q ' o .. . .\ v- ’M, ' u _ . 0 a - . e s e \-. "' ‘- “' . w I t 5 . ..... . . O 7- \ -- . c. . ‘f v .‘ .7, - _ m 1 - ' a ‘ , . . ‘- «U— _ . ‘ i - A _ ’ L .I not \ ' . - Q . ‘V ..- .0 v I - ' d n . C A - ‘ as I- ans—”.2 L- . 1 “ba— - I ' r . t . A t N t ,- — I j . fl 0 . ll for maximum live load stresses the position of loads must first be considered bdfors equation can be applied. Either the method of moving up the loads or the average load method may be used. These two methods are given in detail in chapter III of"Ths Theory cf’structures“ by charles no Spofford. Equation 3 may be applied to determine the maximum live load stresses after th value of V and S; in equation 80 had been determined for the po- sition of loads producing maximum stress at that section. .Another method which may be employed to determine stress in 02113 is the method of moments. This method will not be given here because the reader can find at a glance the value of SV by taking moments about points 15 and I6 respectively. Stress in verticals as U234 is readily seen to be oaqual to the vertical component of‘Uzlz plus the load at Joint U2. Thus, the stress in vertical U132 can be determined by the me- thod of Joints. 3 I 2'1 7 1/8'3 x / \ I 3n I L3 4' 2 l'ig.6 Consider Joint L2, Fig. 6 as a free body. Resolve stresses 3 in sub-diagonals 1111.2 and 1121.3 into their vertical and horizon- tal components. applying principles of equation Id, it is readily seen that the value of vertical component of stress in 1111.3 : 1/2“. Likewise a, in 1121. : i/2u3. an. total stress in vertical 011.2 is thesum of these two components, plus load '3 shish is at Joint L2. the above result may be ' expressed by equation. 3 : 1/2'I + 17‘3“: * 1/2'3 Oeeeeeeeeeeeea 21 l‘or equal loads equation 21 becomes, 3 : 2' aeeoaeeeeeeeeeeeeeeceeeeeeeeOOOQ 82 Bquation 28 gives value of stress in 021.2 produced by uniformly distributed dead load. In cases where the stress is due to concentrated live loads this equation must be slightly modified before its ppplicaticn can successfully be made ft: correct resutls. Let It, 15, I5 and 11;, in lie. 4, represent moments at points 1.1.12. L3,-and L‘respectively due to the applied loads to the left of those points. Let 1 salength of panel. Ehen by general moment equation. - - n‘ - all}; + 115 'z' . I l aeeeeeeeeeeeeeee (" 1 .u' - ' H' a . 2 as + ‘ .O...............(b’ .. 1 I ' 'I = - 2H; + H2 eeeeeeeeeeeeeeeeed(°’ 1 ~15- .‘D-OIOOCOQUO IIOODOIIDIOa. aaaalsea .“ Sr} 0.00.00.00.33... \"';Ides-.e aseoe \;,eeeeoeeeeeeeeeeoe “I -Mh»-.. ,~.'t-. . . Os. mah— O ’4‘ Substituting equations (a), (b), and (c) in equation 22 we have, '. | 3:)“. an; 21 lots: for complete discussion of the above principles see “stresses in named structures' by 11001 and nine, article t1, page 138. . l The stress in upper end of end polst 1.001 can be deter- minedby pas-ing section 4-4., s‘ig. e, and considerirg that portion as free body. Applying general equation 1 we have, 3: ‘L2 1‘ where 1L2 ; moment at point L2, and r = perpndicular distance from point L2 to the lim d action d 1.001. -15.. O‘DOOCOCOODIOOOI IV CUIOLUBIOI It was my great scrrcw that limited time did not permit me to indlude into the present writing of my thesis all that I originally planned to write. It was my endevcr to write. in ad. diticn to this work, on the methods d stress analysis in trus- ses with multple web systems and on the methods of determining stresses in cantilever bridges. The subject, after all. prcvmde to be too broad to attempt in my present work. Because. knowing a subject and being able to write on the subject so that others may know it, are ’two different things. so be able to deddsibs the short and comprehensive form a subject so broad involves a neces- sity.dn.my part, to study the subject thcacughly. It would also add at least twenty pages or written matter to the present work. fhis conditions are a total impossibility for just now. I must conclude this thesis set it is being content with the fact that I had a splendid opportunity to become shmeqhat familiar with the subject of stresses in advanced types of structures all be able to appreciate the vast amount of work and the difficulty one en- counters in attempting to write on the princples of stress ana- lysis. . I do, however, feel that so far as I was able to write . I have fully fulfilled my expectations. I have summarised a fdew fundamental principles which, if properly applied, will enable one to determine stresses in any statically determinate type of bridge truss. .17. 7 SUMMARY 01' 5005 USED MR STUD! LN) WWO]. The Theory of structures. by Charles.M. Spoffcrd. stresses in Bramed structures, by 11001 and Kim. .lcdern.rramed structures. by Johnson, Bryan, and rurneaure. ( vols. I and II ) Handbook to: Engimers and Architects. by Edwin: (rcreign Language book.) -18- y ‘— ROOM USE ONLY I c . IAII’IIIIII- .IIII. ‘. 4 . Ii ‘ .u ,1! 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