ANALYSIS OF STATICALLY INDETERMINATE BUILDINC BY THE APPLICATION OF MAXWELL’S THEOREM OF RECIPROCAL DISPLACEMENTS JOHN WILLIAM ROWLEY B. S. University of Missouri, 1921 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN CIVIL ENGINEERING IN THE GRADUATE SCHOOL OF THE UNIVERSITY OF ILLINOIS, 1922 URBANA, ILLINOIS r M Digitized by the Internet Archive in 2015 https://archive.org/details/analysisofstaticOOrowl /37ai3 tAp 1^22 R79 UNIVERSITY OF ILLINOIS THE GRADUATE SCHOOL 1922 i HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION BY_ JQ.H1L. WIL LIAM ROWLEY ENTITLEDALALY-S1S_0F_ : _BU ILDIIIG 5 MTS BY_THE APPLICATIQH QF-MAXY/ELLJ-a- THEORF.I PLACEMENTS BE ACCEPTED AS FULFILLING THIS PART OF THE REQUIREMENTS FOR THE DEGREE _QF_ SCIENCE I2I CIYIL ENGINEERIIIG In Charge of Thesis Head of Department Recommendation concurred in* Committee on Final Examination* •Required for doctor’s degree but not for master's 509405 I TABLE OF COIJTEI^TS I. INTROBUCTIOU Pag !• Preliminary. 1 2* Scope and Purpose 4 3. Acknowledgements • • 4 II. OUTLm OP METHOD 4. General Application of Maxwell’s Theorem of Reciprocal Displacements 6 5. Application to Class I, Columns Hinged at Their Bases 8 Case A, Horizontal Reactions 8 6. Application to Class II, Columns Fixed at Their Bases. 15 Case A, Horizontal Reactions • ...... 15 Case B, Vertical Reactions 20 Case C, Resisting Moments at the Bases of the Columns 26 III. APPLICATION TO THE DETSRIIIMTION OF REACTIONS 7. Determination of Reactions Due to Vertical Loads 33 8. Determination of Reactions Due to Comhined Wind and Vertical Loads. 36 9. Bents Analyzed 39 10. Results. 40 IV. TEST OF PAPER MODEL OF BENT 11. Description of Test .42 12. Results of Test 45 \ I I s I / ( ) k i I I ' I II V, DISCUSSIOIT MD CONCLUSIONS Page 13* Effect of Variations in Different Characteristics of the Bent on Reactions Due to Vertical Loads. .47 14. Errors in Connnon Assumptions for Various Combinations of Wind and Vertical Loads. ... .50 15. Conclusions 53 ■*? » ' 0 * inr' :j '|•f'!i^^^W ■r^'^ Jrrol * - ■ ' J . -o ' .. 'f ;> . t> \ ;^ 4 itv.v. .. . \ ^ v^v” to^ •. iv "i.'Ci. i'ji no J’lsaS * ^rr^\iS’.c -: iio'~,':oO ..i . xO'i'i • r^ri.' ' 'r'-. • :noi ' J.'X: >■■’!''■ 0 . va. ^ 1^ ' ' f •> . I ( Ill LIST OF TABLES VERTICAL DEPLECTIOITS OF THE VARIOUS P^OJEL POINTS AND Page COICPUTATION OF THE REACTIONS AT THE BASE OF EACH COLUMN- 55 Page !• Design of Bent, Span Length 40 ft 96 p Z 2 * Computation of Span 40*- Col, Hgt,21*; Class I, Case A, and Class II, Case A,... ••••••• 3, Computation of ZJ: , Span 40*- Col.Hgt, 21*; Class II, A 97 99 Case B, and Class II, Case C. 4. Reactions at Column Bases Due to a Vertical Load of 1 Lh. at Various Panel Points; Span 40 ’-Column Height 21*; Class I, Hinged Columns, and Class II, Fixed Columns* 101 5* Design of Bent, Span Length 20 Ft* 102 6 * Design of Bent, Span Length 30 Ft* 103 7* Design of Bent, Span Length 50 Ft* * • • 104 8 * Design of Bent, Span Length 60 Ft* • * * * 105 9* Reactions for Uniformi Vertical Load, of 1 lb* per Foot Length of Truss 106 10*Reactions for Vertical Load of 1 Lh* at Point 1 * . * . 107 ll*Reactions for Vertical Load of 1 Lb* at Point 4 * * * * 109 12*Horizontal Reactions, Class I, Case A,ColT 2 mns Hinged; Combined Wind and Vertical Loads **•••*.*• Ill 13*Ratio Hinged Columns;Wind Load Combined with Various H Vertical Loads 114 14*Horizontal Reactions, Class II, Case A, Columns Fixed; Combined Wind and Vertical Loads •**•*..• 115 IV. . TT^ Page 15. Ratio Fixed Columns; Wind Load Combined with H Various Vertical Loads 118 16# Resisting Moments, Class II, Case C; Combined Wind and Vertical Loads ••••••••• 119 17. Points of Contraflexure , Fixed Columns; Wind Load 20 Lbs. per Square Foot; Various Vertical Loads . 122 18. Comparison of Reactions Obtained by Model and by Theory; Span 40»-Col. Hgt. 21* 127 T T Ij / i: I I Y 1 . 2 to 7 to 21 to 17 to 21 . 22 . to 27. LIST OF FIGURES Line Diagrams of Bent. 40-Ft. Span and 21-Ft. Coltunn Height 6. Diagrams to Accompany the Solution for the Horizontal Reactions on the Hinged Columns, Class I, Case A. . 10. Diagrams to Accompany the Solution for the Horizontal Reactions on the Fixed Columns, Class II, Case A. . . 16. Diagrams to Acconpany the Solution for the Vertical Reactions on the Fixed Columns, Class II, Case B. . • 20. Diagrams to Accompany the Solution for Resisting Moments on the Fixed Columns, Class II, Case C . . . Line Diagrams of All Bents Analyzed 26. Line Diagrams of One Bent of Each Span Length . . . Diagram of Paper Model Page 128 .128 132 136 139 142 143 145 ■* f VI. LIST uF GRAPHS Page !