A STUDY OF THE CIRCULAR-ARC BOW-GIRDER THE CIRCULAR- ARC BOW-GIRDER BY A, H. GIBSON D.SC., ASSOC.MEM.INST.C.K., M.I.MECH.E. Professor of Engineering in the University of Si. Andrews, University College, Dundee AND E. G. RITCHIE B.SC. ,/lssistant Lecturer in Engineering, University College, Dundee NEW YORK D. VAN NOSTRAND COMPANY 25 PARK PLAGE 1915 Printed in Great Britain PREFACE THK problem of the state of equilibrium and of stress of the circular-arc bow-girder, i.e., the girder forming a circular arc in plan such as is often used to support the balcony of a theatre, is one affording some difficulties of solution. These arise mainly from the fact that in addition to the bending moments and reactions involved in the case of the straight encastre girder, twisting moments are called into play at each section and at the ends of the bow-girder, and these moments affect very considerably the state of equilibrium of the girder. The general problem was solved in a paper read before the Eoyal Society of Edin- burgh by Professor Gibson in 1912, and the first portion of this book is based on the principles laid down in that paper. The solution in any particular case becomes easy if the end fixing moments and the reactions are known, and values of these have been calculated for the more important cases likely to occur in practice. This investigation shows that the values of the various moments and reactions for a given loading depend on the relative values of the flexural rigidity, E I, and the torsional rigidity, C J, of the section. A knowledge of the geometrical properties of the section and of its material enable the former of these to be predetermined with some accuracy, but the authors have been unable to find any published data as to the values of the torsional rigidity for such commercial sections as are usual in structural engineering. With a view of obtaining such data experiments have been carried out by Mr. Ritchie on a number of commercial sections, and the result of this work forms the foundation for much of the second part of the book. Chapter I. outlines the introductory theorems necessary for a thorough understanding of Chapter II., which deals with the equilibrium of the bow-girder. In Chapter III. the torsion of non-circular sections is considered, while Chapter IV. deals with the stresses involved by such torsion alone or combined with bending, and Chapter V. deals briefly with the general principles of design of a bow-girder exposed to both bending and twisting. It is hoped that the treatment is sufficiently complete to enable any one familiar with the general principles of design of the ordinary straight plate-web or lattice girder to adapt these to any specific case of a bow-girder under uniform or concentrated loading. In view of recent failures of structures in which straight beams exposed to some torsion have collapsed under seemingly inadequate loads, the data of Chapter III., emphasising as it does the extreme weakness of the commercial I, angle, or T section under torsion, should be of interest. Appendices have been added, giving a list of integrals which will be useful to the reader working through the investigations of Chapter II., and also giving a table of the geometrical properties of some commercial sections. A. H. G. E. G. R. DUNDEE, September, 1914. 331443 CONTENTS CHAPTER I. AIMS. I'A'.K 1. EQUILIBRIUM OF THE STRAIGHT GIRDER ... ... 1 2. CURVATURE, SLOPE, AND DEFLECTION 1 3. ENCASTRE -AND CONTINUOUS BEAMS . . 3 4. ENCASTRE BEAM WITHOUT INTERMEDIATE SUPPORTS ...... 4 5. ENCASTRE BEAM WITH INTERMEDIATE SUPPORTS 5 (!. ENCASTRE BEAM WITH UNIFORM LOADING EFFECT OF SUBSIDENCE OF ONE SUPPORT 7 7. BEAMS WITH UNSYMMETRICAL LOADING ........ 8 8. RESILIENCE OF A GIRDER UNDER BENDING ........ 10 0. CASTIGLIANO'S THEOREM . ......... 11 JO. RESILIENCE UNDEK TORSION . . . . . . . . . . .12 11. DEFLECTION DUE TO SHEAR FORCES 13 CHAPTER II. 12. THE CIRCULAR-ARC BOW-GIRDER . .14 1P>. CIRCULAR-ARC CANTILEVER WITH SINGLE END LOAD 15 14. CIRCULAR-ARC CANTILEVER WITH UNIFORM LOADING . . . . . .16 15. CIRCULAR-ARC ENCASTRE GIRDER WITH SINGLE LOAD 18 16. CIRCULAR-ARC ENCASTRE GIRDER WITH UNIFORM LOADING 28 17. CIRCULAR-ARC ENCASTRE GIRDER WITH UNIFORMLY LOADED PLATFORM . . 34 18. GIRDER WITH UNSYMMETRICAL LOADING ........ 37 1!). BOW-GIRDER WITH INTERMEDIATE SUPPORTS 37 20. BOW-GIRDER WITH UNIFORM LOADING AND CENTRAL SUPPORT . . 37 21. BOW-GIRDER WITH TWO INTERMEDIATE SUPPORTS 40 22. BOW-GlRDER WITH THREE INTERMEDIATE SUPPORTS 43 23. EFFECT OF SUBSIDENCE OF SUPPORTS 45 24. EQUILIBRIUM OF A COMPOUND BOW-GIRDER 46 25. SHEAR FORCE AT A SECTION 47 26. EXPERIMENTAL VERIFICATION OF FORMULAE 47 27. APPLICATION TO SECTIONS OTHER THAN CIRCULAR 48 CHAPTER III. 28. THE TORSION OF NON-CIRCULAR SECTIONS 50 29. EXPERIMENTAL RESULTS . 52 \iii COXTl'XTS CHAl'TKR IV. new. ''AUK :?(>. SHKAI: STKKSSKS IN A BKA.M OF CIIMTKAK SKITION . . . . r >0 :'.l. STUKSSKS IN XoN-(. 1 iucn.Ai: SKCTIOXS ..... . . ">'. :> HM.IPTICAL SKCTIONS ... . <'0 :;:;. Ui; 37. I AND Box SKCTIONS ..... .70 CIIAl'THR V. :>. (JKNKUAK I'lMNCIPKI'.S OF Dl'SKlN OF T1IK BO\V (Jlltl'i;!: ... . 7o AiMMvXDIX A. LIST OF INTKUKAKS ............. 77 APPENDIX I '.. PROPERTIES OF CO.M.MKKCIAK SKCTIONS. ...... 7s IXDKX 7! A STUDY OF THE CIRCULAR- ARC BOW-GIRDER CHAPTER I (i) Equilibrium of the Straight Girder. IF a girder straight in plan and horizontal when unloaded is exposed to a series of vertical loads, each section is subject to a bending moment M, whose magnitude varies from point to point. Under the influence of this moment the girder is bent, and, so long as the loads are not sufficient to produce stresses in excess of the elastic limit of the material, the radius of curvature K of the profile of the neutral axis at a point where the bending moment equals M is given by the relationship 1 M R = EI ....... (a) where I is the moment of inertia of the section about a horizontal axis through the centroid of its area, and where E is the modulus of direct elasticity of the material. If y be the vertical displacement of the neutral axis at a point distant x from some datum point in the axis, it may readily be shown that = (approx.) so that, so long as the deflection of the beam is confined within practical limits, d M (2) Curvature, Slope, and Deflection. From (lj) it follows that if, at any one point, the girder is horizontal after loading, the slope -r- at any other point at a distance I will be given by r l dil | M . t . ~ = -777 . dx . . . . . (c) dx El Jo M and will therefore be represented to scale by the area of the yry diagram between Ji/J. the two points, while if the slope at the first point is not zero, this area -will measure E.G. B 2 A STUDY OF THE CIRCULAR- ARC BOW-GIKDKE the difference of slope at the two points. On integrating both sides of expression (<), the deflection y of the second point below the first is given by fW , ,~ // = 7- dx (a) \dx/ Jo i -i |.V , I /;/ ' dx ' */ In a given beam under load the slope changes from point to point, and the difference of slope at two points, a small distance Sx apart, is given by -p ( j-j Sx, or by f-^Jj M or pj . Sx, where M is the moment acting on the element included between the two sections. If the rest of the beam were to remain straight the deflection at a distance I from the element, due to the bending of the element under this moment, would be equal to M , -7TT . OX . I El and if the slope at one end of the element were zero this would be the actual relative deflection at a distance I. Since every section of the beam is exposed to a bending moment, any element at a distance x from the point whose deflection is being con- sidered contributes its quota pj . x . Sx to the resultant deflection, so that the actual deflection at the point /, relative to the point at which the slope is zero, is given by i M j -TT7 . x . dx EL o or by Ax ....... (e) M where A is the area of the -^j diagram included between the two points and x is the distance of its centroid from the point /. If, instead of being zero, the slope at the point o is equal to i, the deflection at /, relative to this point is given by il + A x . . . (/) Special Cases of Deflection. In certain standard cases the maximum deflection is very readily calculated, and is as follows : Deflection. Beam of uniform section and of length / with single load W at centre . ajW -777- r Beam of uniform section uniformly loaded with u- Ibs. per foot run . ^5 ... IJ'/ :i Cantilever of uniform section with load W at end -rrr u-i 4 Cantilever of uniform section uniformty loaded ..... ^ - AND CONTINUOUS BEAMS (3) Encastre" and Continuous Beams. A beam simply supported at its two ends has, everywhere, a curvature whose con- cavity is upwards. If, however, it is built in to supports at its ends, these supports prevent the beam adopting the slope natural to it when free, and a fixing moment is W, W < w k~~ & T & -> " 3 ) r > f i t 1" FIG. 1 . called into play at each support, these moments tending to make the beam concave downwards. The effect of the fixing moment is transmitted to every section of the beam, and at any such section as, say, X in the beam of Fig. IA, for equilibrium M, = M a - (E a x a - WM - W 2 x 2 ) . . . (g) = M b - (R b x b - Tf-V?,,) P. r FIG. 2. P 3 while at X in Fig. IB, which represents a beam with uniform loading of magnitude w Ibs. per foot run, ii' r a B 2 4 A STUDY OF THE CIRCULAR-ARC BOW-GIRDER If, as in Fig. 2, the beam has one or more intermediate supports, whose upward reactions are PI, P 2 , PS, the moment at X, say between supports (1) and (2) is given by J/, = J/ a l# a *a + (j) -^f\ ' ' ' (k ' } Under such circumstances, the magnitudes of the fixing moments M n and M b ; of the reactions R a , R b , at the ends ; and of PI, P 2 , PS, the reactions at the supports, require to be determined before equations (g),(h),(j), or (k) can be used to determine the value of MS at any given section. (4) Encastr Beam with no Intermediate Supports. Considering, for example, the case of a beam built in at its two ends and carrying a uniformly distributed load of w Ibs. per foot run (Fig. 3), 2 M x = M a - R a x + -r-. U 1 FIG. 3. Since from symmetry R a = -> we nave WLX If, for simplicity, the section of the beam be taken as constant so that I x = constant = I, we have, on integrating, ( ]l- JLIjif x . dX JltJ. V where A is a constant of integration. Since the slope is zero where x = 0, i.e., at the left-hand support, it follows that A = 0, and since the slope is also zero at B where x = 1, we have ivtw _j_ A 4 ^ Substituting this value of A/ a in (m) gives dx 1C Tl 112 ENCASTRti BEAM WITH INTERMEDIATE SUPPORTS 5 or, on integrating this, jv_ {Px x^ _ W } " El 112 + 24 12"* I' The constant B is determined from the fact that the deflection y = when x = 0, so that .6 = 0. (5) Encastre" Beam with Intermediate Supports, or Continuous Beam on more than Two Supports. Let A, B, C, (Fig. 4) represent three adjacent points of support on an encastre beam, or on a simple continuous beam with uniform loading iv Ibs. per foot run. To determine the moments M a , M b , and M c , and the reactions R a , R b , R c . Take the origin at A. Then between A and B, M x = M a R a x-\-~ (n) .'. At B M = M - r/i 2 2 . (o) B FIG. 4. Similarly, working back from C to B, Writing (>?) as we get and dy _ war (P) a O 2i^ Since // = when a: = 0, it follows that D = 0, while since y = when x = 1, we have Tl f- ^1_ -D ^1 I ^'^1 I /^7 Q "* Q n (* I 41/1 ~T~ v*l v '' C ~ '2 6 + 24 [ From (5) the slope at B is given by and on substituting from (s) this becomes f dy\ 1 f /i \d~xj ~m I "2 . o 1 8 J 6 A STUDY OF THE CIECULAE-AEC BOW-GIBDER Similarly, taking C as origin and working back from C to B, we should get b hi ( t .2 3 8 the minus sign being taken before y-_, because x is now measured in the negative direction. Equating these expressions for (-- J , and eliminating terms containing R T>* 6EId this 5.M. = + 2 . The reactions R a and L' 6 under the new conditions are determined from the equa- tions w/ 2 M b = M,, - li n l + ~ M - M b . wl ~ ~z _w 12EH ' 2 I s and /4 = / A',, TT IZEId (7) Beams with Unsymmetrical Loading, or with a Series of Concentrated Loads. When the loading of an encastre or continuous beam is unsymmetrical or consists of a series of concentrated loads, a semi-graphical treatment based on the considera- tions outlined on p. 2 is preferable. In Fig. 6, let A, B, C be three adjacent supports in a continuous beam, and let AGB, BHC represent the bending moment diagrams for such a loading on two simply supported spans AB and BC. Let ADEB and B EFC represent the fixing moment diagrams, and G^, G\, G 2 ', G 2 the positions of the centroids of the areas A I>EB, AGB, BE EC, BU( '. Let the area AGB = AI ADEB = A,' ,, BHC = A 2 BEEC= A 2 ' Then, considering the span AB, taking .4 as origin, since the supports at .4 and 7) are at the same level i h being the slope at B. BEAMS WITH UXSYMMETKICAL LOADING 9 Similarly for the span CB, taking C as origin, since the supports at C and B are at the same level 1 o r K- a" A the negative sign being taken, since x is measured in opposite directions in the two cases. 