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AND A‘PPLIE'D SCIENCE 33' ') Inho R. 333:” Candid“. for the car» of ambulat- of science Jun. 1930 THESlS mopfi gaggqgrg Introdnntlan consideration. rumpus. Analysis of Arches [0601. construction of Models Polarizing Statement of Theory InVIutxgatlanI conclusion- Bibliography 1018“? Wm l oonoroto oroh oonotmotod with find on“ to ototlool- ly lndotomlnoto ond lo onolynod by taking lnto oocount tho olootlolty of tho matorlolo Tho oroh lo moroly n can“ boon following, genorolly 1n bridgoo. tho our" of o. proton. olllpoo, oogmontol or o oo-iolrolo. rho oolootion of o onrvo in loot ouoo lo matter of oconomy. ' In orohoo of stonoo it to found that tho lino or proosnro pouoo through tho middlo third to ovoid tho tondency of tho Jolnto to Open. In tho ploln oonoroto and: tho lino of proo- onro ohonld oloo you through tho middlo third. duo to tho plootlo notorlol nood in tho oroh. Lo thoro olll nloayo ho oomo tonolon produood in tho oroh 1t 1o nooosoary that tho oroh to nonollthio to withotonl tho tmolon nnd olloo tho lino of proosuro to pose outoldo tho middlo think But «It. to tho loo otrongth of oonoroto, roinroroing to uood to pot-alt o stator taxation of thin lino which to of llttlo oooonnt thoorotioony oinoo tho diroot oomprooolon in tho oroh oouolly oontrolo to mob on oxtornt that tho onoublo otrou in t!- oonoroto pom-1t- of but o. mall unit tonollo otrooo in no otool. 1‘ 1o obviouo tint tho otool oddo grootly to tho rollohlllty of tho oonotmctlon and :23on pooolblo . highor vol-kin; otrooo in tho oonoroto than could properly ho ao- ployod in tho doaign of tho plo1n oonoroto otruotzro. rho higher waking otross produces a thinnor ring and lightor shut- unto. It! oonoidoring tho action or any otruotnro undo:- o . loot mother 1t lo ototloolly dotorninoto or indotornlnoto 1t lo nooouory thot tho goonotrio roqnlr-onto mood on tho ltrnoturo to fulfill“. ouch oo bovine oonotont gun longth, oto. no on of tho coupomnto 1n ony dirootlon of oll torooo ootlnc on tho oholo otmtuxo or any port d tho otmoturo to oquol to ooro. ond tho sum of tho nononto of tho molo otruo- tnro or any port to oquol to ooro. rho rotto o! otrooo to otroin in tho difforont lol ports of tho otmoturo mot bo oo rolotod to ootloty tho preportioo of tho motorlolo. m Iho mlyolo of on oroh by tho olootlo thooxy oonoioto ln tho finding the thruot, ohu. and bending nomont at tho om ond ot internodioto oootlcno In tho arch Ans or ooh rib, and than finding tho otreaooo resulting thoroflon. lt lo ovldont that any notorlolo of construction, oo- pooiolly oonoroto during tho tirot too nontho otter fobrlootion do not follow the Inn upon which the theory of olootioity 1o hood. Son of tho futon 1n the onolyoio vory. oo tho ylold O! I ltroacod body Ilth tlmo om tho 1'8.th of stress to otroin. tor thio nood I..G. atroub hoo propoood mothodo of onolyoio tornod no tho ”thoory of pluticity" ond tho'pro- poood thoory of olooticity." rho uothodo provido o gonorol ototo- loont of tho dioplooomonto ond deformationo in otrnctnroo to fit o motor mbor of proportioo oi‘ ntoriolo. m. invoc- tieotiono undo by L.G. Stranb oro boood on oxporimonto of pro- cioo ond occuroto noonremnto. who now theorioo oro dovol- opoc oith tho idoo of oupplomontinc tho ordor of theory of olooticity. Obviouoly. tho noo thoorioo tond to m. tho on in- odoquoto tor otodying otroasoo in mmo .11: indotornimto otractnros. any undouhtodly mm for o ooro ,ociontitio otmiy of orchos ond till omtuolly romlt in bottor ooonomy in urine ’notoriol. oto. on. conotruotion or orchoo by tho onor thoory hoo not boon inodoqunto but hoo boa: ratio by tho noo or rointorcomont. It io oloor thot tho otrongth of mtoriolo or orchoo ond otructuroo 1o mmh diocuosod. B, graphical nothodo tho linoo of thruoto con bo ohon tor mriouo loadingo in orohoo. tho trophicol mothodo chow Quito occurotoly tho distribution of lovo through the orchoo, but not owtly. Dr. actor'o om- porinontol orronganont using polorioed light for otrooo dio- tribntion and dotormination by tho uoo of trunogoront nodolo con bo nood conveniontly in illuotroting what occurs in oollnloid Iodol orohooo Dr. Ookor'l photo-olutic method offordo o, new: of diroot amouramont or otrooa in mboro or oolloloid. mo noono hoo boon nod to odvontogo in tho mechanical linoo for .W toot piocos. can. otc. boo to our orudo orrongo- unto out lock of tino I. oro unoblo to set quontitotivo roonlto. but I. study 0: tin otrooaoo can bo bod ond tho information wood from thoao poloriution tooto oxford o bot”!