• Horizontal Reactions for 1 lb. per foot span of Truss, Vertical Load} Ratio - vs« Column Height. 146 2* Horizontal Reactions for 1 lb. per foot Span of Truss, Vertical Load; Ratio- vs. Span Length 147 3. Resisting Moments for 1 lb. per foot Span of Truss, Vertical Load; Ratio- ^ vs. Column Height ....... 148 4. Resisting Moments for 1 lb. per foot Span of Truss, M Vertical Load; Ratio - vs. Span Length 149 6. Vertical Reactions for 1 lb. Load at Point 4; Right-Hand Column, Vertical Load; Ratio - Colupi Height . . 150 6. Vertical Reactions for 1 lb. Load at Point 4; Right-Hand Column, Vertical Load; Ratio- vs. Span Length • • . 151 t 7o Vertical Load 30 lbs. per square foot, Wind Load 20 lbs. per sq, ft.; Ratio - ^ vs. Column Height. ... ... 152 g^Vertical Load 45 lbs. per sq. ft; Wind Load 20 lbs. per sq. ft.; Ratio - vs. Column Height 153 H 9. Vertical Load 60 lbs. per sq. ft; Wind Load 20 lbs. per sq. ft; Ratio - vs. Column Height ......... 154 10. Vertical Load 75 lbs. per sq.ft.; Wind Load 20 lbs. per sq. ft. ; Ratio - S^s. Column Height . 155 11. Vertical Load 100 lbs. per sq. ft.; Wind Load 20 lbs. per sq. ft.; Ratio - Hr vs. Column Height 156 H 12. Vertical Load 30 lbs. per sq.ft; Wind Load 20 lbs. per sq. ft.; Ratio - & vs. Span Length 157 13. Vertical Load 45 lbs. per sq. ft.; Wind Load 20 lbs. per sq.ft; Ratio - §R ys. Span Length 158 H 14. Vertical Load 60 lbs. per sq. ft.; Wind Load 20 lbs. per sq.ft.; Ratio vs. Span Length 159 H VII. 15. Vertical Load 75 lbs. per sq.. ft.; Wind Load 'dO lbs. per Sq.ft.; Ratio - SR vs. Span Length 160 H 16. Vertical Load 100 lbs. per sq. ft; Wind Load 20 lbs. per S^. ft. ; Ratio vs. Span Length 161 H 17. Vertical Load 30 lbs. per sq.. ft.^. Wind Load 20 lbs. per sq. ft.; Ratio - |R ts. Ratio - 162 18. Vertical Load 45 lbs. per sq. ft.. Wind Load 20 lbs. per sq. ft; Ratio - ^ vs. Ratio - 19. Vertical Load 60 Lbs. per sq. ft.. Wind Load 20 lbs. per sq. ft.; Ratio - Hpvs. Ratio - Colnmn Height 164 H Span Length 20. Vertical Load 75 lbs. per sq. ft.; Wind Load 20 lbs. per sq. ft.; Ratio - |R vs. Ratio - 1^5 H Span Length 21. Vertical Load 100 lbs. per sq. ft. ^ Wind Load 20 lbs. per sq. ft.; Ratio - vs. Ratio- 166 H Span Length 22. Points of Contraf 1 exare ; Vertical Load 30 lbs. per sq. ft.. Wind Load 20 lbs. per sq. ft.; Ratio-X vs Col. Heightl67 d 23. Points of Contraflezure; Vertical Load 45 lbs. per sq.ft.. Wind Load 20 lbs. per sq.ft. ; Ratio-^ vs. Col. Height .168 d 24. Points of Con traflexnre; Vertical Load 60 lbs. per sq.ft. Wind Load 20 lbs. per sq.ft. ;Ratio-Y vs. Col. Height.. 169 d 25. Points of Contraflextu’e; Vertical Load 75 lbs. per sq.ft.. Wind Load 20 lbs. per sq.ft.; Ratio-^ vs. Col. Height 170 26. Points of Contraflexiire; Vertical Load 100 lbs. per sq.ft.. Wind Load 20 lbs. per sq.ft.; Ratio-X vs. Col. Height 171 27. Points of Contraf lexure ; Vertical Load 30 lbs. per sq.ft.. Wind Load 20 lbs. per sq.ft.; Ratio-i vs. Span Length 172 28. Points of Contraf lexure; Vertical Load 46 lbs. per sq.ft.. Wind Load 20 lbs. per sq.ft.; Ratio-Z vs. Span Length 173 d VIII 29* Points of Contraflexure; Vertical Load 60 ITds, per sq.ft., Wind Load 20 Tos. per sq.ft;Ratio- ^ vs. Span Length... 174 30. Points of Contraflexure; Vertical Load 75 Ihs.per sq.ft.^ j Wind Load 20 Ihs. per sq.ft; Ratio-^ vs. Span Length... 175 31. Points of Contraflexure; Vertical Load 100 Ihs.per sq.ft.^ Wind Load 20 Ihs. per sq. ft; Ratio-^ vs. Span Length... 176 32. Points of Contraflexure; Vertical Load 30 Ihs.per sq.ft., | Wind Load 20 Ihs.per sq.ft; Ratio - j vs. Ratio - ! Column He ight -irjrj Span Length * ••••• | 33. Points of Contraflexure; Vertical Load 45 Ihs.per sq.ft., | Wind Load 20 Ihs.per sq.ft; Ratio-J vs . Rati o 178 i d Span Length j 34. Points of Contraflexure; Vertical Load 60 Ihs.per sq.ft.^ Wind Load 20 Ihs.per sq.f t;Ratio-J vs. Ratio 179 * * d Span Length 35. Points of Contraflexure; Vertical Load 75 Ihs.per sq.ft., | Wind Load 20 Ihs.per sq.ft; Rat io vs .Ratio -" |;^ ' an^§en§t ' h ' 36. Points of Contraflexure; Vertical Load IQO Ihs.per sq.ft., j „ _ _ Y _ Col. Height i Wind Load 20 Ihs.per sq.ft; Rat io--^ vs • Ratio -span Length IX TABLE OF SYMBOLS, A - area of section of member. C - unknown horizontal force at the base of either column when forces or moments are applied at the base of the right-hand column . d^- deflection at any point x of the truss or col'jmn in any direc- tion. d - distance from base of column to foot of knee brace. E - modulus of elasticity. F - unknown force developed at the top of either column when forces or moments are applied at the base of the right-hand column. H - any horizontal reaction; also the total horizontal component of the wind forces acting on the bent. - horizontal reaction on right-hand column. Hl - horizontal reaction on left-hand column, h - height of column. I - moment of inertia of column section. k - lever arm of force which produces unit moment at the base of the right-hand column. 