10 A STUDY OF THE CIKCTJLAR-ABC BOW-GIEDEE Equating the two expressions for the slope at 13 gives fi^i__zulii _ _ 2 ~* 2 ~ ^z''-i / { \ fcj / 2 Again, taking moments about .4 and C of the fixing moment diagrams on each span and, on substituting these values, equation (/;) becomes M,J, + 237, (^ + / 2 ) + 3/,./ 2 - 6 {^i + ^| - . . (*) 'i '2 This is the most general form of the equation of these moments and is applicable to any form of loading, 9 trl 3 9 irJ 3 / / 1T T !' a UJt-i . II in (i ' Writing Ai = - -gL ; ,1 8 = ---; x, = ^ ; a- 2 - ^ gives the equation for uniform loading, which is identical with (u), p. (5. If some or all of the supports sink, B falling d^ below A and d. 2 below C, equation (z) becomes MJ, + 2.V, (I, + g + M,.l, - C> j4jfi + ^2 1 'l '2 (a) (8) Resilience of a Girder Exposed to Bending. If, under the action of a bending moment M, two originally parallel vertical sections of a beam, enclosing an element of length bx, become inclined to each other at an angle rj of Si, the work done by the moment in bending this element is equal to M - '. (This 2i assumes that the moment is applied gradually, and, at any instant, is proportional to the curvature obtaining at that instant.) .'. "Whole work done in bending beam= ^ ---. M TI r ^1 \ W + - ^ \ S// x + Tr 2 8// 2 {. & } .-.&U= ir^'/i + Wy t . (y) Now suppose the structure unstrained and gradually loaded with (W\ -f ^^\) and TT' 2 , these loads during application always maintaining towards each other the ratio of their final values. The final deflection must be the same as before, while the resilience is given by U' = i + Tr 2?/ 2 ) + y = i {2^+517 + ^8^ But U' U must equal &U. .-. 8[7 = ?/ dU TF . (7) a- -i i ( H Similarly -^ =y 2 , derivative of U with respect to any one load equals the deflection of the point of application of that load. (10) Resilience of a Beam Exposed to a Torque. If a beam be exposed to a torque whose magnitude at a given point is T, successive plane sections suffer rotation about the longitudinal axis of the beam, and the relative rotation of two sections, distant Bx apart, is equal to 86, where Here (' is the modulus of transverse rigidity or the shear modulus of the material and J is the polar moment of inertia of the section, or its moment of inertia about an axis through its centroid perpendicular to its plane. 1 5j/3 Y'2 The work done by the torque during this relative rotation is T - ^ bx, so that Z ZCf7 over the whole length I of the beam the work done by the torque is given by ? 7-2 < j ''' 1 SeelChapter III. fur the effective value of J in any particular case. DEFLECTION PRODUCED BY SHEAR FORCES 13 Where a beam is exposed to both bending arid twisting moments, its resilience is the sum of the works done by these moments, and this, by the principle of work, is equal to the work done by the applied loads daring distortion. (u) Deflection Produced by Shear Forces. In addition to the deflections produced by the bending of a girder, there is some slight deflection due to the fact that each vertical layer is exposed to shear stress. In a straight beam, exposed only to bending and shear stresses, the deflection due to shear is always a small fraction of that due to pure bending, being greatest in a built up beam of I section in which the web is comparatively thin. 2 In such a beam of normal proportions and span simply supported at the ends, the deflection due to shear is seldom more than 4 or 5 per cent, of that due to pure bending. In an encastre beam of this type the proportion may be as much as 20 or 25 per cent. In the type of bow-girder to which this treatise is particularly devoted, the deflection is mainly due to torsion, and moreover the proportion of the whole deflection due to torsion is greatest for those beam sections for which the shear deflection is greatest. Even in an extreme case, in a bow-girder the shear deflection does not amount to more than 4 or 5 per cent, of the whole, and will, in general, be much less than this. It has, in consequence, been neglected in the following dis- cussion. Where, as in a large built-up bow-girder of I section with very slight curvature, it may be advisable to make allowance for the extra deflection, this may most easily and with sufficient accuracy be taken into account by using in the calcula- tions a value of E about 20 per cent, less than the true value for the material . 2 For a discussion of this point, see Morley's " Strength of Material?," p. 226, or any text-book on the same subject. CHAPTER II (12) The Circular-Arc Bow-Girder A GIRDEK built in to supports at one or both ends and forming an arc of a circle in plan, is subject, at each section, to both bending and twisting moments. At the supports, fixing moments of both kinds are called into play, and until these are known the resultant moment tending to cause rupture at any section is indeterminate. The following investigation is devoted to a consideration of the general state of elastic equilibrium of such a girder under various systems of loading. The investigation is based on the theorem (p. 1) that in a straight beam, fixed horizontally at some point, the slope at any other point is given by the area of the M rjj diagram between the two points. Where a girder is circular in plan and is sub- jected to both bending and twisting moments this theorem requires modification. Let M and T e be the bending and twisting moments at a point P distant 6 (in angular measure) from the support A (Fig. 9). Then a given slope at P in the direction of the tangent at this point produces a slope of cos (6^ 6) times its magnitude at Q in the direction of the tangent at Q. Also an angular displacement 7 at P, due to a torque between the support and this point, produces a slope y sin (6 1 9) at Q, in the direction of the tangent at Q. It follows that if distances along the arc of the girder be represented by s, the resultant slope at Q, assuming the slope at the support to be zero, is given by /arc 61 (dy\ = I jM* \ds/0i ] EI0 f A arc 61 /*arc 61 Mfi cos (0! - 0) ds + ^ sin (0i - e ) ds Here I 6 and J 6 are the moments of inertia of the section at 0, about the axes of bending and of twisting. Where the beam is of uniform section, this becomes (^) = _L Me cos (0! - 0) ds + ~ T e sin (0! - 0) ds ; \ds/8i El I CJ I v */ * or, since K if r is the radius of the arc, . dy 1 dy ds = rd6 ; -=-=-. -y^ ; ' ds r eld CIECULAE-AEC CANTILEVER WITH LOAD W AT FEEE END 1 5 (13) Circular-Arc Cantilever with Load W at Free End. Let a (Fig. 9) be the angle subtended by the arc. Then, M e = W X CR = Wr sin (a 0), T g = W X RP = Wr{l cos ( a 0)} ; ... W) = !IT | sin (a - 8} cos (0, - 0) d = T 2 (1 cos V), Jo CO T e = wr*(l cos ) d = wi* (0 sin 0), Jo CIECULAE-AEC CANTILEVEE WITH LOAD W AT FEEE END 17 as the moments produced at P by the loading on that portion of the beam between P and the free end. h/\ _ wr wr where a is the total angle subtended by the beam. Integrating this expression and simplifying gives A fa 0i . sin2a sin20i) sin 0i sin 2 a sin 3 0i m (a _0 ; )_ CO s - sin (a 0i) a cos (a 0i)+ 0i cos0i a 0i sin 2a sin20 ] ''} sin 0i sin 2 a sin 3 fa sin 2a sin 20) sin0sin 2 a sin 3 0"1 ^ sin(a 0) acos(a fa sin2a sin20T "~ sin 6 sin 2 a sin 3 2AT 2 2 cos (a --. 26'e/ - | (cos 3 a 2 2 cos . sin &i sin 2a >i) + (a 0i) sin 0i + cos a cos 0i + - 5 a 0i sin 2a sin 20i a cos 3 0i) cos 0i snr a -\ ^ . cos 0i sin 2 a 0i) + a 2 - 1 2 + (a-0i)sin0i sin 0i sin 2a . , , a + cos a cos 0i -^ - + | (cos d a 0, . sin 2a sin + cos 0j siir a -- ^ | 2 cos (a 0i) 2a sin (a 6 sm0 lS in-za , - ;3 a _ COS 3 ^ B.G. 18 A STUDY OF THE CIECULAU-AHC BOW-GIHDEK At the free end t = 0, and the deflection becomes ICI A [V a . sin 2a~| typj 1 cos a f (cos j a 1) sin- a + - -\ 7 . tn A P 1 - ., a sin _>"] -)- I 1 cos a 2a sin a -j-a + 5 (COS 8 a 1) + sin a - H 1_( / L !u 4 J e.g., if a = 5, the deflection at the free end becomes a 4 f-5594 . -1035 " = "' .^r + ^r (15) Circular- Arc Girder, Built in at Two Ends, with Single Load W. Let the arc subtend an angle (n 2 <), and let (Fig. 12) be its centre ; Alt the line of supports; AOW = a; BOW T = @; It,, and 1\,, the vertical reactions at A Fio. 12. and 1? / 3/ ft and Jl//,, T,, and 7',, the bending and twisting moments at the supports A and B, the axes of these moments being respectively parallel to and perpendicular to OA and OB. The bending and twisting moments at any point between A and W, distant 6 from OA, are now given by M e = ^f ll cos e ll,,r sin B + 7',, sin 6 (4) TS = T n cos + Ii a r (1 cos 0) 3/ fl sin . . . (5) while the moments at a point between B and W, distant 6 from OB, are given by similar expressions, with suffix b taking the place of suffix a. Before these moments can be calculated for any particular case, the values of the six unknowns, M (l , M,,, 7',,, T,,, It,,, 11,,, are to be ascertained; and for this, six relationships between these unknowns are necessary. Taking moments about B, of the forces and couples acting in a vertical plane we have, for equilibrium, E,, ('2r cos <) T a cos M /r sin cjb IF;- cos + cos (a + 0)1 -f 'J'i, cos -f ^ sin ^ = CIRCULAR-ABC GIRDER, BUILT IX AT TWO ENDS 19 en Again, taking moments about the line AB, (M a + M b ) cos < - (7 7 a + T b ) sin < = Wr { sin (a + )- sin }. . (8) while, equating the torques at the weight, as obtained by working from both ends of the girder, T n cos a + R a r (1 cos a) M a sin a = T b cos ft R b r (1 cos ft) + M b sin /3 ............ (9) The other two necessary relationships are obtained by expressing the fact that both slope and deflection at the weight are the same, whether the latter is considered as being at one extremity of the arc A W, or of the arc BW. The slope at any point X between A and W is given by cos - sn - and, on substituting for Jl/ fl and T e from (4) and (5) and integrating, sin 6,\ - (R a r - T a ] 0, sin ^ cos Similarly at any point between B and IF, distant X from OB, " cos X + sin 0j I (E b r T b ) 6 l sin t /4r) 1 sin X + 2/4 r (l cos X ) M 6 -[ sin X X The slope at the weight is obtained by writing 6 l = a in the first, or X = ft in the .second of these expressions, and is thus given by / 2 r n .,-777 I M,< i a cos a + sin a } (R a r T a ) a sin a r 2 r ^7-7 (7 T ft cosa) sna acosa n I (10) or by cos sn ~ T ~ sn r 2 f "1 + 0777 (T 6 74r)/3sin/3 + 2/4r(l cos/3) A/ 6 {sin/3 /3cos/?l ZCe/ l_ ' J according as the point IF is considered as forming part of span AW or of span B W. On equating these two expressions, with the sign of the second changed since c 2 20 A STUDY OF THE CIECULAE-AEC BOW-GIEDEE is measured in opposite directions in the two sections, a further relationship between the unknowns is obtained. Deflections. Assuming the supports to be at the same level, integrating obtain the deflection gives (between A and W) to r^_ Cr IK I] L a Jo - (R a r- r a )0sin0J dd a r(l cos 0) 3/ ft ! sin 0cos M IV / (I ( ( J 1 . 