Skewed Rigid Frame Bridge Citation Cabell, John Bell and Gant, Edward Victor (1940) Skewed Rigid Frame Bridge. Master's thesis, California Institute of Technology. doi:10.7907/m5y6-hd75. https://resolver.caltech.edu/CaltechTHESIS:12072024-132454125 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: 1 January 1940 Record Number: CaltechTHESIS:12072024-132454125 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:12072024-132454125 DOI: 10.7907/m5y6-hd75 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16907 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 11 Dec 2024 18:19 Last Modified: 11 Dec 2024 18:30 Thesis Files PDF - Updated Version See Usage Policy. 4MB Repository Staff Only: item control page

SKEWED RIGID FRAME BRIDGE Thesis by Jobn B. Cabell Edward V. Gant In Partial Fulfilment of the Requirements for the Degree of Master of Science California Institute of Technology Pasadena, California 1940 Al.If INVESTIGATION OF THE VARIATION OF :MOMENTS .AND THRUSTS IN A SKEWED RIGID FRAME BRIDGE WITH A CH.ANGE IN HEIG""tlT OF THE SPRINGING LINE The purpose of this investigation is to attempt to establish a relationship between the moments and thrusts of a skewed, rigid frame bria_ge and the height of its springing line as the latter is varied, the angle of skew being held constant. In changing the height of the springing line special treatment had to be made of the legs of the structure in order to maintain a constant thickness of the elements at the springing line and at the abutments. Reference to Fig. 2 shows the manner in which this was accomplished, the thicknesses of the intermediate elements being varied to maintain constant values for the thickness at the to:p and bottom of the leg. The same dimensions for the barrel were used in all cases. In the investigation a structure with a right span of 55.9 ft. and a skew angle of 38 degrees was analyzed. The span and angle of skew were held constant as the height was varied, the latter being taken as 12.9 ft., 17.2 ft., and 21.5 ft. The analysis considered elements 4.3 ft. in length. Thus the changes in the height of the springing line which were investigated allowed the addition of whole elements to the legs. The curves plotted from the analyses are shown on the following pages and indicate the variation in Rx, Rz, lvlx, and My for a unit load. :placed at points .A and B shown in the diagram, for earth pressure of 30 # /sq. ft./ft. depth acting on the left end of the bridge only, and for a temperature change of plus or minus 50 degrees F. ~ 1 ( ,.-or !!..::_:-,x. lq_adlnq) I OIAG W, ;t (for verf load)- I ;.yu:;s¢ U•XSIN BEARING kssu~co 1 F/,l'ED IN A¥L Ollj)EC10NS EXCEPT 4.ROLIMO THIS Ax,s-J ~ I A.-_L ......£l6H[_:,5PAN L _ PL,4N 0 /AGl?AM SHOWING FORCES USED ACT/IYG A ND QUANTITIES IN DERIYATION FORCE.SAND MOMENTS SHOWN POSITIVE ,__ _ _ _ _ _ _ _ _ S.S:9 ' Ss ore Span c_-~cc...,·.,__ _ _ _ _ __ Skew~J3° 0/AGl?AM SHOW/JYG RIGHTAN6L£ PROJECT/0/YOr r/?AM~ LOAl}//V(j AIVD SCALED 1Pt/AtYT/T1ES VSELJ /IV AIY'ALVJIS ,. I 4' --t- F1G 2 THEORY OF .ANALYS IS The skew arch is a structure cont a ining forces wh ich cannot generally be made parallel to a single plane. They must therefore be analyzed in three dimensions instead of t wo as is done in the case of a right arch. As the structure is on a skew, forces and moments are introduced which either do not exist at all in the right arch or are present only in a small degree. The problem is a three-dimensional one, and therefore a system of reference axes for working in three planes will be adopted. Reference is made to Fig. 1. The horizontal axis perpendicular to the abutments is called the X axis, and the vertical axis the Y axis. The axis parallel to the abutment is called the Z axis. Each axis is perpendicular to the other two and each of the three planes formed by the axes is perpendicular to the other two. The total reactions Ry are the