'L - length of any truss member. L - length of span of bent. M - resisting moment at the base of either column. Mp_ - resisting moment at the base of the right-hand column. - resisting moment at the base of the left-hand column. N - either the horizontal or vertical reaction. P - stress in any truss member due to various conditions of loading. P-j^ - stress in any trues member due to a horizontal force of one pound applied at the base of the right-hand column, when the JA_. \ -f .'D '“I"' ‘ ,0 1 f'f . Jr :/ , ' . . >rM ! . ; 'f A V, . u , ‘ ; I T .*4 V;. c.'i:' r ■; ?i«N .. ,xn.;d . {{J-aneX ,.(0.i 7>njtrf'atn{>T ,j::- friDno^st :r>niJt8.t5 0T ^r> ino .io.a U}. & CIO n trod Gild- tod# to 0 * *' i / 8- ■ .',f X columns are hinged. Pg- stress in any truss member due to a horizontal force of one pound applied at the base of the right-hand column, when the columns are fixed. Pg- stress in any truss member due to a vertical force of one pound applied at the base of the right-hand column, when the columns are fixed. P^- stress in any truss member due to a moment of one inch pound produced at the base of the right-hand column by a force ap- plied at the end of the lever BK. Q - numerical term in the expression for the value of Pg. R - the unknown vertical force acting at the base of either column when a unit moment is applied at the base of the right-hand column. S - coefficient of F in the expression for the value of Pg. t - total height of bent. T - coefficient of R in the expression for the value of P^. U - the stress in any truss member due to a unit force applied at any point where the deflection in some direction is desired. V - vertical reaction at the base of either column. - vertical reaction at the base of the right-hand column. - vertical reaction at the base of the left-hand column. W - total load at any panel point of the trues. w - wind load in pounds per linear foot on the windward column. Y - distance from base of column to point of contraf lexure of eithea column. Yj^ - distance from base of right-hand col^jmn to point of contra- flexure. I XI Yl - distance from base of left-hand column to point of contra- flexure . - deflection of any point on the elastic curve of either column away from the tangent at some other point. ^JL - deformation of the bottom chord of the truss. 1 . I. INTRODUCTION 1. Preliminary ,- A mill building bent of the ty^pe shown in Pig,l is a statically indeterminate structure. The deflection of the truss under vertical loads effects bending in the columns and develops forces at the points where the truss is connected to the columns. These forces cause stresses in the knee braces and in the members of the truss which are statically indeterminate. There are two classes of this type of structure herein considered. The first is ClassI, columns hinged at the base. For either vertical or wind loads there are four unknown reactionb the horizontal and vertical reactions at the base of each column. The three equations of static equilibrium are not sufficient to determine these reactions, and an additional equation must be obtained. The second class is ClassII, columns fixed at the base. Here there are six unknown reactions, the vertical reaction, the horizontal reaction, and the resisting moment at the base of each column. The three equations of static equilibrium must be aug- mented by three other equations to determine these unknown reactions. In designing practice the effect of the deformation of the truss under vertical loads on reactions and stresses is neglected in both classes given above. The stresses in the raent- bers of the truss due to vertical loads are then the same as if the truss were supported on solid walls. The columns are designed to resist bending due only to wind forces, and to resist direct compression due to both wind and vertical forces. The knee braces 2 are designed for the stresses due only to wind loads. In the case of hinged columns, the fourth equation necessary to determine the reactions due to wind loads, is I obtained in practice by the assumption that the horizontal reac- | tions are equal. The three additional equations necessary in | I the case of fixed columns are obtained by assuming