2G'J as the deflection at a point distant 6 1 from A. On integrating and simplifying, this becomes [ A^ sin X - (R a r - TJ (sin 0, - d, cos 7 T f( _ jR a r)(sin X 1 cos 0^ + 27^' (^i sm ^i + M a (0 1 8in0 1 +2eo80 1 2) Similarly for a point between B and W, distant 6 l from B, J sin X - (/V - T fc ) (sin X - 1 cos (12) 2EI (T b - /V) (sin t - #1 cos ^) + 27^(0! sin 6 r 2 r( h 2CJ L + 3/ 6 (i sin 0! + 2 cos ^ - 2) (18) At the weight, X becomes a in (12) and /3 in (13) and these expressions give (A to W) r 2 f "1 JJTJ- M a a> sin a (E a r T a ) (sin a a cos a) J r 2 r(!r rt E a r) (sin a a cos a) + 2/O' ( a sin a) 2CV L + ^/ ( a s" 1 a + 2 cos a 2) (14) and (B to IF) sD 2A7 M b /S sin /3 - (72 6 r - T 6 ) (sin /3-j3cos r(Tb - J? b r) (sin /3 - /3 cos /3) + 2 sn 1 (15) On, equating the identities (14) and (15) the final relationship is obtained, and from the six equations (6), (7), (8), (9), (10 = -- 11), (14 = 15), the six unknown fixing moments and reactions may be determined in any particular case. These moments depend somewhat on the relative values of El and of CJ, except where the load is in the middle of the span. An increase in the ratio El : CJ is accompanied by an increase in all the fixing moments. The effect on the values of M a , of M b , and of the end reactions, produced by a large variation in this ratio, is very small, especially when the angle a is large. The effect on the end torques is more pronounced, particularly for small values of a. In order to facilitate the application of the results of this analysis, and to make CIKCULAB-AKC GIEDEE, BUILT IN AT TWO ENDS 21 10 1/sL/ues of (i fi-MA FIG. 13. Values of M A , M B and E A for a bow girder built in at both ends, subtending an angle 180 2, and carrying a load If at a point distant a from end A. A STUDY OF THE CIRCULAR-ARC BOW-GIRDER 30 40' 50" \fa.lues of K 60 70 80 FlG. 14. Values of J/ A , J/ H and A' A for a bow "inlor built in at both ends, subtending an arc (180 2}, and with a single weight II' distant a from the end A. CIECULAR-AEC GIBDEB, 13UILT IX AT TWO ENDS 23 it more useful in practice, the foregoing equations have been solved for a series of values of a and of <, and the values of the end moments and reactions have been calculated for a series of values of El: CJ. Owing to the comparatively small effect of this ratio on the end bending moments and reactions, values of these have 40' 50' Values of (L FIG. 15. Values of 7\ -f- Wr for a bow girder built in at both ends, subtending an angle l.SO ? 2, and carrying a single load W at a distance a from the end A. only been calculated for the extreme cases likely to be found in practice viz., forl^T: CJ 1*25 (its approximate value is a solid circular section) and for El: CJ = 100, These results are plotted as curves in Figs. 13 and 14, and for intermediate values of the ratio the moments and reactions may be obtained with a sufficient degree of accuracy by interpolation from these curves. Owing to the relatively greater variation in the end torques, values of these for 24 A STUDY OF THE CIRCULAR-ABC BOW-GIRDER a series of values of El: C'J have been calculated, and are plotted in Figs. 15 and 16. By substitution from these values in equations (4), (5), (12), and (13), the values of 10 4-0 50 Va-lues oF CL FIG. 1<>. Values of T K ~ \\ r for a bow girder built in at both ends, subtending an angle ISO '1$, ami with a single load II' at a distance a from the end J. the bending and twisting moments, and of the deflections at any point of the girder, may be obtained. Special Cases. Semicircular Bow-Girder with Single Load W in any Position. Here a -\- fi = 180 ; < ; and the foregoing equations simplify. The values of the various con- CIRCULAR-ARC GIRDER, BUILT IX AT TWO EXDS 25 stants for such a girder have been calculated for the case where El = 1*25 CJ, and are given in Table I. TABLE I. a 1 ? I 15 30 l^o 45 135 60 llo 75 1 D< 90 IL W 1-00 990 940 870 764 640 500 B, W o-o 0104 060 131 236 361 500 Ma Wr o-o 239 128 542 590 571 500 Mi Wr o-o 0200 0725 165 276 395 500 T,, Wr o-o 0251 0662 115 155 181 182 T,, Wr o-o 0118 0382 082 128 161 182 In the particular case where a = 90 = T5708 radians (i.e., weight at centre of span) from symmetry M n = M h = -5 Wr T a = T,, From (10) the value of ~ at the weight fa = ^j is given by From symmetry tbis equals zero ; and in this case both M a and T n are independent of the relative values of El and CJ. The curves of Fig. 17 and 18 show respectively the values of the bending and twisting 26 A STUDY OF THE CIRCULAR- ARC BOW-GIRDER moments at each section of a semicircular girder clue to a single load IT at any distance a (degrees) from one end, and ordinates of the envelopes to these curves shown in CIECULAK-AKC GIEDEE, BUILT IN AT TWO ENDS 27 0= jo fan /erf t H a g dotted lines give the maximum positive or negative moments produced at any point by a concentrated rolling load of this magnitude. 28 A STUDY OF THE CIRCULAR- ABC BCTW-GIRDEB Circular- Arc Girder, subtending an Angle less than 180, and carrying a Single Weight at the Centre of the Span. Let 2a = (- 2) be the angle subtended (Fig. 12). The moment of the weight about AB = }Vr (1 sin ), and as, from symmetry, M a = M b ; T u = T b ; equation (8) becomes HV or also 31 a cos d> T a sin d> = (1 sin ) 2 II V 2 cos (1 sin ) -f 7',, tan , 7? = It = On substituting these values of 37,, and R a , equation (10) becomes ~Tn- 3 r/i sind. . r d 2 cose/) I si ^ ,"1 From symmetry this equals zero, and, on substituting for a and and equating to zero, the value of T a is obtained. Except in a semicircular girder ( = 0), this value depends on the ratio of El : CJ. The following values have been calculated for the case in which this ratio equals 1'25. 15 30 45 60 u a Wr 50 410 314 223 140 T n Wr 182 099 045 0157 0032 Knowing M a and 7',,, tha deflection at the weight may IK; obtained by substituting these values in equation (14), p. 20. (16) Circular-Arc Bow-Girder, Built in at both Ends, with Uniform Loading ic Ibs. per Unit Length. Let 77 2< be the angle subtended by the arc (Fig. 12). The total load = UT (- 2$) Ibs. The centre of gravity of the load is at a distance from the line of supports given (16) / 77 2 = 2T- jcos -^ - . (17) 30 A S'lTDY OF THE CIRCULAK-ARC BOW-GIEDKIJ Taking the origin of at the supports, M e = J/,, cos 9 li a r sin + T n sin + wr\l cos 0) l = (M n - in- 2 ) cos - (U ( ,r - T a ) sin d + ?rr 2 . 7* = T a cos + AV(1 cos 0) 3/,, sin i 60" 70 80' 3iT FIG. 21. Values of M a and of r a in a girder with uniform loading, subtending an angle [180 -24*]. EI:CJ = 1Q. ,2 r yj \(M a wr^sin 1 (R a rT a )(sm 6 ^cos^) 2w?- 2 (cos^ : (T a R a r) (sin 1 1 cos ^)+ {(M a wi*) (6 l sin x +2 cos ^5 2CJ cos - (21) From symmetry ^ is zero at the centre of the span where 6 l =~ ^,, and A by substituting this value for ^ in (20), and by also substituting for M a its value f /7T T \ I it;r 2 |l (^- <^ _ ^j tan | and equating to zero, the value of T a may be obtained, A STUDY OF THE CIRCULAR-ARC BOW-GIRDER after which the values of M 6 and T e for any point on the girder may be obtained by substitution in (18) or (19). The values of M lt , T a , M , T have been calculated from the foregoing equations for one-half of a uniformly loaded girder for a series of values of $, and of 6 for each value of . These values depend slightly on the relative value of El and of CJ , and in Figs. 19 and 20 values of M, t and of T,, are plotted for a series of values of El : CJ. Fig. 21 shows the variation of M,, and of T n with tf>, for a given value of El : CJ. The curves of this figure are calculated for the case where this ratio equals i-o 80 60 N^ degrees. Special Case. Semicircular Girder with uniform Load. Here $ = 0, and we have : M lt = M b = wr 2 : E a = R b . wr : CIECULAE-AEC GIRDER, BUILT IN AT BOTH ENDS 33 ^ [(7' - R a r)0i sin 0, r sn sn - cos ~ r)(sin *i ~ ^ cos i> ~ 2 (7 T rt ZV)(sin ^! 0J cos X ) sn (21') Values of d measured From one support . FIG. 23. Twisting-moment diagrams for one-half of a uniformly loaded circular-arc girder, subtending an angle [180 20] . Substituting for M a and R a in (20'), writing ~ for 0, and equating to zero, gives T a = ivr 2 X ? (~ - 2) = -298wr 2 , 77 \4 / and on substituting in (18) and (19) M 6 = wr\\ 1-2728 sin 0), T e = W7\1'570S 1'2728 cos 0). 1 This makes M 6 = when sin = = -7850 ; *.., when 6 = 5143', and 1-2728 makes T e = when = 2240', and again when = 90. Af fl is a maximum when ~-~ = ; i.e. when cos = 0, and therefore at the supports. T,? is a maximum when E.G. D 34 A STUDY OF THE CIRCULAR-ARC BOW-GIRDER cW = 0, i.e. when sin 6 = '7850, or when 6 = 5143'. 7T Writing 6 1 = ~ in (21') and substituting for T a and R n , the deflection at the A 4 P7272 -053~| ^centre,- _ ^.j +26VJ ' centre is given by (17) Circular-Arc Bow-Girder, Subtending an Angle (180 2(/>), Built in at the Ends and Carrying a Uniformly Loaded Platform. Let w Ib. per unit area be the load on the platform whose area will be r 2 f ) -^ -j TT 20 sin 20 \ . Imagine the latter to be divided into a series of strips ^ I ) parallel to AB, each of these strips transmitting its load to the girder at its ends. The length of the particular strip resting on the girder at points distant 6 from A and B, FIG. 24. is 2r cos (6 + 0) (Fig. 24). If this strip covers a length S.s = rSd of the girder, its width;is r&6 cos (6 + ), and the load on it is 2zw 2 cos 2 (9 + )#. Its moment about AB = 2;r 3 cos 2 (0 +){ sin (9 + (/>) sin <} 80, /. Moment of whole load, about ^1 = ~* = Zivr* Jo +^){ s i n (6' + )- si , 2 sin Since, from symmetry, M a = M b ; T a = T b ; it follows that Af cos - T a sin = wi* [^ - ^ (rr - 20 - sin 20)} ., Ma = ^{2^* - ^ ( W - 20 - sin 20)} + T a tan 0. Again, since the total load is ' I cos 2 (0 + 0>70 CIRCULAR-ARC BOW-GIRDER, SUBTENDING AN ANGLE 35 .-. R a = K b = - -\TT 20 sin 20 \ . 4 ( r j The bending and twisting moments at a point x r distant Q l from OA are given by r M ei = M a cos X (K a r T a ) sin X + wr 3 cos 2 (0 + 0) sin ft 0) dO. T 6i = (T a R a r) cos M a sin 6 + R a r r 3 cos 2 (0 + ) {1 cos (ft - 0)}d0, the last term in each case representing the moment, bending or twisting, about the point x l (Fig. 24), of the load between A and x r On integrating these terms and writing d for d lt the general expressions for M 6 and T g become ~ l){cos 6 si a fi> -r\ a . M. = M a cos - (R a r-l J sm ^ + = ( a - a r cos M sin ^ ivr 6 . . - + sm(9 , [_ _ (i _ cos (9) sin 2) ?-' J ~t~ i^n - cos cos2 . sin2<^ . cos 0] cos 20 -- - cos . . sin2 2\ . sin + g-f - gj + 9 sn i + cos i - (24) From symmetry the slope is zero at the centre of the beam where #1 ;= ^ 0, and, on substituting this value for #1 in (24), and also substituting the values of M a and R a as given on pp. 34 and 35, and equating to zero, the value of T a may be obtained. E.g., Semicircular Girder (0 = 0). D 2 36 A STUDY OF THE CIECULAE-AEC BOW-GIEDEE In this case, on putting < = in (24) M a {0icos 0i + sin 0i} - (R a r T a )0i sin ( 2EI + 'in 3 ' (7 /4 \ ^ ./ll* f\ rt I * " t\ \ f\ \ + ^sm0i-cos0 1 ^sm0i+0ij ; 2CJ (!T fl -JR^ sin i cos + 27? ft r(l - cos 0i) - in- 3 -! 0i - sin 0i (-^ - ~ cos 0i (24') At the centre, where 0i = -., the slope is zero, and M tl = -g- ', K a wr 4 - wr* Tl (IT ^_ T a \TT 7"| , n-r 5 f / T _ TT\ TT __ j. , lg"| _ 27 L3 " \4 _ icr s J 2 """ 9j ^ 2C'J L Ur 3 4/2 3 It follows that, on substituting in (22) and (23) o fl 4- Sin 2 r,f\r,A ' a Me = wr 3 \ '7074 sm ( o (sin 0cos . TT = ivr . T n + T '7074 cos ^ I D 4 The deflection at any point 0i is obtained by writing 0i = in (24') and integrating between the limits 0i and 0. Thus, ZCJ M u 6j sin 6 1 - (R a r - T a ) (sin 8 - ^cos X ) 4. ^{10 - 10 cos X - 2 sin 2 0i - 30 X sin 81} v \ ' . f (T a - R a r) (sin 0i - 0i cos 00 + 2R a r (0i - sin 00 + M a (0i sin 0! + 2 cos 0i - 2) 9 I 2 77 16cos 0i - 16 + 30. sin in l \ J I (25) At the centre, where 0i = ^/centre - "?t P1815 , -01211 T i . JH cy J GIEDEE WITH UNSYMMETEICAL LOADING 37 (18) Girder with Unsymmetrical Loading. Where the loading of a girder does not admit of being represented by a simple trigonometrical expression, or where the girder is not of uniform cross section through- out its length, a solution is most readily obtained by dividing the load, including the dead load due to the girder itself, into a series of comparatively short lengths, and by calculating the moments due to each of these portions of the load separately, by an application of the reasoning and results of (15). In practice a first approximation would be obtained by assuming a likely value for the cross section and weights at each point, and by then applying these results. A second approximation would then be made taking into account the weight of the girder calculated from the sections found necessary by the first approximation, and this would in the majority of cases give results sufficiently near for all practical purposes. (19) Bow-Girder Built in at the Ends and Resting on Intermediate Supports. Assuming all the supports to be at the same level, the reactions of the intermediate supports may be most readily obtained by expressing the fact that the upward deflections at these supports caused by their reactions, are equal to the downward deflections produced at the same points by the loading. (20) Girder with Uniform Loading and Central Support. Let P be the reaction of this support. Let 180 2$, or 2a, be the angle sub- tended by the arc of the girder. The upward deflection at the centre due to the reaction is given by equation (14), in which W '= P, and in which M a and T a have the values given by the curves of Figs. 13 16, for the corresponding value of a or (90 ). The downward deflection at the centre due to the load is obtained by substituting a for d it and by substituting the corresponding values of M a and T' a as given by the curves in Figs. 19 21, in equation (21). E.g., ct = 90 ; = (semicircular girder). The upward deflection at centre = m. [I -e^o-182)] + ^ [oi82 - -5oo) + 1 - 1 + 1 (i _ 2)] . p,.3 f-4674 -03821 L 2A7 2CV J ' The downward deflection at the centre, due to the loading -7272 , and on equating these -727207+ -053EI _ r '4674C'J r The value of this depends slightly on the ratio of El to CJ. Taking this ratio as T25, gives ' 7928 A STUDY OF THE CIECULAR-AEC BOW-GIEDEE Again, 7? a -f R b + P = trier ' ^a = #6 = -^- \TT Also = -801/cr. .17,, + M b = 2T 2 - Pr = -46/n- 2 /. .1/ H = .V 6 = -23v 2 . The value of T a is the difference between the values produced by the load and by 20 80' 90 30 4-0 50 60' 70 Ka/i/es of 8 measured from the end support:. FIG. 25. Bending moments in a uniformly loaded circular-arc built in at the ends and having a central support. (Full-line curve.) the upward reaction P. The first of these is -Z98wr* (Fig. 20) ; the second is -182/V (Fig. 16). .