| concoption or o otmuro ohm placing rointoroonont. W1 no 31313.1. 01" whoa by tho olutio theory method ll tho moot cannon and will bo briefly mtlinod. A concroto oroh with {nod ondo io otnticnlly Motormimto. Tho onolyoio anoint. in fixture tho thrust. ohonr ond moment ot tho crown and intormodioto points. moi-o on throo unknown ot oooh flipper! ond throo minoono for tho deflootimo or tho orch mating lit in .11. it tho oupporto. tho Hominid“ tho diroction on! tho point of application oro tho tirot throo unknomo and oro dotominod by tho princ 1plo or ototioo. mo othor throo consist of tho dicplncozzont horioontnl. vorticol and mint. mo horilontol lo the chargo that ocean in tho open lmgth and io donotoc byAxao . who vertical io tho rolntivo displaoamnt or ono and to the other on! io donotod byAy=0 and tho angu- 1 lar dioplocosnant io that ohich takoo placo at tho too ondo dioro tho angle botooen tho tangents to tho arch axio romaina unchanged and is denotod by A k=0 . no method of procodnro folloood in tho arch-r113 design no givon by Boll io no follow: 1. 2a 3. 4. 5. 7. loom a thicknoas of arch ring at tho crown and springing. Lay out the curve assumed for tho introdoo. lay out a curvo for tho oxtrodoo to givo ao noarly ao poaoiblo tho aasnmodring thicknooo at tho opringing. Draw arch axio botooom tho oxtrodoo and intro- dOOo Divido tho arch axio into an oven nmnbor of di- visiona ouch that tho ratio 3/! is constant for all. Oompnto tho flood and lioo lodo. Oompnto no, to and lo at tho crown for tho dit- i'oront conditiomo of loading. Drao tho forco polygono for tho difforont con- ditiono of loading and the corresponding oqnili— brim polygon, or lino of proosnro. Dotoroino tho thruato, shoaro, bonding Incont- and ““11"“ ciatancoo at tho contor of tho 3/1 divioion of tin arch ring for tho arm-m conditions of loading. 10. canto thruot and moot for variation in t-poratnro. ll. Oupnto throat and oo-onta for rib-ohorton- No 12. Wino tb throato. ohoaro. and auto duo to tho ditforont condition of loading oith tho thruoto, ohoaro. unto duo to tqporatnro and rib-ahortoninc. . la. canto tho anim- atroaooo - ans-onion in tho oonoroto and tonsion in tho otood - dno to tho than” and “to. lo attolpt ia horo nado to dmiop tornlao dodncod in toxtbooko. a. oquatioao at. tho ouao ao thooo givoa by “lo boflootion tomlao aro fir-t aim booaooo at can» too in Ito arch thoory. no natorial oonoidorod to rob-- forood ooncroto. tho tornlao apply to onrvod boano with a radiuo of curvaturo largo couparod to tho dopth. "no arch io aoauaod with ono ad fixod‘. to angular chan‘o at tho crown ao rowooontod by limo liodonotodbykorxa Z: :1- é‘T-X <1 lino 8 chow tho oano bonding with tho oo-pononta or dotlootion givon. nanoly. tho horiaontal and vortical noto- nonto. norioontal norm" 1: A! a Z: 5 Vortical nomont a AyF-Zig Y 5 a 5 x— 44 x F / v T C a ta ”+14%! . ,/7-" 9 , l ‘2’,’ \‘4 ’N ,0; MI A' c z/ _ __ _ _ 9 X a Y Dina-o8 limo-o 8 represonta o canonical arch out at tho oroon to ohow tho acting torooo at tho an notion. 3° ' FEW." #1: 27-1. + In)?! a [:1an ~ m8] '0 o Zhl- :3)! 22: no I 2h + '3’ '- mag: an no ulnoo of tho thrust. ohoar. and monont boing found from. tho abcvo oquutiono. tho moment or any ooction of the loft contilovor io I o llc+licy+m '- IL and, at any notion to tho right cantilovor, ' "Clo-{'Hcy-Yoo-‘i no 3% . tctpmm zflth? - (21):] loo-mgr... tho bonding Iomont at any point is lam +30, WWW o in ‘ Inf; - (27)] ”.11. 8. 10 mm W So for. an analytical nothcd is giwon by which tho archos aro dooignod. rho nowlodgo of tho oxioting conditions of otrooo distribution is still vaguo and so to supplant this. throo typical archoo woro mado of tronoparont natorial for tho purpooo sooing tho oxioting oonditiono in tho loadod archoo. it thio point rocourso is mdo to oxporinant with polarisod light. - Such oxporinonto woro nado with tho oliptical. parabolic and tho circular arch. Tho archoo woro cut from a rough pioco of colluloid plato. Tho plots was oandpaporod with throo grados oi' sandpapor to oncothnooa. rho shapo of tho archos woro first nado on ordinary papor and than pastod on to tho plato from tho throo nodolo woro out .1: with a coping saw. rho rough surfacos woro snoothonod and tho arohos polished, tint on the cunt” and