that the j horizontal reactions are equal and by assuming the location of j the point of contraflexure in each column. | I I The point of contraflexure is usually arbitrarily i j assumed as midway between the foot of the Imee brace and the base of the column, it i-s evident that the columns are not rigidly fixed at their bases, the point of contraflexure is often assumed to be located at a point a distance above the base i equal to one-third to one-fourth of the distance from the base I of the column to the foot of the ioiee-brace. Sometimes the j location of the point of contraflexure is obtained from formulas | based on other assumptions. Three of these methods are as follows t 1. Each column is assuned to be fixed at the base and to be hinged at the top, and the horizontal deflections at the ! i foot of the Imee brace and at the top of the column are assumed to be equal.* The location of the point of contraflexure is then found from the equation Y = ^ x where Y is the distance *See "The Design of Steel Mill Buildings” by Milo S Ketchum, Third Edition, Pages 87 to 91. ' , r . -.1* ' » T • -^.i* -WU o' ^x..-. o.?;j •.• -x : .>i.^ .'• ■ •/; . ' ir:‘:ci;o’i ck , lu '.lo o o.i? -nl ' ’ ✓ ■J-'ii , 'I' OX i'i’.'iv" r j ar^ Sfi 'ti. ! ’j;.0*j. O'. '■ C :; L; .t .'.;■ u .' 0» ^Cfio rt O' ^ 'j - .1 _' r , r o' I.":? no.U'fJjftr/. C5:' C.'' ^ i. J’ O ■. 'Iq .;’ fto’Tl ' i^O <♦'> ..'-i ' : '•,'ti t’'-£fCiX:t ' fj.Vi ao-jcr?**- Oif'2 ?'r: ■ ;• r/l’ T ■ ; -y.i : v(f 59r;i;.' J\'v axt '5oxj.X !io ^■ y I'^'I 0£^j* *;> ^:r; 5-, li-cfp;;! i;.:OLiu Qi. , ‘wlio!^ ii'o. ; XI' j-;tC50 :.o .'n"ou o '? ’'. ?;v \:7 V.'X ; nr ,’h J' - .X^J 4 ^^i^r ;0 Iv ■J' l,^ aX? ■ ■ '\ - j * -C •> . oi'i Xc J'oc'j or^t ae'f!' .t'iJii .[sr. ^', f?o'‘ ^ c:iL rXoo s 9t 1 i: orcar; . o >o -T •• xioo' 1:0 X •■ : o- , . '{xv.l' J- ' f)Oxx~ i Ou a/:-^ 370(‘ • oo;::' < g Ic-^ > 7 • ‘B4)tr.a'0X o. o7 ov i;- o:U .xc- ■ 'j- 1. •;77s'-o . o# /vtir- : .. jf^o O'J X '•■pa 0..J :,3: ; -f ;3.;yo:' . -.'ft., X rv. 7. > OU't lo J OOX 0 : ©ib" 'co * ^ '^U ^ - ' ■ • V- Xo-il -tjc. r>i ©-...•x'j-XIr •'•.} >'!<.• :) onT'-^' KOlvKooX ...;.f' :T'’ o*x ' 4l;0; t. • 4 ■ ; ' ©ir-,':' ' , v; ; ? cOiiTci ; "iti Jx fff: o:'' .nr' ©; d oili' ..' • (‘ft 0 .' yo-.trt; 5 '> >-’1: ftrxf f'-^' ' •■* .7 ' '.'J ' c r. * 7 i ' oi^. 1 ‘‘B 6 X • l." Q ■• ili'tK X'' , ";t't o ' t ' ocf Xo.iXJti"' .' JT . n. 'j’lo!/ o. -' ■ . c ‘ e X''':*r‘* ©ox-i'rd ot'" '. o '* lr*' 7 col ©■; Xo j- ftib" 1 o r: 1 ^ col ox'" x a'i" aX ,r r« r ' . • r WW..' nj ■y O'r,'. ■ ' - (%. r, ....... r.^ '{ rr •-: -. v' ; ox;c;*i; X •'.' (;>- 0 X ^ Xv '’ 0 ' i .' / I *) i : 3\>0 . rx o: '-: •'I If' 3. from the base of the coltnnn to the point of contraflexnre, d is the distance from the base of the column to the foot of the knee brace, and h is the height of the column* * E. Each column is assumed to be fixed at the base, and the portion of the column from the foot of the iaiee brace to the top of the column is assuned to remain a straight vertical line.* In accordance with these assumptions, the point of oontraflexure is midway between the foot of the knee brace and the base of the column* 3. Bach column is assumed to be rigidly fixed at its base and top**. In accordance with this assumption , the point of contraflexure is located midway between the base and the top of the column* This method places the point of contraflexure higher than the other methods* A recent investigation*** has shown that, for the type of bent given in Fig.l, the ass trap tion^ that the horizontal reactions due to wind loads alone are equal^ is materially in error. This investigation has also shown that, while the common assumption regarding the location of the point of contraflexure is not materially in error, the bending moments in the columns ob- tained from the assumption of equal horizontal reactions are materially in error* '^See "The Design of Steel Mill Buildings" by Milo S.Ketchum Third Edition, pages 87 to 91* *5fcSee "Influence Lines for Bridges and Roofs" by Burr and Falk, Third Edition, Page 53* ***See Thesis, "Analysis of Statically Indeterminate Building Trusses by the Application of Maxwell* sTheorm of Reciprocal Bis- placements"by S.R.Offutt, University of Illinois, 1980* 4 . The writer has been imable to find any other investi- gation of the effect of the deformation of the truss under vertical loads upon the reactions at the column bases. 