-. T a = {-298 - (-182 x 1-54)} r 2 = -018T 2 . This value may be obtained alternatively by substituting the foregoing values of M a and of E l( in equation (20) with Oi = > and by equating to zero. The values of M g and of T e at any point between the end and the support and distant d from the end then become, on substituting in equations (18) and (19) M g = u-r 2 {l '11 cos '783 sin d}, T e = MT 2 {-801 -783 cos 6 + '77 sin d d}. GIEDEE WITH UXIFOEM LOADING AND CENTRAL SUPPOKT 39 If El : CJ = 10 the values of the end moments and reactions become P = T47 wr ; R a = H b = -885MT ; M (t = M b = '265T 2 ; T a =T b = -OSOirr 2 , and equations (18) and (19) become Me = wr*{l '735 cos "805 sin d\ To = HT 2 {-835 -805 cos 9 + '735 sin 6 9}. Figs. 25 and 26 show the bending and twisting moments at each section of one- 20 30 40" 50 60 70 80' Values oF Q measured From the end Support. FIG. 26. Twisting moments in a uniformly loaded circular-arc built in at the ends and having a central support. (Full-line curve.) half of such a girder with a central support and with El -r- CJ = 1'25, while for com- parison the moments with the same loading but without the central support are shown by the dotted line curves on the same diagrams. Where the girder subtends an angle less than 180, the problem may be solved in an exactly similar manner by making use of the requisite relationships from the fore- going curves. 50 40 A STUDY OF THE CIKCULAK-AKC BOW-GIEDEE (21 ) Circular- Arc Girder, built in at the Ends, with Uniform Loading, and with two Symmetrical Intermediate Supports. Let the angle subtended by the girder be (180 2<), and let the supports (at C and D, Fig. 27) be distant y from each end. Let the upward reaction at each support - P. Let M a ", T a ", RJ' represent such end conditions at A as would be produced by these two reactions alone, and let M a ', T a ', R a ' -represent such end conditions as would be produced by the load alone, with supports removed. Under these conditions the downward deflection at C and D due to the loading would be, by equation (21) 2/v = -(^V r (( ')(smy-7cosy) ACJ ~(T a ' -R a 'r) (sin y - 7 cos 7) + (M a ' - T 2 ) { y sin y + 2 2 cos 7 - 2} + 2/4'r(y - smy)-2T 2 (^-+cos7- 1) . -] (26) where R a ' = utri- J , and A// and T a ' for the particular value of $ obtaining in the girder, are given by the curves of Figs. 1921. The upward deflection at C and, from symmetry, at D, due to the two upward forces P is obtained by substituting y for 61 in equation (12), which becomes ~ |V tt " y sin y - (R u "r - T a ") (sin y - y cos y)] r*_ r(T a " - R a "r) (sin y - y cos 7) + 2fl rt 'V (y - sin 7) "1 *~ 1CJ L + M a " (y sin y + 2 cos y- 2) (27) The values of M a ", R a ", T a " for use in this expression are the sum of the corresponding values produced by each of the two forces P acting at points distant y from A and from B, and may evidently be obtained by adding the values of M a and M b , R a and R/ t , T a and T b , as obtained from the curves of Figs. 13 16 for a girder having the correct value of <, and having the force P at y from A. On substituting these values, each of which is given in terms of P, in equation CIRCULAR- ARC GIRDER, BUILT IX AT THE EXDS 41 (27) and equating to (26), the resultant expression contains P as the only unknown and enables this to be calculated. E.g., Semicircular girder with uniform loading and with two piers at 60 from the ends of the span ( = ; y= 60). From Figs. 19 and 20 the values of M a r , and T a ' for substitution in equation (26) are M a ' = wr 2 ; T n ' '298/rr 2 ; while 74' = 1'5708/rr, and, on substituting, the down- ward deflection at the supports (y = 60) is given by , P564 , -0371 The values of M a ", T tt ", and 74" for substitution in (27) are, from Figs. 13, 14, 15 and 16 M,," = (.17,, + M b )t =0> y = 6 rp = (-588 + -278) I'r = '8Q6Pr. T a " =(T a + T b \ = o, y = GO> = ('156 + -127) Pr = '2837V. R " P "'a * > and, on making these substitutions, v s P 539 . ' 035 1 // '" ' L2A7 "*" 2Gvd * Equating these two expressions for // 00 gives r.564(7j + -o37/';n L-539CV7 + -035 Elj ' and taking El = 1'256'J", this makes P = I'OS^r. The reactions at A and 73 are then given by 74 = 74 = 7C - 74" = in- (j - 1-05Y= -521/rr. \ij / while the moments M a and A// ; are given by ,17, = M n = M a ' M fl " = wr z (1 - "866 X 1'05) = -091-/- 2 . The torques T a and T b are given by T b = T a = T a ' - T a " = jrr 2 {-298 - '283 X 1'05} = '001/rr 2 . The state of affairs at any point on the girder is thus given by the relations (equations (18) and (19) ) : Between A and C M e = M a cos - (R a r - T, ( ) sin 9 + WI A (1 - cos 6) = wi* jl - '909 cos 6 - -520 sin 9\ T e =(T a - R a r) cos B + R, t r - M a sin 6 - icr 2 (d - sin 0) = wr 2 {-521 - '520 cos 6 + "909 sin 0-0} Between (7 and the centre (0 being measured from OA) Me = M a cos (74?- T a ) sin + wr* (1 cos 0) Pr sin (0 60) = wr* 1 1 1-045 sin 1 T e = (T a - 74r)cos + /4>- - ,l/ ft sin - /rr' 2 (0 - sin 0) + 7 J ;- j 1 - cos (0 - 60) j = r 2 1 1-571 1-045 cos ; . Fig. 28 shows the bending and twisting moment diagrams for such a girder, while for purposes of comparison these have also been drawn as dotted line curves on Figs. 25 and 26. From these it appears that the maximum values of the moments with and without supports have the following ratios, the bending and twisting moments for the span without intermediate supports being taken as unity. 42 A STUDY OF THE CIRCULAR-ARC BOW-GIRDER Number of Intermediate Supports. none one at centre two at 60 Maximum bending moment 1-0 26 09 Maximum twisting moment 1-0 11 035 10 o Ol 90 FIG. 28. Diagrams of bending and twisting moments for uniformly loaded semicircular girder, with two intermediate supports, distant 60 from each end. The following table shows how the fixing moments and reactions vary with the ratio of El : CJ in the foregoing example. El CJ P UT Ra_ ivr M wr- T /- 1-25 1-05 521 091 001 100 1-06 511 081 -002 SEMICIRCULAR GIRDER, BUILT IN AT THE ENDS 43 From these figures it appears that a considerable change in this ratio has very little effect on the magnitude of these moments. Semicircular Girder with uniform Loading and with two Piers at 45 from Ends of Span. In this case, the end constants and pier reactions for EI=1'25 CJ become P = l-460?tr ; M a = - -QSlwr 2 ; E a = R b = -llltcr T a = -010/rr 2 . As before, between A and C M e = M a cos e - (R a r - T n ) sin d + v 2 (1 cos 6), T e =(T a - R a r) cos 6 + R a r - M a sin - wr* (d - sin 6), while between C and the centre Me =M a cos 9 (E a r - T a ) sin - Pr sin (0 45) + wr z (1 - cos 0), T e = (T a - R a r) cos + R u r - M a sin + Pr {1 - cos (0 - 45) } - wr* (0 - sin 0). (22) Semicircular Girder, built in at the Ends, with Uniform Loading, and with three Intermediate Supports. Let the supports be arranged symmetrically, P 1 and P 2 being the reactions at the outer supports and Q that at the central support. These reactions may be obtained by expressing the facts (1) that the downward deflection at the centre due to the loading is equal to the sum of the upward deflections at the centre due to the forces JP lt P 2 , and Q, in their respective positions ; and (2) that the downward deflection at PI due to the loading is equal to the upward deflection at this point due to forces P 15 P 2 , and Q ; thus if, for example, P x and P 2 are each at 45 from the ends, we have Downward deflection at Q due to loading J-7272 , -053) j/i-r* -{ _ _i_ - _ '- \<2EI + 26VJ- Downward deflection at P l or P 2 due to loading I 2A7 h 26V/' these values being obtained from equation (21') by substituting the values of 0, viz., 90 and 45, and of M a and T a from Figs. 19 and 20. Again, the upward deflection at Q due to force Q . f-4674 , 0882~l ~'~ from ^ 14 ^ and Fi S s - 13 and and the upward deflection at Q due to the two forces P l and P 2 ( = P from 13 and Fis - 13 and L Also the upward deflection at P l due to force P x = Pt * L~2El + 2C7J from ^ 14 ^ and Figs ' 1B and 14 while the upward deflection at P 1 due to P 2 = Pr 3 + from (13) and Figs. 13 and 14, 44 A STUDY OF THE CIRCULAR-ARC BOW-GIRDER and the upward deflection at P l due to force Q and Figs. 13 and 14. Collecting and equating deflections at the same points gives UT(-7272CJ + '053EI) = Q('4674CJ + -0382EZ) + Z J (-422(X'3 F'4674 , 0882"! force P I : " " h ~ p. 34 p. 37 L 2EI ' 2GV /. P = wr 05 3"| 2GVJ 7272 + -053 4674 Thus, for example, if the cantilever be of uniform section, of moment of inertia /' I 3 and of length /, so that k = Q7 , this becomes 7 272 + -053^ P wr The following table shows how, in the case where I = r and 7 = I', the yielding of this support would modify the end moments and reactions as compared with those experienced with a rigid support or with a cantilever so erected and designed as to deflect under load to the level of the end supports. These figures apply to the case where El = 10 CJ. 46 A STUDY OF THE CIECTJLAE-AEC BOW-GIRDER /' / R,i tor M* ,772 Ta wr i Eigid support 1-47 835 265 030 Elastic support . 828 1-157 586 147 No central support 1-571 1-00 298 It will be noted that, due to yielding of the support, the end moments (and approximately also the average moments) in the girder are increased roughly in the FIG. 30. proportion in which the value of P is diminished. Owing to the comparative shortness of the cantilever it will generally be more economical to design this so as to take the load P corresponding to a rigid support, and to erect this with sufficient camber to allow its deflection under this load to bring it down to the level of the end supports. The same reasoning in general holds however many intermediate supports may be used. (24) Compound Bow-Girder. The state of equilibrium of a compound bow-girder of the type illustrated in Fig. 30, may be obtained by an application of the methods used for the simpler forms. In the case shown, with intermediate supports at the points of inflexion B, C, E and F, the whole of the reactions and the end moments M a , M g , T a , T g , are unknown. From symmetry, however, with uniform loading M g = M u ; T g =T a ; li f =li b ; R e = 7i,, ; R,j = R a , so that in effect the only unknowns are M a , T a , li u , H,,, R c . Knowing the radii i\, r 2 , r 3 , and the angles d v 6 2 , 3 , the total load on the girder, and the position of its centre of gravity, may readily be obtained as in art. 16, p. 28. COMPOUND BOW-GIKDER 47 Calling wl the load, let x be the distance of its centre of gravity from the line joining AG, and let x l and x 2 be the distances of supports B and C from this line. Then taking moments about A G gives irl Again ~R a + R b -\- R c , so that if R b and R c are known, R a and T a may be deduced from these equations. This leaves in effect three unknowns, M a , R b and R , and in order to determine these, three further equations are necessary. These are to be obtained as follows : (1) Span A B. Write down the expressions for the slope and deflection at B in terms of R a , M a , and T a . These are the same as equations (20) and (21), pp. 30 and 31, with i\ taking the place of r. Equating the deflection at B to xero gives the first of the required relationships. (2) Determine values of M b and T b from equations (18) and (19), p. 30, in terms of R a , M a , and T u . (8) Span EC. Obtain the slope and deflection at C in terms of M b , T b , R a and R b , and of the slope at B. Equating the deflection at C to zero gives the second of the required relationships. (4) From equations (18) and (19) determine 3/ c and T c . (5) Span CD. Obtain the slope at D and equate to zero. This gives the third relationship. (25) Shear Force at a given Section. The vertical shear force at any section of a bow-girder is the same as would be experienced at the corresponding section of a straight girder subject to the same loading and to the same reactions. Thus, between an end support reaction R a and the first concentrated load W v the shear force is constant, except for the weight of the girder itself, and equal to R a . Between this load and a second load W 2 , the reaction is R a W\. In the case of a uniformly loaded girder, carrying w Ibs. per foot run, the shear force at a distance x, measured along the arc, from the support A is R a icx for all points between the end and any intermediate support. If there be an intermediate support at a distance x from the end A, and if its reaction be P x , the shear force at a point distant x from A, between this intermediate support and any third support, is given by R a + 1\ - wx and so on. (26) Experimental Verification of Formulae. In order to verify the formulae of this chapter by experiment, measurements of deflection have been made by the authors on a series of bow-girders fixed at one or both ends and loaded either by single concentrated loads or by a uniform load. Some of these girders were of circular section, others of angle section. Values of El and of CJ were obtained by deflection and torsion experiments on straight lengths of the same sections, and these values have bsen adopted in the calculations. 