than on broad- cloth for tho finishing touchoo. Supporto woro Ila. or wood and spandrol rolls woro constructod ovor tho archoa as shown in tho pioturos. Tho loading from the walls woro oonoontratod to fiwo points oqui-diotant from oach other. Thio was dono by ooldoring tin-load to tho shapo of a lump at oach point of ‘ concontration to a strip of tin that was nailod to tho curood aux-taco of tho wall. i'hoso points woro than ocrapod to sin 12 such that from.oooh point an equal load would be diotributrd. Tho accuracy of loading from.thcso points is difficult to dotorisino. Homer, tho points are filed by comparing tho bands producod by polarisod lights Various othor difficultios aro oncountorod which also computing tho snot strossos in this can practically impossiblo. Ono difficulty onoountorod is tho sis. of tho archos. Tho: aro as follows: xix-mu span 5 inchoo ri so 2-1/2 inchoo thicknoss at cm 3/8 inohos springing 3/4 inch W span 5-1/2 inchoo riso 1-1/2 inchos thicknnos at arm 1/2 inch springing 1/8 inch mm filth 5-1]? inchos rm io/s inoi‘ios thicknoss at crown 1/3 inch springing 2/3 inch 13 rho thiornoss of tho material in all is about 1/4 inch. this shows insodiatoly that tho archos. oven though small. aro onggeratod greatly in thickness. so tho strains shown by tho polarised light will also be greatly exaggerated through an area that night other wise show a wariation in the lines of strossos where one lino would coincide with another and balancing tho rosulto to sons extent and porhaps leaving no stress at tho particular point. Another thing that onters arches and is not shown in tho nodols is tho roinforcing which in itsolf takes up stress. Eowovor. this experiment will help to visualiso that happens insido tho arch. While in tho construction another difficulty arose but was remodiod. It no noticed that initial strossos woro produced at cortain spots such as. for instance, at tho springing lino. Each of the arches had these initial strossos at that ond of tho arch. From this was inforrod that tho original plate from which tho arches woro out an initial stress was produced. It may be possible that in the course of construction that some parts woro unoqually balanced by tampon- ture heat. hence causing a stress in such a portion. the stresses woro oliminatod by placing tho celluloid models in water of rising tauporaturo to a point noni- the plastic stago of tho oatsrial or noar 180 degrees. The nodols were allowed to cool in this wator for a period of some five or six'hcurs, thereby bringing the material slowly back to its 110an state loosing prac- 14 tically no strains in the saterials. W l silplo box is used for experimental arransuonto that contains the source of light and tho reflectors. A few rows of blue incandescent laps placed above a diffusing screen reflecting plane polarised lifit at a block mirror. on. aim:- is m at as' to the sirmiu screen or to the horisontal. rm thissirror the plane polarised light is passed through the transparent Iodel to an analyser and to the eye of the observer. rho analyser is uother black lirrcr set at 45 decrees to the vortioal. as into that appears on the onalyser appears eolored according to he strained condio tioss in the arch. in onstrained part will appear black and other parts appear in differut tinted colors according to the count and kind of strain. whether the strain is tension or compres- .1... this mum is much simpler nun that uood by Dr. deter in his expert-atom Stretch). , / gibfilack Mirror 15 W Light, according to phys ice, is an electromagnetic disturbance. The light waves travel in all directions. Plane polarised light travels only in one direction in one plans. It may be clearly understood as passing through a narrow slit, immersing with a single bows. If another plane containing a similar slit is placed in the beam of light coming from the first plane and is made parallel to it, it will then allow the same beam to pass through sithout interference. but if the second plane is placed perpendicular to the first it I111 allow no light to pass through. The coloring effect produced in the models may'be understood as rotating the second plane 90 degrees. being black when the slits are at right angles and passing through the colors until it coincides with the beam and giving'the shite ray of light. Plane stress in a homogeneous mediun.produces no color, as the incoming rays of