2. Scope and Purpose. - The method of solution pre- sented in this thesis may be used to find the reactions at the bases of the columns of any structure , consisting of a truss of any type supported on two columns, and connected to each column at the top of the column and at some other point by a knee brace. The quarter-pitch Pink truss supported on two columns is a common type of structure used in mill buildings, and has been chosen for this investigation. The writer hoped to present empirical curves, from which the reactions at the column bases of any bent of this type due to combined wind loads and vertical loads might be obtained, but he found that^ because the reactions due to vertical loads were dependent upon so many characteristics of the bent, this was impossible with the limited data available. The purpose of this investigation is to study the effect of variations in different characteristics of the bent upon the reactions due to vertical loadSj and to determine whether or not the assumptions regarding horizontal reactions and the location of the points of contrafleocure are materially in error^ when the bent is subject to both vertical and wind loads. 3. Acknowledgements . - This investigation was made under the supervision of Professor W. M. Wilson, Research Professor of Structural Engineering at the University of Illinois, who gave ' T: . .T' ■ h. - • o t C'i; n-fir nr^oi •!o'»n.r r^"rfr'A.; -©^; o .r; J..* ':u*io^.oj& Oiv-'" ,i£iCM3’ rsirilfXcc' 3/lJ- nor-*; :■:.{) ■ : )l.Lr£0'-, t .ovT jb^r.'i o :qo' t n r rl iJ't 0 • ' I J- •■ '•■• '> ■'■ ' I •■ •- ■ wV? Usf‘V;^‘ £il ■ ■>. ' ,. . -.0. L- c'MJj • CO 6f-l , r.'Ico o;Vv‘\ ao 'Vi- ., c ':*/ '\ . : rc oxof: >. ... iTc^crj;- -' j> ' i • .■ / • i;'"-? - rs.i.’. i l^'f. •• rtT ur. :•%. tTr<"r/r? ;i -*T^'.^•r/ •/. w ; -- V- , c;:r,. j: M i;rc JJ ;,. ni pediT 0''-‘.i'»csrr ■ ■- “ ?-ic.”f;.oo •: »j. .f / ■.: i ■ T V'-'r Oi-fA ff{=*9,’ OJ*. 00'^, On. 'Ss .'i'.’. »ii'- , .f .' 'j ; ■.li' •s'Y : -O'i'v -.c'X'l: , v'nfo f •:■ • o:( ■ cn'ol ' 'LiOi- ■ ’iur;' ot ..v.‘ . ! If', ti:" sr-J" 0 -oef :'■ tu h'vjo'y o ^ ■ ofj lo ‘ 'rc.r : r c ; r^' ;• ' .J , .;•■ j'rf ^r.-'oT- I 7 o;^- sa'i' c,-vIto c^'7 \...i.: o:. : si: Ir- r *;■ .-ii-'..' , ‘X«- u -:•<• * !'•* ‘v r oiV •*.: 7".: o:r ‘:i i: I; - i. -r:v:sk lUAy 'to -i: -Ovx.'i a:. ; 0 ^ .J- ::;o cci "-K o- •:. iio ^ae-r«X'J'i;s ri ^icroX : ’ • 7 '\v Ic f v:o.;-' (.LiwiC.-O’' ot Z):r:- . ;-x». oI J ‘ .M.-:; •:!■ ’"oreiVL ' rrcvi^u o.i’- 0;';^. j; f •a.uJ'- ■^‘jrr : ;' 'o-?t ' ' \^c. »r A* *rf ri;h oi'ja .v.«x “ 'XO ' ;'0'T OC- , ".;JIJ;,' / .j^ion^ivaAoX't'r.fc^. . 7 ■‘ ot'I '‘.o ari.- • :7'tocr.7 ■ pjt ^ 0:i , : :: rtXI2 ■?“(' ..r .t -^n o /i nv' E>rtt >• ar. C 1 .- \t f ‘ ‘i rg i ! . * ' -u. i jijtf.iijy.i-j.u fgi 6 the writer many helpful suggestions. The method used in the analysis of the bents was formulated by C. A. Ellis, formerly Professor of Structural Engineering at the University of Illinois. The values of the reactions due to wind loads were taken from a thesis written by Mr. S. R. Offutt*. The vertical deflections of the various panel points of the bents analyzed were scaled from the Williot diagrams prepared by Mr. Offutt for his thesis. The writer also used sane of his figures and tables. The writer wishes also to acknowledge his indebted- I ness to the Department of Theoretical and Applied Mechanics for j 1 the use of the microscope and micrometers belonging to the Department, which he used in his test of a paper model of a bent. *See footnote on page 3. . -i-.. .. 1 ..:r»oci VI . :* ■^0 ‘if '< ,r- '/;• 81 ' ' , ••^V '•' ' ? £%j ■' C/ ' • 1 .1 r; .* i-', rr':e; -*■ ~. ■* • u'j'v" ■.; :^v:i I " ■ '' •■ ■■' - 'a. , ^.'O ii.i ' ' a ; »b X : f ' ^ ,-^r .. .> L ■ V ■v> i 0*J i OV. a ' r"' aj ';I.,rtn 0 . ' ^ J ' J * r r\ ' V -►J .4 ^ V‘ ^ * Vti • f) •; ,[i ^ ■• 3 i - ^ O JL4. J. T'C.V :) ’ c "‘ »N I » ^ . Ocf; a. A 'a; ') , ‘'Oii .', 0 1' 0- .a OJ'.' .i o;,.r '*c : -w Iv. . .. '.: /■ * ' ''c Hr . f ^ . T J* '■> ‘-I i J..' 1 V* '1 ’ iil-' '.' 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':• .-ut-o'-X- X*'iUic:- x*5 otf a , " f!>fi;.'.. - - ■;'• . ■0': :.kX a ’ :i=i’ U^o ^ - r- • \ o' E ^rtlorr^ Itc. t-nJ :.: -L., •.: I att <3X";^’4 il Irvioq; t . 'tr * ■ ■ ^ r ‘ - i .i\: o 4 i;roq ofio to irrl I vd > ’ot -;-5 fi .‘nioq ' . . t i-jti '? ct o-r,j :: t n :oa ':o :\io..’0 0' I Zi, I.u> ;at: 0, •: ril.j; --. J4 ':t or^t nod' , Xv Jj.fC'a''-' f> X v rti 0 -?x'X 0 J iJi c t Dii.foy; 1 c bJooX I ■''Ij'iVr .- ct u'jb t-' t *. r-..!'o« •■ i X - ^.H .. 4.' S. O j i « n O j 0 1. ili'-lOs) > 'i v c I- *0'XX-.” iXioI L'I: eot.; L;D'rc:'.: '■ * 7 .'v.’; » ?