48 A STUDY OF THE CIBCULAK-ABC BOW-GIRDER The following are the results of the experiments : TABLE II. Series Type of Conditions. Angle subtended Deflection (ins.). section. by arc. Measured. Calculated. a Circular Circular arc cantilever 90 1-469 1-475 with weight at free 135 4-475 4-475 end b Ditto with uniform 90 510 502 loading c Semicircular bow gir- a = 30 043 043 derfixed at endswith 45 117 115 single load at a from 60 202 204 one end deflection 90 307 307 at weight d Circular arc girder with 120 075 074 single weight at centre e )J Ditto with uniform 180 310 306 loading f Angle Circular arc girder 90 on 012 with weight at centre 180 124 116 !l Angle Semicircular bow'gir- ( der with single load ) Deflection at weight 036 032 at 45 from one] end Deflection at OG8 072 centre From these figures it appears that there is a very close agreement between experimental and calculated values in every case. (27) Non-circular Sections. The foregoing formulae are of general application to a beam of any section of which the El and CJ are known. The former of these products is usually known or can be NON-CIRCTJLAK SECTIONS 49 determined by calculation with a close degree of approximation for any commercial section. While the geometrical polar moment of inertia J of any section may also be calculated, the product of this J and the shear modulus C of the material does not, how- ever, give the effective value of CJ for use in these formulae, except in the case of circular sections. The reason for this and the question of the effective value of J for non-circular sections is considered in some detail in the following chapter. B.O. CHAPTER III (28) The Torsional Rigidity of Non-Circular Sections. ON the assumptions that the displacement of every point in a section under torsion is proportional to its distance from the centroid of the section, and that a section originally plane re- mains plane after straining, the angle of twist of a straight member of length / is given by TL CJ (28) ;jl where J is the polar moment of inertia of the section, as deduced from its geometrical properties. If the section is circular, these assumptions are fully justified by experiment so long as the stresses involved are within the elastic limit of the material. But this is not the case for any but a circular section. In any other section radial lines originally straight do not remain straight after straining, and sections originally plane become warped under strain. For example, Fig. 31 shows the shape assumed by each section of an elliptical shaft, and Fig. 32 indicates the deformation of a square section under strain. The net result of this is that a given torque produces a greater angular displacement than is indicated by formula (28), and the angle of twist is given by Tl Tl CkJ CJ' . (28A) where J' is the effective polar moment of inertia of the section. In a few simple cases, where the profiles of the section are the graphical representations of definite mathematical functions, values of J' may be deduced from considerations of strain, and Table III. shows such values as deduced by St. Venant. 1 1 Todhnnter and Pearson, "History of Elasticity," Vol. II. Fro. 32. THE TOESIONAL EIGIDITY OF NOX-CIECULAE SECTIONS 51 TABLE III. Type of Section. Remarks. Effective value of /(= /'). Solid ellipse Major axis 2a Minor axis 2& Hollow ellipse Major axes, 2a and 2aj Minor axes, 26 and '2b 1 Square Side = s 14s 4 Eectangle . Lengths of sides, b and d Any symmetrical section, including rectangles, in which the ratio of outside dimensions in any two directions in a cross- section is not very great A area of section ./ = geometrical polar moment of inertia It becomes apparent from St. Venant's investigation that there is always greatest distortion at that part of the section of a shaft or beam under torque, where the surface is nearest the axis. The distortion, and hence the intensity of stress, becomes very great at the apex of any re-entrant angle, becoming infinite where the apex of this angle in FIG. 33. coincides with the centroid of the section. On the other hand, the distortion and stress in the neighborhood of projecting points is very small, so that while such projecting areas at a distance from the axis add largely to the magnitude of the polar moment of inertia, their effect on the tortional resistance of the section is usually inconsiderable. Thus such sections as are usual in I, or channel beams, and which offer a very efficient distribution of material to resist simple flexure, are relatively inefficient to resist torsion, and their inefficiency becomes more pronounced as the distance of their main members from the centroid of the section is increased. As having an interesting bearing on these points the results of investigations on the following sections may be cited. These are (Fig. 33) is 2 52 A STUDY OF THE CIRCULAR- AEC BOW-GIRDER (1) Square section. (2) Ditto with slightly concave sides, and round corners. (3) Ditto ditto ditto and acute corners. (4) Star-shaped section with four rounded points. T 'L Writing 6 = where J', the effective moment of inertia of the section, equals kj, St. Tenant showed that the values of k for these sections were : Section i 2 3 4 k 843 819 778 537 FIG. 34. The concavity in section 3 was about fa of the length of the side, and this small degree of concavity reduces the value of k by approximately 8 per cent. As shown by the value of k for section 2, this concavity has more influence in diminishing the torsional stiffness of a beam, for the same moment of inertia, than the rounding of the corners has in increasing it. The large effect of a greater degree of concavity, accompanied by the massing of material in projecting points of the section, is well marked in section 4. As com- pared with a circular section of the same cross-sectional area and weight, these sections, offer only '891, '867, 828 and '674 times respectively the resist- ance to torsion, notwithstanding the fact that the moments of inertia of their section are respectively 1*05, 1*06, 1 P 07, and T25 times that of the circular section. St. Tenant's investigation of the form of section shown in Fig. 34 is also of interest. This section consists of two isolated masses of material symmetrically situated with respect to the axis of twist ; and on the assumption that this represents, the section of a beam subjected to torque, the investigation shows that the value of k is only '0185. This section approximates more or less closely to the case of an I beam in which the material is mainly concentrated in the flanges, the thickness of the web being small. Comparison between this value for k, and the values obtained by experiment on I sections (see Table V.), is instructive. It is evident that a structural member consisting of two flat bars connected by a lattice bracing must of necessity be excessively weak in torsion. For complex sections, and indeed for the great majority of commercial sections, the difficulties involved in a mathematical investigation of the value of J' are insuperable, and such values can only be determined from torsion experiments. (29) Experimental Investigation of Torsional Rigidity of Commercial Sections. Such experiments have been carried out by one of the authors and are described in the following pages. In all, twenty-one beam sections were tested. The details and EXPERIMENTAL INVESTIGATION OF TOKSIONAL RIGIDITY 53 dimensions of these are given in Table I\ r . With the exception of the solid circular and rectangular sections, and the welded tubes, which were of wrought iron, all were of mild steel. TABLE IV. No. Section. Dimensions. Moments of Inertia (ins. units). Width. Depth. Thickness of Flange. Thickness of Web. Area. sq. ins. ,, ,, J. 1 I 5-01" 8-02" 605" 30" 8-02 91 10 13-10 104-20 2 do. 3-01" 3-00" 325" 200" 2-43 3-70 1-20 4-90 3 do. 1-75" 4-78" 324" 190" 1-91 6-70 0-26 6-96 4 do. 1-66" 3-16" 23" 17" 1-222 1-92 177 2-097 5 do. 99" 1-95" 165" 22" 6825 336 0281 3641 6 do. 76" 1-50" 165" 14" 4141 1328 0124 1452 7 Channel 97" 2-00" 227" 22" 7825 413 0618 4748 8 Angle 1-175" 1-175" 250" 5245 0615 0615 1230 9 do. 1-00" 1-00" 185" 3363 0275 0275 0550 10 Tee 1-58" 1-58" 231" 21" 650 1450 0739 2189 11 do. 99" 99" 135" 145" 2573 0236 0108 0344 Solid 12 Rectangular 87" 1-96" 1-70 5460 1075 6535 13 do. 51" 1-62" 827 1810 0180 1990 Solid 14 Square 96" 96" 920 0702 0702 1404 Hollow 15 Rectangular 872" 1-432" X -0360" thick 151 0479 0223 0702 Hollow 16 Square 1-500" 1-500" X -0502" 296 1035 1035 2070 Solid 17 Circular 1-01" dia. 801 0510 0510 1020 18 do. 876" dia. 601 0288 0288 0576 Hollow Circular 19 (Welded) O.S. dia. 1-305" I.S. dia. 1-05" 473 0826 0826 1652 Hollow Circular 20 (Solid-drawn) O.S. dia. 1-005" I.S. dia. "923" 124 0144 0144 0288 Hollow Oval 21 (Solid- drawn) 862" X 1-74" X -045" thick 1788 0515 0173 0688 The method of carrying out the torsion tests was as follows. The beam under test was mounted between the centres of a six-foot lathe, centre-pops being made on the ends of the beam at the centre of gravity of the section, to receive the lathe 54 A STUDY OF THE CIRCULAR-ARC BOW-GIRDER centres. To one end of the beam was clamped a lever from which was suspended a hanger fitted with a knife edge, and carrying the load. Two pointers, each three feet long, could be clamped to the beam at any desired position. These pointers moved over scales, clamped to the bed-plate, and graduated in degrees and minutes. Readings were taken to the nearest minute. The other end of the beam was clamped to the head of the lathe, the gear being locked to prevent rotation. On the addition of each increment of load, scale readings were taken at both pointers. In order to eliminate the effect of friction at the centres, the torque lever was elevated slightly, and allowed to decend slowly, depressed slightly, and allowed to rise slowly, the angle of mean position being noted. Observations were made for both loading and unloading, and the mean angle of twist per unit of load so obtained. The value of the product of C and J' was then found from the formula. CJ' - Tl ~e where the symbols have the significance already ascribed to them. In each case the experiment was repeated over a span of about half the original span. In no case did the two values of CJ' so obtained differ by more than 3 per cent. The values of the product of E and / were also determined by supporting the beam on two massive knife-edges firmly bolted to the bed-plate; Load was applied to a hanger fitted with a hardened point, suspended from the middle point of the beam. Deflections were measured by means of a micrometer microscope sighted on to a silk fibre fixed to the beam. These deflections were observed to the nearest *001 inch. Readings were taken for both loading and unloading, and the mean deflection per unit load calculated. The value of El was then found from the relationship In order to obtain the values of the two moduli E and C, specimens were cut from the thickest part of each section, turned down to a diameter of about '18 inch, and cut to a length of about 9 inches. The values of C were then found by means of a small torsion meter, and the values of E determined by supporting the specimens on knife-edges and applying a load at the middle of the span. The values of the constants so found have been tabulated in Table V., which also shows the results of the torsion and bending experiments on the beams. The Bending Tests show that in general the experimental and theoretical values of E I agree closely. In the few cases where a fairly large discrepancy exists between them, it is probably due mainly to the fact that the section was not perfectly uniform throughout the length of the beam. These figures indicate roughly the discrepancy that might be expected from calculations based on the ordinary suppositions that a beam is of uniform section throughout, and is perfectly straight from end to end. One point of considerable interest is brought out in the above tests. It will be observed that in the case of the I, channel, and other sections, the values of E obtained are not equal for both axes of bending. In the case of the large I section, for instance, the observed values of E when the web is vertical and when the web is horizontal are respectively 30'7 X 10 6 and 26'4 X 10 6 in.