polarised light coincide with the lines of stress. P-S‘l‘r‘ess q— sf'r'ess Figure 5 16 All systoss of plane stress lie in one plane and are represented by double system of curves intersecting at right angles at all points. These lines may be called Principal Stress Lines and are denoted by p and q. There is also a three dimensional stress but in our study we consider only plane stress. Tangent at every point on the line shows direction of the principal stress. is stated by Dr. Coker. "Light is thought of as being separated into two polarised coinponents with vibrations at right angles and the direction of too vibrations coinciding with the prim ipal stress. if principal stresses are unequal one of the vibrations is retarded with respect to the other; this rolar tive retardation being preportional to the principal stress difference at the point considered.” Mathematically expressed: 0(rq)t when 1)“ one principal stress my other principal stress b. thickness of specimen or Optical constant for material used The colors produced in the specimen may be an order of colors or a series of colors. is p and q differ more and more this series l? of colors produced may pass through more than one order. And as the relative shift of different colors becomes greater. there is a consequent dimming of colors. The w ite results fromra supar position of all colors of equal intensity. is p and q increase (p-q) colors pass through definite series. In celluloid when p-q : e or poo-black. sith thedifforenco increasing the colors follow in this order: straw, orange, red, blue, green, and again straw, orange. red, blue. green, etc. In celluloid these colors represent a certain stress for each color. When the amount of stress for each color is found that color will stand for that stress in celluloid of any shape and form, except where the model might be of such.a shape as round and the colors become too confused. 18 Elms far. a method of arch analysis has been given and a statement of the theory of photo-elasticity. A knovledge of the magnitude of the loadings and deflections is required for the ‘ approximate computations of the magnitude of stress in the various points of the arch. is mentioned before, we are unable to make quantitative measurements and our time does not permit. so conclusions are drasn by inspecting the color fringes. It might be sell to state tso methods used in determining stresses. on. simplest 'but less accurate method is the comparison of colors known in a stressed specimen to the specimen in question. The other method is the compensation method. Various photographs were taken of the arch when placed in the polarised light. The photographs were taken with Eastman Kodak Company's panchromatic cut films. The film differs from ordinary film in that it will show a greater contrast of colors. lhe printing was done on contrast paper. The photographs show hands of uniform darkness and brightness in the arch and has this disadvantage. that black does not represent the same streest neither does gray nor white. The first photograph. Figure 7. as taken of the para-- belie arch. inserted in the instrument with a sixteen pound weight hanging out on an arm as in Figure 6. 19 >f Figure 6 The loading vas attempted in such a manner that it would be concentrated equally at each of the five points. However. this sas not accomplished. A rough estimate of the loading shows that our leads are someihat more than the proportional loads generally used. lrom.the photograph in Figure 7 it will be noticed that the .maximum shear occurs at the points of application. The stress produced in the arch.is mainly a compression stress. With.the varying cross-section the compression also varies. The com- pression stress is disturbed at each of the five points where there is a concentration of stress. The maximum load at these points reveals that there is a reversal of stress and the spots in the arch are spots of high stress concentration. The stress is reversed into a tension stress so that at the bottom of the ardh at each point. to the form.of the curved band, tension is produced, being maximum than directly beneath the point. The amount of tension.then follows the curve to a minimum.and changes over to a compression stress. It may he illustrated as follows, with tension on the inside of the arch and compression in the arch and out side as shown in limo 8. Figure 7 //////[{‘<<