--*■. f* (:iCi;./ir 3 o OIt;'I} "t ‘-4!;. ■ '■' • .' !' I0 > j ji ’ g ap . i ; g g B m. ' g w twa - j.Ly- ■■ .r;ar, ~ : -r I. • •■•• 7 Thus, the deflection diagram of the structure when subject to a horizontal force of one pound at B is the influence line for the horizontal reaction at B* Similarly^ it may be shown that the deflection diagram of the structure when subject to a vertical force of one pound at a reaction point, is the influence line for the verti- cal reaction at that reaction point. j Similarly, I«Iaxwell*s theorem may be applied to finding the resisting moment at any reaction point. For example, I suppose it is desired to find the resisting moment at point B oi the structure given above due to a vertical load of one pound at | I I point 3 (See Fig.l9). Proceed as follows; | i Apply a force of l/k pounds at the end of a perfectly | rigid lever arm M of length K inches^ rigidly attached to the column at B. The resulting moment at B is one inch pound. Find d]£ + 4 , the horizontal displacement at the end of the lever arm due to the moment of one inch pound at B. Find d„, o the vertical deflection of point 3 due to the moment of one ' inch pount at B^or due to the force of l/k pounds acting at X. By Maxwells theorem, dg equAls what the horizontal displacement at the end of the lever^ would be due to a force of l/k pounds acting vertically at point 3, or equals l/k times what the displacement would be at X due to a vertical load of one pound acting at point 3. Then the horizontal displacement at the end of the lever X due to a force of one pound acting vertically at point 3 equals kdg* Since kd^ is the horizontal displacement at at" the end of the lever X due to a vertical force of one pornl acting - .igjr., -t. ' 'J: l' 7 't:>'r*-:-‘ ; o;l' . ' j. . ' i> ;■:* 1 ‘ f)‘.'X^;,» '. •/■•i.';;' <■ • • :: .1 ;*•'. -icrrc-q O: r, ;• CrrO’.'. !••■■*•-' .1 .'. . : J-r . I'o ^ .u' 0.. i.S‘- to'*, r:iP I'. T . I ^ t .t! ■ n.-o'r, jd v'- ^ ’ . 3iio ro 0 0 ^ 't t} t*. U'UtJ ’ /r^.T^ a:i ^‘ ;• t .‘ ; -Mt y o: "V. 0 t.L: OI. . -■■ :'>o: £i, L' '::trocr , ^ ft ‘ .■> I o • A i< i II Oj » j- r-'' •■ • C>:’ ■•: ■.. :' , YJ.*£\ i j I IjF y , A.r^'n’ -K*-^ • :o; «' ■ i “ ^ vjr- t., i ■ •■ ■"■ - o r. J-.’j.’! r ' .; ■’.: .t>'‘ci'i ."r - .J-'D font:. •...* . ■ ^ J ©rory:;:; li. ^ ■) ’:r b-^'S ' o’:‘'icv ■? o~rO'< o’rs'ois'x^. di} A J, .' '•<:•• >;m '■ a:.*t:r.q ?'o aaT&'' - ;. • '/ L'if; c: Ij.Jo ^ - .h^ ^ Ti:.-'’ Tf'T:. I . f.'jAi ; nc .ij. c. J- ■ joitcr. o ’'-^ . t. ru:,;!X^c f- '.•-L.'os ■; i: & f.Sii-icrr-: s '20d sr- ■ . A '*■ >;fo “ ''CV9l J. J . .X - ji'. r':' . 1. ii'rtnl o.io do -odf :vu V ; Tii t-.j Cl? 'f - ?S f-fti-r '> i:0-J x^’ T'-i; f’OoX' Xivjni;|-Aijr e.;:? X t •■ T'-Tii-o - 3t\X xc eoToi o? to^»: ) *x;u;: : r.i.TLOo X Tno;-xj;-.c o.dr « dw elXTi O ,f:'r>T00j‘; i" v.'~ • ^ j'\l oo'tod • r.j- g>fr5 Slf’a-'^ToTsx crft ll.o j-' r ', < TW i wif. i M f i i i j '1 . — ■ X ' g ' Jg. ' i . ,4 1. . , .X . ■ ^ J •-. . .# . 1 . .' >1. '.I ' '!■■• /'X4>^.,rrUdo.?. o.'io !lo ib 'Ol I uxJT '■/ od* 6^*r .»> xf if ‘- I) ... x'C . . J .01' 'JO Iv.'" j • *'4i ' J J ;.v . j'j’X.j . L'-' Irii; .. 1 i^Orl 0'.::" • i j- \^xal.‘-on ; ■, » ^ .T' Cvl . ■ ./ 1 J T i rriXo- fi ii/o.., ;jno If OO'lcj^ ■' od O-.'d YoX 'o-iTt j.o ' ii) X> .TTv . iTou ctXX t' I vl):< . 6:1 ol • ' 0 y-'.:oof, ) -04 o,T' ■'. 0 ootc X i .viJlV Y - O f 0 ■•■■ T •^0." -j.':] :-c, u-t'i.u.n ; 8 point 3, and d^^ + is the horizontal displacement at the end of the lever due to a moment of one inch pound at B, then the moment at B due to a vertical force of one pound at point 3 is k da M-r= ^ xl inch potind. dK + 4 4 ^ Thus^the deflection diagram of the structure when subject to a moment of one inch pound at a reaction point is the influence diagram for the resisting moment at that reaction point. This principle will now be used to determine the reactions for a typical mill building bent when subject to vertical loads. ^ line diagram of the structure is shown in Fig.l. Table 1 gives the design and general data. There are two classes to be considered, viz: Class I, columns hinged at their bases, and Class II, columns fLxed at their bases. These two classes will be considered in the order named. 5. Application to Class 1 , Columns Hinged at Their Bases. " In this case there are no resisting moments at the bases of the columns^ since the columns are free to turn at their bases. The vertical reactions may be determined by the equations of static equilibrium. Thus, only the horizontal re- actions are statically indeterminate. Case A . Horizontal Reactions .- The influence line for the horizontal reaction at B, which is the deflection diagram of the structure when it is subject to a horizontal load of one pound at B, will now be determined. Pig# 3 shows this deflection diagram greatly exaggerated. 