-lb. units. In the former case, the web pro- vides 14'5 per cent, and in the latter case only '64 per cent, of the total moment of inertia. Generally speaking, therefore, the modulus of elasticity of the metal in the flanges is less than that of the metal in the web ; this want of uniformity being undoubtedly produced in the process of rolling. This is confirmed by the results of experiments by Prof. E. Mar- / .7 / 7 .7J>nj5A iwtnwoaqj, tf 1 1 1 c c. IO h- GO C5 CO co CM do CM IO CM o o IO do CM 00 C-l ~c do r 1 >C CM GO CM '~ CM f C: CM ITM to ec tc O! 3 I *~ jo 9n[t!A paAjasqo do U ~ i C ? C > ^ i e n > : C ^ 01 c j 3 4 3 i H Op CM CM -f P do c~. CO t- GO to to op 9) 1" CM 59 CM H : / r-. n noiyuwm O CM 7l ge ri to ^1 i - y-OpaAa^qo ! '' a O O 9 O o v c a got + 7V7 CM IO CM op do ic O ? co 00 to CO- CM o it i' 3 r i ) it X M uainpads punoji n BUI S UKMJ gOt -1- J JO StIIBA CM CO CO S to ) gOt -r /.O ta H * rt 1-1 1-1 v * 1-1 ' ^ "* co C CO / CO CO Oi oo O5 Si OO Op GO (N CO IS 05 OS CO -f o IN N CO 05 O Oi (M O CO CO oo OS o 00 o o O5 CO OS to o o o s o o o o CO Oi o o PH "~< PH 11 .""' 11 1-1 ^ H ^" 11 '. g: O o i~. 2 -f IN CO cc IN o O o IS o oo IM IS IS o to o 00 CO GO 00 p IN - M CO CO 2c. 2b. Theoret. J. k. Effective JorJ'. 1 1-0 1-0 166 841 140 2 1-416 708 209 550 115 4 2-00 500 354 198 070 6 2-448 408 511 096 049 10 3-160 316 917 037 034 EXPEEIMENTAL INVESTIGATION OF TOKSIONAL EIGIDITY 57 d Since an increase in depth renders a section more efficient to resist bending, the most effective value of this ratio when both torsion and bending are to be resisted, depends on the relative values of the two moments. With zero bending moment the section should be square. With zero torque, expe- rience shows that the ratio of breadth to depth should be between 3'5 and 5'0 for best results. With both torsion and bending the most economi- cal ratio will usually lie somewhere between 2O and 3'5, its value increasing as the ratio of bending moment to twisting moment increases. / Sections. A comparison of the results of the torsion tests on I sections Nos. 1 to 6, Table V., indicates that the ratio of actual to calculated value of ./ .diminishes with an increase in the size of the section. The penultimate column in Table VIII. gives the values of k for these sections. The value of J' in inch units is given with a fair degree of accuracy by the relationship = &> (r T FIG. 35. where A is the area of the section in square inches. The last column of this table shows values of A 2 -^- 60, while experimental values of J' are given in column 6. TABLE VIII. Section Number Table V. Approximate Dimensions. 2fl 2o" Area " A " J J' it A* 60 1 8" X 5" 1-60 8-02 104-0 1-04 010 1-07 2 3 4f" X If" 3" X 3" 2-73 1-00 1-90 2-43 6-96 4-90 058 099 0083 0202 060 098 4 5 3" X H" 2" X 1" 1-90 1-97 1-22 682 2-10 364 024 0094 0114 0260 025 0078 6 1 1" v 3" 1.) A 4 1-97 414 145 005 0344 0029 From these figures it appears that for sections 1 to 4 the formula gives results which are accurate within about 3 per cent. These are all commercial sections. The agreement is not so close for section 5, and is unsatisfactory for section 6. These tsvo are not commercial sections, and the relative thickness of web and of flanges is much greater than in commercial sections, especially in section 6, in which the discrepancy is most pronounced. Probably for all normal commercial I sections expression (29) will give results sufficiently accurate for purposes of design. Angle, Tee, and Channel Sections. An examination of the results of the tests on the angle, tee, and channel sections of Table V., shows that the value of k varies widely 58 A STUDY OF THE CIRCULAR ARC BOW-GIRDER with the type of section. The value of J' is given within about 2 per cent, in every case by the relationship /' = m (30) where m varies with the type of section. Values of k and of m are given in Table. IX. TABLE IX. Section. Mean value of k. HI. Channel . 025 40 Tee . 06 25 Angle 09 18 Compound Girder. Experiments were also carried out on a compound girder of the type shown in Fig. 36. This consists of two 8" X 4" commercial I sections, distant 10'3 inches centre to centre, and tied together at intervals of 2' 6" by plates across the bottom flanges. The value ffTl f ftl rf^H f^Ti f J for this combination ~n is 370 inch units ; the value of J' is 2'05 inch units ; and the value of k is -0055. Calling A the total area of both sections, Axis oF Twist . -no as compared with the value a single girder of A* 60 for *-^ the same total weight per FIG. 36. foot run as the combined girder. Tests on Hollow Box Sections filled in with Concrete. Since in a hollow box section torsion is accompanied by distortion of the webs and flanges (Fig. 46) it was anticipated that by filling the interior of such a section with concrete this relative distortion might be reduced to some extent, and the section be stiffened in consequence. To test this point the hollow sections Nos. 15 and 16, Table IV., were filled with cement grout and, after setting for four weeks, were again tested in torsion. The effect of this is, however, not great. E.g., with section (15), J' without filling was '0483, and with filling '0508, while in section (16) J' was increased from '1645 to '1941 by the filling. CHAPTEE IV MAGNITUDE OF SHEAK STRESSES IN A BEAM UNDER TORSION (30) Beam of Circular Section. IN a beam of circular section the shear produced by torsion is everywhere circum- ferential, and varies directly as the distance from the axis of twist. Thus if / be the magnitude of this shear at a radius r, and fs its magnitude at the surface where the radius is a, we have T f f - J-Js a - The moment of the shear on an elementary concentric ring of radius r and of radial width Br will therefore be 2m* . /. . Br a and on integrating this expression over the whole section of the beam and equating the result to the external torque T, we have /.= (3D Here f s is the maximum circumferential shear in the section. This formula is applicable to both solid and hollow circular sections. (31) Sections other than Circular. In a non-circular section under torsion the assumptions that the shear at any point is perpendicular to the radius at that point and is proportional to its distance from the axis of twist, are no longer true. It has been shown both by St. Venant and by Bach 1 that the maximum transverse shear stress in any non-circular section under torque occurs at that point on the surface which is nearest to the axis of twist ; that the stress is great in the neighbourhood of re-entrant angles and zero in the neighbour- hood of projecting corners. Expressions for the maximum shear in the case of a few of the simpler sections such as the ellipse and the rectangle have been deduced by St. Venant, and are given on p. 72. Autenreith 2 assumes that the stress at a given point P (Fig. 37) on the boundary of any solid or hollow section bounded by a continuous curve convex outwards, is given by 9T T f. = jf (82) where T is the torque, A the area of the section, and r is the length of the perpen- dicular from the centroid of the section on to the tangent at P. The maximum shear stress will thus occur where r is a minimum, i.e., at the end of the minor axis of the section, and the minimum surface shear at the end of the major axis. On the same assumptions the surface shear in a hollow section having a continuous 1 "Elastizitat und Festizkeit." 2 Zeitschrift des Vereines deutscJter Inyenienre, 1901, p. 1099 60 A STUDY OF THE CIKCULAK-ABC BOW-GIKDEK boundary, in which the ratio of inner to outer radius is sensibly constant and equal to y for all radii, is given by 27' /.= . (33) (32) Solid and Hollow Elliptical Sections. For a solid or hollow elliptical section, having semi-major and minor axes a and b, the value of ; at any point P whose co-ordinates are xy (Fig. 37) is given by .34) SOLID AKD HOLLOW ELLIPTICAL SECTIONS 61 In a hollow section having a and b as the semi-major and semi-minor axes of its external surface, the area of section is , . ai bi TT [ab a\bi\ , and since = = / a o /. A =Trab{l y 2 } // 1-0 -9 -8 -7 -6 -5 -4 -3 -2 / Note.- Intercepts of Normals Give Values of y FIG. 38. Diagram showing distribution of surface shear stress in a'solid elliptical section subjected to a twisting moment. Thus in'the general case J * -nab [I -y 2 ] [1 + y 2 ] r 27Vv/ 2 (a 2 b 2 ) + /> 4 y 4 ] (35) 62 A STUDY OF THE CIRCULAR-ARC BOW-GIRDER and for a solid elliptical section (y = 0) this becomes _ arvv - ip) + ''* ~ ird'b* (36) Note:- Intercepts of Normals Give Values of FIG. 39. Diagram showing distribution of surface shear stress in a hollow elliptical section subjected to a twisting moment. The maximum shear occurs at the end of the minor axis where y = b, and is given by 277 /(max.) which agrees with St. Venant's result. The minimum stress on the periphery is given by (37) 22' lniuj y 4 ) SOLID AXD HOLLOW ELLIPTICAL SECTIONS 63 Where a = b = r, each of these expressions reduces to 27V f 77 \1 .4 _ . (38) the expression for the shear at the periphery of a hollow circular section. Figs. 38 and 39 show respectively the distribution of surface shear in a solid and a hollow elliptical section, in each of which a : b = 1'5, while y = '934. These are subject to the same torque and have the same cross sectional area. The magnitude of the stress is indicated by the normal to the surface, intercepted between the sur- face and the curve. In this case the maximum stress in the solid section is 5 times as great as in the hollow section. In a solid circular section of the same area the maximum stress is "82 times that in the solid elliptical sec- tion, while in a hollow circular section having the same thickness and the same area as the hollow elliptical section, the maximum stress is "76 times that in the latter section. While the assumptions made in deducing the foregoing formulae give results in close agreement with ex- periment if the boundary is a con- tinuous curved line, they fail to do so if the section has a discontinuous boundary. In the latter case the re- searches of Bach indicate a state of zero stress at projecting points, and, in an extreme case would postulate zero stress at the corners of a poly- gonal section no matter how closely this approximates to a circle. To obviate this difficulty Autenreith as- sumes that the stress at such a corner depends upon the included angle, being zero for a right angle, and that, at any point in the surface of such a section in which this angle is not less than 90, it is given by FIG. 40. - 2 # fi (* r L sm a . (39) where /; is the circumferential shear stress ; r the length of the perpendicular from the centroid to the corresponding side of the polygon ; a constant ; z the distance from the mid point of the side to the point at which the stress is required ; c half the length of the side ; and a is the included angle (Fig. 40). 