same as for a simple beam of the same span. The total horizontal reactions Rx are equal to each other and depend upon the elastic and geometric properties of the arch. In order to make the analysis general, the assumption is made that reactions Ry and Rx are not uniform along the abutments. The reactions Rx may be represented by placing a force Rx at the center of the abutment and introducing a moment or couple My eg_u.aJ. to Rx•e about the Y axis, where e is the distance from the center of the abutment to the center of gravity of the horizontal pressure diagram. Likewise Ry may be represented by placing a force Ry at the middle of the abutment and introducing a moment or couple Ry•~ about the X axis. The two horizontal reactions Rx form a couple which tends to rotate the structure as a whole. To resist this, the two cross shears Rz form a couple which balances in part the moment :produced by the two reactions Reactions RY and Rx produce moments, shears, and thrusts in the vertical planes perpendicular to the Z axis. Reactions Rz produce shear parallel to the Z axis. Reaction moments My and Mx produce torsion in the vertical and horizontal :portions of the arch, respectively. Reactions Ry with moment arms along the Z axis and reactions Rz with moment arms along the Y axis :produce torsion in the horizontal sections. Reactions Rx with moment arms along the Z axis produce torsion in the vertical sections. Torsion in the inclined sections of the arch is :produced by combinations of all of the torsion moments. The torsion moments in turn produce vertical shears in the YZ plane, which are small and may be neglected. They also :produce horizontal shears :parallel to the Z axis which must be combined with the shears due to reaction Rz. A moment about the Ya.xis is also developed by all the reactions and reaction moments parallel to the XZ plane. This moment is resisted by the entire structure acting as a horizontal beam between reaction supports and therefore the stresses produced are very small. In the derivation of equations to be used in solving for the reactions, the supports are assumed to be so altered by the removal of the redundant reactions that the structure becomes a simple span beam. By using the laws of static equilibrium the stresses in all parts of the structure due to any given loading may be found. Likewise, the linear and angular deflections at the right support may be calculated. Use is made of the fact that the total resultant deflection in any support at the reaction must equal zero. The directions of the forces for positive signs are as shown in the diagram. The internal stresses at any section such as P may be represented by a system of external forces and moments exerted by the portion of the structure to the left of the cut section on the portion of the structure to the right of the cut section. The forces to the right of the section must be in equilibrium with the forces to the left of the section. REACTIONS FOR TWO-HINGED SKEWED ARCH By inspection of the diagram, general expres sions may be written for Mv, Mu, Mz, Tv, Tu, Tz, respectively, in terms of the loading and the unknown reaction components. Since 1~ and Tu are very small they may be neglected. FOR VERTICAL LOADING AS INDICATED ON DIAGRAM l& " ~ X - ft/IA- f:, fl ~ I¼ - I~ :I #y = (;~ s-1;; l)ex - rr; t/ -1- Mx cos/ - 11/y s-;~ /; ;{'1/ - 1-()rj cos/) e,1 - Wcos ;if Ix - x j e ::: ~ s1/2 ¢)e,x- l~t/ 1- It cosf - / ~s-1a f6-r(Jt cosp) e. Tv = - ~ .J/17 I> -r # v1n f T B cos1' =- /{s-/~ I r ~ Cos f. ~ ~ ~ . In which M0 :: moment for simple beam of right span l; In which VO :: shear for simple beam of right span 1. The above e quations for vertical loading can be made to apply to the horizontal earth :pressure loading shovm on diagram (30 # Is ~. ft./ft. depth) without rewriting by making the following substitutions: For simple span moment M0 in expression for Mz : substitute For M0 cos¢ substitute in expres s ion for :Mv: Jf, -r Cos f; - JS/; S/,.; / f ,,r -,x). For V O s1i-?