9 Apply a horizontal force of one poimd at B (Pig* 3). An equal and opposite force must he applied at B* These forces cause bending in the columns and produce horizontal forces at the points where the columns are connected to the truss* These forces acting on the columns are statically determinate (See Pig. 8) and, when referred to the truss, are equal and opposite* Consider the point D to be fixed in position and the point B (Pig* 3) to be free to move to B£, due to the bending of the columns and the defoliation of the truss* Both columns are free to rotate at their bases* The solid outline (Pig*3) shows the initial position of the structure* The broken outline shows the structure with its i nitial form, but dis- placed through the horizontal distance DDi=BBi due to the bend- ing of the column BG* The dotted outline shows the structure displaced to its final position due to the deformation of the truss* Point B has moved from B to Bq_ due to the flexure of the column BG, and from B^^ to its final position B£ due to the deformation of the truss and the flexure of the column BK* The deflection diagram can not be drawn to scale until the flexure of the columns and the deformation of the truss have been determined* First, the flexure of the columns will be detemined. Flexure of the Columns: - Since the forces acting on the two columns are equal, but opposite, the flexure of the columns will be the same in magnitude, but opposite in direction* The oolumn acts as a beam under the horizontal forces. BGiJg (Pig* 4) shov/s the elastic curve of the deflected column BGJ* Braw a tangent to - V. - ' ' 7 «> « ■;;^ 3-1 .'■•.^jic‘: .•;rTO'J t' ' ';dOq[c|' ‘ii ’i'X'C'fi aviJ^O’i-. 1 ■ ■^:." iii n.; I £»i ;<,•*<-' 'j -j • I o . . : • j ■ ' 0 '-‘ n.: O v ; OGtOU.Ti'fC'J 1 : ■ ■• fi/lTJCrj-GG ant o v TO.; ,>'i> h'’ : - gxx • ' ^ - : :sr'ij too ry ' 4*./V ■ - =■ y ^ ■J OT , . n ‘ Oi-iXJv 0 1 noTTfil'ita' r . .' •*•-:■ oii Ov^ ( I t lit „<> lit Zr.-.'i, r ‘ ;rX , • ' . / ) /v.;: ct 0 '.‘ VI OO ' J « oX'^..L':c'JO-ft fif" .h iX-»- - .r' , ■f' tiv..; 4. - V -V i, -Vi .-' -'•tif' • '” " O ' \ l 7. '*. '.<• Kv l:l X -XX'' X .'VtJ .fit X'-. {■‘■i:; , r • .■ . ©4 - ' • - r ' X . ■ ' ■ ' I T' t .0 ■'i? ^ C ,.I • 0:1 0.' t ;{ ^'.-T. h fif? •^) ■; i:Uc- ’ fit '"'"cdn 1 j v.lo. "'i J**! ^':: ■■ •■ . ^ DiU •'to ;:oj;r , .to'' ' ^ o:; o ;':6 tv j-.; X ::ocv "j K ? J vf^O'iX . ■ K' rf*' !».o OT -rr- X? t?.' ‘ of o.*r* j- ■ o ' il i'C. i^.vv ; n a •- • •’■ •''?■ • ' iiiri , .tiioj X' X.;; .V.: 0 ./ j-/' . ot"; , ry, •: ,;.:or- » " T:-: 0 T;rx©X'; r - ' o:’’. .■ .. ' '''••/■.’■ ... ’’k' ‘ 1 • an? IX v>; .. .'. 'oa r‘ t? -iX. oo .ton r;;iv .-• rtv '.; s. if ■ ‘ k'r ( n..a/ ov • ou ■_ rcoX ofi ao * .!c ii : ' 'r-^OJi^h ant I, '* -'‘Xn^. tj'. ■ ■> r. 6 .^ * 1 * 1.1 'i.,t 3t :‘iX ' . ' t c^^vj : n y:£r o '- . ■ ,V^-s.r roo"‘•^ _ i ,: tt ' o-X'-'Kirf 6: ” - : p.rj. .■ ,'OL* i>:‘t ' 1 ' D-.l-'TOri ,. X no :' .Xv'O.. v.f)£>TO . C4'’ 'X- ■'"■ ( ■■J • . . a iu * ... 4 . G - ■ • .'w .-^vi a«j . • o f ». .,'t ar;c--'‘ ut t;-!> Vi'l£.f •; •*'. *■;.# .-.i>,yTC> I )tnOoXiO.:X i>ilt TV -.■rrTi w r '■? g-‘- ' . 10 the elastic curve of the column at point and produce itto intersect the horizontals through D and J* Then Ao er -■ I... ti, ■>■ I ■.. '■• ■* • . ‘J ,) v » > * l-l . VTM U j: i v- j / V • A ^ ♦ i » * 1 '"*’ *•• • • -'s ■i 0 - U ’ •1 io Li.'JJf 1 ) -'. «• • .4 4'; • ■ V x; ■ o 's^: 'to Q 1:: . * .' os. .!. y K 'i : .1 i * ’ j:. ' . ' ma gft 11 . Where ^ is the displacement of any panel point x of the truss in di/e. any direction in inohes^to the given loads. P is the stress in any member in pounds due to the given loading. U is the stress in any member in pounds due to a unit load acting at the point where the deflection is measured and in the direction of the measured deflection. 2^ is the length of any member in inches. A is the sectional area of any member in square inches. E is the modulus of elasticity in pounds per square inch, and is constant for all members when they are made of the same material* P and U are positive or negative according as the stress is tension or compression. The quantityPU ^/a is found for each member, and the algebraic sum of these quantities, when divided by E, gives the desired deflection. The horizontal deflection of point J relative to point Gr(Pig.S) .dll. due to the deformation of the truss is de- sired. ^11= z ^ a. is the stress in any member due to the forces shown in Pig. 2. la is the stress in any member due to a force of one pound acting horizontally at J(Pig*5). To keep the truss from moving to the right under the unit force at an equal and oioposite force must be applied at G(Pig.5) .Since the horizontal displacement of J, due only to the deformation of the truss and not to the rotation of the truss about G, is desired, an upward vertical force must be applied at K and an equal downward force at G to form a couple whose moment is equal and opposite to the moment of the couple due to the unit horizontal forces at G andJ. This couple prevents rotation of the truss about G. The magnitude of each vertical force equals Itimes . V ' • t; on:r 'xu ?. X*.»j:fciq It-: l. • . 