64 A STUDY OF THE CIRCULAR-ARC BOW-GIRDER When a = 180, i.e., for a circular section, this makes /, = constant. When a = 90, i.e., for a square or rectangular section, f s becomes zero when z = c (at corner), and attains a maximum value when z = 0, i.e., at the centre of the side. In these two extreme cases the formula thus agrees with the results of experiment. Assuming that at any point in the interior of the section the component of the shear stress normal to the radius vector is proportional to the distance from the centroid, an expression may be obtained for the moment of the shear on any element, and on integrating this over the whole section and equating to the torque the value of the constant ft may be obtained. This is given by ..... (40> . where A is the area of the section. Since p sin < = z (Fig. 40) equation 39 becomes 867' L /psin 2 For a hollow polygonal section in which the ratio of inner and outer radii vectores is sensibly constant and equal to 7, this formula becomes 36 T | _ fp sin = 0, and is given by - , where, for a solid section, AT li = ^-. (43) 9 2 sm a and, for a hollow section, ift (44) 9(1 + 7*) - 2 sin a (1 + y* + y 4 ) (33) Rectangular Sections Box Sections. In a solid rectangular section (Fig. 41), whose longer side is 2 e and shorter side 2 b, r for the shorter side is c, and for the longer side is 1>. Also sin a = l, so that, for the longer side equation 41 becomes and for the shorter side Thus the maximum stress in the longer side (at its mid point, where = 0) is given by /_,:= 2-57-^ . (46) and the maximum stress in the shorter side by At the corners in each case/, = 0. f 2-57 T. . (46A) ./(max.) " BECTANGULAK SECTIONS BOX SECTIONS 65 In the case of a hollow rectangular or box section in which y is sensibly constant equation (42) applies. The shear at any point in the longer side is given by 18 T L /psin from which Ab[l(l 4- y 2 ) - 2y 4 ] v 1ST /(max.) (47) (48) FIG. 41. while for the shorter side '.-= and 1ST Ac[l(l + i /(max.) sn 1ST + y 2 ) - 2y 4 ] ' . (49) . (50) From equations (45) and (47) it appears that the curves of stress distribution in a rectangular section are parabolic. Figs. 42 and 43 show such curves drawn respectively for a solid and a hollow rectangular section having the same ratio 1'5, of depth to breadth, and the same cross sectional area and weight per foot run. In the hollow section the ratio of inside to E.G. F 66 A STUDY OF THE CIRCULAR-ARC BOW-GIRDER outside dimensions, or y, is '975. From these curves it appears that the maximum stress in the box section is about 19 per cent, of that in the solid section. Comparing diagrams 39 and 43, it appears that the ellipitical section is the more efficient in that the maximum stress is only 72% of that in the box section. In the FIG. 42. Diagram showing variation in surface shear stress in a solid rectangular section sub- jected to a twisting moment. Ratio breadth =.1-5 ; Area of section = 2-4. ordinary box section used in practice the value of y will not in general be the same for the top and bottom flanges as for the webs, nor can it be the same for different points on web or flange since these are of uniform thickness. From the following table, which shows calculated values of il in the formula ./(max.) (51) EECTANGULAB SECTIONS BOX SECTIONS 67 for a hollow box section 4 ft. square and with different thicknesses of metal, it appears, however, that a given percentage variation in y only produces about one-half the same st -ss ia a rectangular box-section sub- Eatio breadth 7= '975. Area of section = 2-4. percentage variation in O. In practice the mean of the values of 7 measured at the mid points of the two sides will give results within a few per cent, of the truth. p 2 68 A STUDY OF THE CIECULAE-AEC BOW-GIEDEE Thickness of Metal. i" i" 1" i" H" H" 7 989 978 968 958 947 937 A 47-75 95-0 141-7 188-0 233-7 279-0 O 1-510 1-517 1-532 1-538 1-542 1-548 The foregoing investigations of Autenreith are based upon a consideration of the stresses involved during torsion. St. Venant, considering the strains produced, obtained the expression rise + 9/n /(max.)- _ 40c . 2/>2 _ _OT = Ab for the maximum shear stress in a rectangular section of sides 2c and 2/>. In this formula 12 = 1-5 + 0-9 -. c Table IX. shows how 11 varies with the ratio of depth to breadth. TABLE IX. Ratio -. o i 2 3 4 6 <; 7 8 9 KI li (St. Venant) . 2-40 1-95 1-80 1-72 1-68 1-65 1-63 1-62 1-60 1-59 According to Autenreith il is independent of the ratio e-i- />, and has a constant value 2'57, so that stresses calculated from Autenreith's formula are greater than those obtained by St. Venant, the difference becoming more pronounced as this ratio is increased. Bach's experiments on the whole appear to show that Autenreith's values are in closer accord with the result of experiment, and for purposes of design these may be adopted with some confidence. The calculated stresses, if they err at all, will do so on the side of safety. (34) I Sections. Little definite is known as to the magnitude and distribution of stress in a member of I section under torque, except that the stress is greatest at the mid point I SECTIONS 69 of the web and is zero at the extremities of the flanges. Since the stress is always large in the neighbourhood of a re-entrant angle, it is probable that it will be large at the junction of web and flange, particularly where the radius of the fillet at this point is small. As to this point, however, no definite information is available. From experiments on I sections made of lead Bach found that rupture always occurred at that point on the web nearest to the centroid of the section, and deduced the expression /^ax, = 4-5^ (52) where A is the total area of the section and t is the thickness of the web. Some confirmation of this formula has been obtained by the authors. Thus considering I section No. 1 (Table IV.), the effective value of J' for the whole section is T04, while J' for the web if isolated from the rest of the section would be approximately 086. Adopting these values, the web may be expected to take approximately = '082 of the total torque, and from formula (46), p. 64, the maximum stress in the web would then be equal to 2-57 X '082 T Ab where b is the half thickness of the web, or ^. a On making this substitution the formula becomes 4-2 T /(max.) At which is in fair agreement with Bach's expression for the same stress. Although the stress at other parts of the section is indeterminate, experiment shows that if the web is made stiff enough to withstand this stress the remainder of the section is amply strong. (35) Horizontal Shear in a Beam Subject to Torsion. What aver be the magnitude of the transverse shear stress due to torsion at a point in a vertical section of a horizontal beam, this shear will be accompanied by an equal shear stress on the horizontal plane passing through the same point. In a beam of box section in which the depth exceeds the breadth, or in a beam of I section, the magnitude of this shear on horizontal layers is a maximum at the neutral axis. (36) Resultant Shear on Horizontal and Vertical Sections of a Beam Exposed to Torsion or Bending. The resultant shear at any point in a horizontal or vertical section of a beam is the algebraic sum of the shears due respectively to bending and to torsion. The shear stress due to torsion has already been discussed. The shear stress due to bending, or to the application of the vertical loads and reactions which produce bending, varies from point to point in a section. If q denotes the intensity of shear due to this vertical loading at a point distant 70 A STUDY OF THE CIRCULAR-ARC BOW-GIRDER ~~! from the axis of bending, and if the breadth of the section at this point be y lt this shear stress is given by l (38) where F is the shear force at the section in question, and z 2 is the distance of the outer fibres of the section from the neutral axis. FIG. 44. In a rectangular section of breadth 26 and depth 2c, while if/2 = c, and expression (53) becomes Tc 2fd^ = //i = 2/> is constant, 2 i J (54) This distribution of shear over the section is parabolic. The maximum value occurs q 77* q T, T at the neutral axis where z\ = 0, and is equal to - - or - , or to 1'5 times the mean o be ' A shear over the section. The minimum value, zero, occurs at the outer extremity of the section where z 1 = c. (37) I and Box Sections. In the case of an I or rectangular box section the breadth is constant over the web and is suddenly increased at the flanges. As a result of this the magnitude of the shear stress in the flanges is much less than that in the web. The distribution of this stress is indicated in Fig. 44. In an average section the intensity of stress in the 1 Moiiey, " Strength of Materials," Chapter V. I AXD BOX SECTIONS 71 web does not change greatly, and the usual assumption that the web carries the whole vertical shear force with uniform distribution gives stresses which are in fair agreement with, and usually slightly higher than those actually attained. In a hollow box section formed by the rectangles 2&, 2c, and 2&i, 2ci, or in the corresponding I girder (Fig. 44), in the flange at a height z 1 from the neutral axis. 3 F 8 [be 3 b lCl 3 ] . (55) while in the web at a height z lt 3 F \b(c 2 " q ~ 8 [be 3 b^ 3 } ! b hi .21 and, at the neutral axis, (/(max.) 3 F $ [be 3 b lCl 3 ] b - . . (56) (57) It should be noted that whereas the shear on a vertical section produced by the vertical loading acts in the same direction at all points in the section, that due to torsion acts in opposite directions at opposite ends of a diameter. It follows that the shear stresses due to bending and torsion act in the same direction in one of the webs of a box girder, and in opposite directions in the other, and that under such combined moments one web will be much more heavily stressed than the other. The nature of the resultant shear stress distribution over the vertical section of such a girder is indicated by the curves of Fig. 45. 72 A STUDY OF THE CIRCULAR-ARC BOW-GIRDER TABLE X. Type of Section. Maximum surface shear stress. St. Venant. Autenrieth. Solid Circular Eadius r 2T TTT 3 27 7 Try 3 Hollow Circular radii TI and r 2 2T yi 2r yi ^[n 4 - r a *] wtn 4 - ^ 4 ] Solid Elliptical T\TrHor Avifi 9,/* 2T 2T Minor = 26 7TC6 2 7TC6 2 Hollow Elliptical formed by [2c 26] [2c 26 ] 2T6 77 [c6 3 c 6 3 ] Solid Eectangular Long Side = 2c Short Side = 26 T15c + 961 T 643^ t-6 2 L 40c 2 6 2 J Hollow Kectangular is r Short side = 26 [7|l+7 2 j -VI ' ^6 Any Polygonal Section Had. of InsfHbpd CirnlA 18 T Included Angle = a [9{l + y 2 } 2sina[l + y 2 + y 4 ]] ' Ar I Web Thickness = t z CHAPTER V (38) General Principles of Design of the Bow-Girder, FROM the data of Chapters III. and IV., it appears that where a beam is exposed to any appreciable torsion, the box section is from every point of view the most suitable, and, for beams of considerable span, or carrying heavy loads, is the only practic- able section. For comparatively small spans ; for spans in which the radius of curvature is large and the angle sub-tended by the arc between successive supports is small, or for moderate loads, the I section may be permissible, but in general its use is to be deprecated wherever combined torsion and bending is anticipated. In any case, where not barred by other considerations, intermediate supports are, as shown by the results of the investigations in Chapter II., of the greatest value in reducing the applied moments, and especially the twisting moment at a given section. In a box section exposed to twisting and bending, a general consideration of the problem indicates that most economical results are to be obtained where the ratio of depth to breadth has a value somewhere between 2'0 and 3'5, the former value applying to encastre beams without intermediate supports and subtending an angle in the neighbourhood of 180, and the latter for beams adequately supported at intermediate points or subtending angles not exceeding 45. The following may be taken as affording a first approximation to the relative dimensions of such a girder designed for heavy duty : Angle subtended by arc between supports. 180 150 120 90 60 30 depth 2-0 2-25 2-5 2-75 3-0 3-25 breadth Having assumed a suitable section for the girder, the tensile and compressive stresses due to the bending moment, and the shear stresses due to the vertical loading, are to be determined for each section of the girder, as in the case of a straight girder, the value of the bending moment being obtained from the data of Chapter II. The value of the twisting moment at each section having been calculated in the same way, the shear stress due to this may be determined by an application of the results of Chapter III., and this shear stress is to be added to the shear stress due to the vertical loading, to give the actual shear at a given point in the section. In the box or I section both components of shear have their maximum value at the neutral axis. The shear in the flanges of such a girder, due to the vertical loading, is sensibly zero. That due to torsion is in general also small, and where the flanges are of adequate thickness to withstand the direct stresses due to bending there is little question as to their ability to 74 A STUDY OF THE CIRCULAR-ARC BOW-GIRDER take care of the additional small stress due to torsion. Having obtained the resultant shear in the webs, these should be designed by the ordinary rule applicable to the web of a straight plate-web girder subject to the same stress. 