¢ in expression for Tv: substitute - ~ v//J / - I S-/; a.. cos- p DE..~IVATION OF EQUATIONS FOR REDUNDANT REACTIONS Consider the deflections at right supp ort due to moments Mz and Mv at any section such as Pin the diagram. ( Deflections due to :Mu and to the thrusts T z, Tu, an.d Tv are comparatively small and may be neglected.) Let M~:: moment at P about Z axis due to all forces acting; IvL,, = moment at P about v axis due to all forces acting; moment at P about v axis due to unit thrust at right reaction along Rx; moment at P about v axis due to unit thrust at right reaction along Rz; moment at P about v axis due to unit couple at right reaction along Mx ; moment at P about v axis due to unit couple at right reaction along My ; moment at P about Z axis due to unit thrust at right reaction along Rx; moment at P about Z axis due to unit thrust at right reaction along R2 ; moment at P about Z axis due to unit couple at right reaction along M1X; moment at P about Z axis due to unit couple at right reaction along My; G ; shearing modulus of elasticity; e = tangent of the skew angle; s = a division of the arch a..""{is; k = ratio of modulus of elasticity for direct stress to modulus for shear :: E/G. The total deflections at right support along the lines of action of each of the u.nJmown reaction components, due to all forces acting, are eQual, respectively, to zero. -.::;-- Ir/.,,F mY(x) f Cl -<.. -<;;: 1"1~ lll~-rrJ = 0 I . Ci~,<_ That is: J ..l[. s 6 -z G-<- m1110:xJ 1-7'~ _m.u~J = 0 .L _s. 7 I¼, m~1o.xJ -== o c F JJI.. ~ "'?° I¼ z _! l'/v mv/eJ -f r F 1::::..<.. N. .2.. 5· l'¾-m.-r"r1 -f i ·G ~ .[ :r l"lz- m~/'c,~) = o E~ F I I ~ (rr ven1✓"col loa✓J =ff:- S-//J )ex - {JC/ 1-li Cos/ - If»~ I ;1-ltect1sf ~ ) = - ,€j r I ~ I~ ( ,, l o o ~ ) = f; e . . r S//7 f - /r; 0 r / > t , e t ,. J f - / J 1-ff;,,,rco.s-/ -/S/2 :Vhf/~ -:.::!/e 11 j = _ f;'. j I- 1c;7-1 _ S-/2 3 SuilsM. tu ting the above values in equations I, II, III, and IV, simplifying, cancelling, and. collecting terms, the final eq_uations are obtained as follows: I f: ( e'lr z A'Y:" 'JP Li) - ,e z f I-/~ e k =- - £ ,rs7/ ek 0 4';.: " t cos I_ /~ elr£ A';/Jy> e'l.k -y--/¼ .Pnp Cosz X f- z/~2 z F .L s-✓n , P llL. f?e£,t'S-/;t Q>sp_ l (c.za:;;:sp + J½Z~~s~p -;Ji 7 'j-<- ff ~e S/n~ cosfP= - r z ;:n ,_~--½Z X /- 7 /~ Cos'"p ~ F ~n ¢ -r l ~Z S/~t Cosp - /J4 7 s-~ l-¢; = - e 7 ~-<.. e / t S/n? cos-¢ ,<_ ;=- Note that left-hand members of above eQuations app ly to any loading. Right-hand members of e q_uations as above given apply :particularly to vertical loading, for which M0 e~uals the simple span moment. For horizontal earth pressure use substitutions given on previous page. For temperature reactions substitute for right-hand members of eq_uations I, II, III, and IV, respectively, the terms: ~eel!-/ o/ one/ o. rl r/se) KS ( c equals coefficient of expansion) 0 ~ .G '--........_ r----_ -- Ur It 1.,ood ~ol L .4 ', I '--- r---__ .2 Um !leijhl 12 1 14' /6 1 load of A o/ S. £ : 18' 20' Rx P!ofted Aio1n.sf He19hf o/ Spnnqin9 22' line <'z, 200 0 0" Q( ( \'0 ~o /60 V 1.20 ~ 80 / --- 40 12' / Ternp.erCtt i:lre He/9hf o/ S L,, /4' 16 ' /8' zo' 22' , .6 .. ~ "'~ ...;, ..z ~ o b-,.,,, ~ ~ I'---_ H ei9hf /,.?' 14' of ~ ..5. ~ - 1 18 ' /6 ' /(} ' /6 ~ (,/nil L oad ol A . 2o ' 22' /60 120 40 /2 1 22 ' r-2 0 /8 1 ' 12 ' Uni! /V/,,r 22' Loa d or 8 P io /fed A90/17,5 t Hei9hf oj ..5prin9i179 line f/600 f-/200 f-800 ~ l - - - - - . . . + - - - - - ~ - - - -- s - 1 - - -- - - + - -- - - + - -- - - --t-- - - - - - - j ~ f '-.. t-400 0 -400 ----~ 14 1 V /6 ~ / 8 ' He /9hl of .5. /.;. 20' 22' -t-40 Un i f J.,ood of_8 0 ~ -4.0 12' 16 ' /4/ {Intl 18' Lood af A . ? ,? , 2 0' V / -c3 . 0 !it~hl of ..5. l, re pr~ f800 -~ 0 tZ ' 14 1 I~ , ~ !'8 ' 20 ' ~ ~ ?2 ' -800 ure -16 0 0 V / -2400 I V' ~ Hei9hl of S L , .