1 Under torsion such a girder tends to buckle as shown by the dotted lines of Fig. 40, and particular attention should be paid to stiffening the webs against this action. Under normal circumstances this may be accomplished by the use of angle or tee stiffeners, between flanges, reinforced if necessary, where the torsion is greatest, by the addition of a cover-plate to the web. The pitch of the stiffeners should, strictly speaking, diminish as the torsion increases. Where torsion is large the pitch should not exceed the depth of the girder, for girders less than 2 feet 6 inches deep, and should not exceed about one half the depth for a girder 6 feet deep. Special attention should be paid to the design of the riveting at the junction of web and flange, since this has not only to with- stand a shear of magnitude equal to that of the vertical shear at this point, but has also to resist the tendency to relative distortion indicated in Fig. 46. This latter effect also involves the use of somewhat heavier angle sections than are usual in the straight girder. Where joints in the web plates are necessary these should be placed where the sum of tor- sional and load shear is a minimum. As an example the preliminary design of a bow girder of uniform section of 30 feet radius, built in at the ends and subtending an angle of 120, and carrying a uniform load of 2 tons per foot run, may be considered. The values of M e and TO for such a girder having El : CJ = T25, , are given by the curves of Figs. 22 and 23, $ being 30. From these curves it appears that MO has its maximum value ( 42 /rr 2 ) at the support, while at this point T = '048 irr 2 . The maximum value of T g ('052 wr 2 ) occurs at approximately 30 from the support, but since at this point MO is zero, and since the at support, vertical shear force is only wr I5~~^"~gj as a g a i nst wr 5 the latter will be the point of maximum resultant stress. Preliminary investigation indicates that a box girder 5 feet deep and 2 feet wide, with flanges \\ inches thick and webs \ inch thick will be somewhere near the required section. For such a section I = 104 X 10 3 (inches) 4 units; while J= 110 X 10 3 units. From Table VI., k for the given ratio of depth to breadth is '413, so that J' = 45'5 X 10 3 (inches) 4 units. Assuming E = 30 X 10 6 Ibs. per square inch and C = 12 X 10 6 Ibs. per square inch, the effective value of El : CJ becomes 5*73. From Figs. 19 and 20 it appears that the values of the end moments M n and T., for this value of the ratio when = 30, are M a = '435 in- 2 and T a = '067 wr 2 . The effective load per foot run, including the weight of the girder, is approximately 2*2 tons, so that the moments become 1 See " The Design of Plate Girders and Columns," Lilley, or any similar woik. GENERAL PRINCIPLES OF DESIGN OF THE BOW-GIRDER 75 M = -435 X 2-2 X 900 = 880 ft. tons T = '067 X 2-2 X 900 = 133 ft. tons while the shear force F = 2'2 X 30 X X \ % = 69 tons. -LoU Flanrfes. Adopting a working stress of 6 tons per square inch in tension and com- pression, and assuming an effective depth of 57 inches, we have 6 X a f X ST = 88 A4: .'. a* = 61'8 square inches where a f is the flange area. Assuming this to include | the area of the webs ( = ^ X 57 = 7 square inches approx.) the required area of flange plates and angles is 54'8 square inches. This might be made up of 2 plates, f" X 33" = 49'5 square inches 2 angles 6J" X 4" X '55" = 11'5 Total 61-0 From this is to be deducted the area corresponding to two rivets, and assuming these to require 1-inch holes, this will be approximately 5 square inches, leaving an effective area of 56'0 square inches, or slightly more than is required. Webs. Calling a w the area of the two webs, the maximum shear stress due to 69 vertical loading = - - tons square inches. The maximum shear stress due to torque - #to 1'547 T = rj (p. 65, equation 48), where A is the effective area of the section to resist A f) torsion and b is the breadth across the webs. Allowing ^ inch between the edges of angles and of flange plates, 26 becomes equal to 33 10 = 23 inches, while A =. (a w -\- area of a 23" width of flanges) 99 X 23 a w -\ 33 = a w -\- 69 square inches The resultant shear stress in vertical and horizontal planes at the neutral axis is then given by 69 1-54 X 133 X 24 . K + 69)X 23 Equating this to the working shear stress, say 3 tons per square inch, and simplifying gives a w 2 - 25-8a w 1587 = 0, from which a, = 54'4 square inches. If t be the thickness of the web plates this makes It X 57 = 54-4 . . t = '477 inch or, say, | inch. Rivets. Assuming the centre line of the riveting at the junction of webs and flanges to be 3 inches from the edge of the web, or at a distance 25'5 inches from the 76 A STUDY OF THE CIECULAE-AEC BOW-GIEDEE neutral axis, the shear stress at this point due to the vertical loading is, by equation (56), p. 71, equal to 0'90 ton per square inch of web section. The shear stress at the same point, due to torsion, is, by (47), p. 65, equal to 1-547- J M\ 2 I ~M~ I 1 ' (7) } , i 25-5 where c -=io- so that this stress equals '2775 X 77 Ab _ -2775 X 1-54 X 133 X 24 (57 + 69) X 23 = '47 ton per square inch The resultant horizontal or vertical shear at this point is therefore '90 + '47 = T37 tons per square inch. Considering one of the web plates, the horizontal shear force corresponding to the shear stress over a horizontal length p inches is 1'37 pt tons = '685 'p tons Then if p be the pitch of the rivets and R the safe working resistance to shear of one rivet R Adopting a working stress of 5 tons per square inch for rivets in shear, and using -inch rivets (area "602 square inch), gives 5 X '602 J>= ~- = 4'4 inches. To allow for the stress on the rivets due to the tendency to distortion indicated in Fig. 46, the pitch would be reduced to about 4 inches, or alternatively two rows of rivets with a correspondingly greater pitch would be used. Stiffen ers. Considering the web as a column whose effective length is \/2 times the distance between adjacent stiffeners the allowable mean shear stress depends on the ratio of this length I to the least radius of gyration " r " of the plate. For a J-inch plate r ( = p= j = -144 and / -f- r = 6'92. In the case in question the mean stress in the web is approximately (3 + T4)-^2 = 2*2 tons, and for this stress Moncrieff 1 has shown that the maximum permissible value of I -=- r is about 265. This makes I = 265 6*92 = 38'3 inches, in which case the distance between the stiffeners would be 38'3 -f- \/2 27 inches. As the shear diminishes, this distance is to be increased to suit, up to a maximum of about 3 feet 6 inches. Over the end bearings the stiffeners should be designed as columns of sufficient strength to transmit the total load. Intermediate stiffeners would be about 4" + 3" + f" angles. For a more detailed examination of this point and of details of design the reader is advised to consult any modern work on the design of girders. 1 J. M. Moncrieff, Trans. Am. Soc. G. E., Vol. XLV., 1901. See also "Structural Engineering," Husband & Harby, Longmans & Co., p. 154. TI APPENDIX A THE following list of integrals will be found of service in solving the various problems involved in the circular-arc bow-girder. f f I cos dd = 6 sin -f cos ; sin (19 = sin cos . J Q _ sin 20 1 .C e [D - [BD 2 M 2 ] 2 4Z)kZ [D d]* - bd 2 ] 2 4BDbd [D - d]' 12 [BD - bd] + 12 [BD - bd] [DB* - f/fr 2 ] 2 - 4BDbd [B - b]* 12 [BD bd] ^ [bD 3 + Bd 3 ~\ INDEX A. Angle sections, torsional rigidity of, 57 Appendix A, 77 B, 78 Autenrieth, investigations of, on the torsion of beam, sections other than circular, 59, 72 B. Bach, researches of, 63, 69 Beims, bending of, 1 best section to resist torsion, 73 continuous, 3 having more than two supports, 5 curvature, deflection, and slope of, 2 distribution of stress in, 59 encastre, effect of settlement of one support, 7, 45 uniform loading of, 3 unsyinmetrical loading of, 8 with intermediate support, 5 with no intermediate support, 4 Box-sections, distribution of shear stress in, 70 torsional rigidity of, 56 C. Cantilever, circular-arc, with single load at free end, 15 uniformly loaded, 16 straight, deflection at free end of, 2 Castigliano's theorem, 11 Channel sections, torsional rigidity of, 57 Continuous beams, see Beams. D. Deflection of circular-arc bow-girder, 14 cantilever, 2 straight beams, 2 straight cantilever, 2 Deflection produced by shear forces, 13 Distortion of a beam section under torsion, 74 Distribution of shear stresses in a beam, 59 E. Effective polar moment of inertia, 50 Encastre beams, see Beams. Equation of three moments, 6 Experimental investigation of torsional rigidity of commercial sections, 52 verification of formulae for circular- arc girder, 47 F. Fixing-moments in circular-arc bo w- girder, 14 in encastre and continuous beams, 3 Flexual strength of beams, experimental in- vestigation, 52 Formulae, for deflection of bow-girder, Chapter II., 14 straight beams. 2 for shear stress in a beam under torsion, Autenrieth, 59 for torsion of beams, St. Venant, 72 G. Girder, box section, distribution of shear stress in a, 70 stiffening of a, 74 circular-arc bow, Chapter II., 14 carrying concentrated load, 18 carrying uniform load, 28 carrying uniformly loaded platform, 34 compound, 46 effect of depression of supports, 45 equilibrium of, 14 general principles of design of, 73 shearing-force at any section of a, 47 uusymmetrical load- ing, 37 with intermediate supports, 37 with one central support, 37 with two symmetrical supports, 40 semi-circular-arc bow, carrying concen- trated load, 24 semi-circular-arc bow, carrying uniform load, 32 semi-circular-arc bow, carrying uni- formly loaded platform, 34 " semi-circular-arc bow, supported by cantilever, 45 semi-circular- arc bow, with two inter- mediate supports, 40 semi-circular-arc bow, with three inter- mediate supports, 43 straight, Chapter I., 1 curvature of, 2 deflection of, 2 distribution of shear stress in a, 47 80 IXDEX Girder, straight, equilibrium of, 1 resilience of under bending, 10 torsion, 12 H. Hollow sections, distribution of shear stress in, 65 effect on torsional rigidity of concrete filling, 58 torsional rigidity of, 60, 64 Horizontal shear stress in a beam, 47 I. I sections, shear stress due to torque, 68, TO torsional rigidity of, 57 Inertia, moments of, for various sections, 78 J. J, effective value of, in commercial sections, 54 M. Maximum shear stress in a beam section under torsion, 69 Moments, bending and twisting moments in a bow-girder, 14 end fixing moments in circular-arc girder, 14 of inertia of various sections, 78 N. Neutral axis, shear stress at, 59 P. Polar, moment of inertia, relation between actual and theoretical in commercial sections, 5-1 Principles of design of a bow-girder, 73 R. Relation between curvature, deflection, and slope of a beam, 1 Resilience, flexual, of beams, 10 torsional, of beams, 11 Resultant shear stress in a beam subjected to combined bending and twisting, 69 Rigidity, torsional, of non-circular sections, 50 S. Sections, deformation of, under torsion, 74 moments of inertia for various, 78 most suitable type, to resist torsion, 73 Shear stress due to a torque, in sections having a continuous boundary, 60 due to a torque, in hollow sections, 65 in I sections, 68, 70 in solid polygonal sections, 63 horizontal, in abeam under torsion, 69 in a beam, due to vertical loading, 47 under combined loading and torsion, 69 under torsion, 69 in sections other than circular under torsion, 60 Shearing force, at any section of a bow-girder, 70 deflection produced by, 13 St. Venant, investigations of, on the torsion of beam sections other than circular, 51, 52 Supports, effect of sinking of supports, 7, 45 T. Tee sections, torsional rigidity of, 57 Theorem, Castigliano's, 11 of three moments, 6 Theory of bending, 1 Torsional rigidity of non-circular sections, 50 W. Webs, design of. in bow-girder, 75 BRADBURY, ACNEW, & CO. Ll>., I'KIXTKKS, LoNlniN AVIi TOXI1KIIM ;K. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. MAY 30Sep'52pF PEB201953LU JAN 17 1955 Lfl K LD 21-95w-H,'50(2877sl6)476 YE 012 6 - UNIVERSITY OF CALIFORNIA LIBRARY