3 o- < Lyfi « - 'TX' ‘ ... ' c.) ^ ^;«j(ic^.U ni lie-: tX Xitr. .fi Lvnor^s' OC..I OJ e£r(\ f f .' oX t.* •'. . 'iicjoq nL 'jra.-L'cr/r *i;rj'.v £x.f seQ'j . r; *fi ^2 Q 0 ; ■ L'i ijita r • •; u or{;t 2 r yi<>q^g^^ ‘ : 3 f/i ?•.•: / j , '.’’V <' . i . t 'W ;.L t: h\'. i- :fl* 37 &X ‘W... :;: a'T i.' ' >>«, < fT^. J c ■ Jl;: ' ^ 1' ,' ■- •■’ ■: ' , i-: U Ti-^oitv Xo_ -TO ■.Jl'j;..n:. O'" 'ro.or.'v/ o.* ..^opi f-fir-rui o:: . O' v-n-s . 'V*.; ..•,>.-t> To'i: Jb.l»T.>'j. <-l2 . iJ iv;: i.' ;: .7 t>'iC^u;06 *XQ I • O t X, , i .i - , 'j'‘/-.'. 7;> o.'.oxit 'i ;)i- T ; O'.' r ► . , ‘j j -iOv? !) j i .1 X 1. . j. . 0^.'U '.^ii’j.* i. .. tf'j:. V ‘10 } L . ' :>.f tt.: O*.'. J. ' Or> A>S>7i.,';0L' ,|. . ;fx^2■J<'x 0 ; I; ',. 1 :; ..4 V :'r., ::i'r^ ^fi j;^ 1 ».• X.. o.r^'" aO cj' »ir,& 'X>'T ... \:x:' cil e c orf^ £’ ^ '‘if oj- ?;jjkX7c • iiS'/'ii 1 '.'; q003i o-i* . .’X 17,.'^. XX.ri 2*:^. •i-.:orl ■:0 0 ": C’ X XwO-' , c f>;i X .;m, iX:- ^ ' * s i'-oX JiL:.' O ' to'lj.f'''' trt!^ *■'1 ,: •• *^’ ir OO- j; j.i» 1. -' .. ; :I ;w;r ^.. 'X 01.T , { .. .XS).^ X ^ Dcf X : ». ^r. / X f>.i' c.' .r:. 7 X' ir-.xri'l' t: i .'■ X C' n 0 X X r.?! 0 ;> 0 ' C' YlilO OtX' Of." i 7 ... X' J . 'i ■ 7 'V' •.-'sj.u , ./'.‘iXbU r.j i , ' - atiJTi;.*' odt ’. ' '-'J : ‘ i'!". - . O.r ' ■• v''I. a-:'‘''.’ir7.'o0 .fir -•. ) *)-Cs .ii j -J *y X ,k C^»J* ‘1 c:‘ O.C^ -T ■ » ■i? f'j ail- no i, . I ::_ X i,l f''/',rrr r: :XO.ct X 0'2 q? oXqiJC 5 ,'f , j.i '/ tt * f X i ' .' , . J*- .trji ,- 'f •:.i-...r 0 OO'.;-'. ' : ■' ■ r 7 i j: . 12 the distance from the foot of the Imee ^racC to the top of the column, divided hy the span length. Then U]_ becomes the stress in any truss member due to the forces shown in Fig. 6. , A , and E are as given before. Values of U^, P]_ i /A, and Pi Ui Z M various members are tabulated in Table II. A 2 , 216.93 . 1 dii= inches. hi The deformation of the bottom chord JL of the truss, which is designated A.JL, is the summation of the strains ^1^ o f the ' AE bottom chord members. From Table II, for the bottom chord members = inches. ilraphical He thod : The Willidt-Mohr* displacement diagram shown in Fig. 6 is constructed by using the strains. Pi t , given in Table AE II. The Williot diagram is first drawn, assuming the point G to be fixed in position and the member GJ to be fixed vertically in direction. The Mohr rotation diagram is then drawn to correct the error in the assuiTied direction of GJ, using the condition that the point X does not deflect vertically away from its initial position as a basis for drawing the diagram. Deflections are measured from any point on the Mohr diagram to the corresponding point on the Williot diagram. Deflections are up or down and to the right or left according is the point on the Williot diagram is above or below and to the right or left of the corresponding point on the Mohr diagram. The Mohr diagram is shown in dotted lines. The Williot diagram is shown in full lines. *For theory and method of construction of the Williot-Mohr diagram see ’’The Theory of Structures” by Spofford. a : t j- 'I > y r '*1 n -5 f c ‘ ■ ' : " '• ‘ ‘ •' h j -'j-.t < - . . . ? c • * i i j 0 * ff ■ I ;. . *. lOV i J V.‘ ’ ' '. 1 J * n •:■ ( O' ( J. ■ 1 1 :>'V ;.' ' ’.r r • « K-'l ’ .W 13 The algebraic method requires V PiUi Z to bo wiritten for each panel point , whereas, all the deflections of the truss are obtained at once by the graphical method. The deflections dj and d(j of the column DJ may now be found. dig = 8 l/2(dii+ \ ) = inches. (Pig.d) de =4+di5 = dj = dg + dii= — inches. The total deflection of B (Pig. 3) is dB = 2dG + 2dii + A,J1 = 8g,48S.04 inches. The vertical deflection of any panel point of the triss due to a horizontal force of one pound at point B, is obtained by scaling the distance in a vertical direction from the point on the Mohr diagram to the corresponding point on the Williot diagram. Then, the deflection diagram of the structure (Pig. 3) can be drawn to scale. As previously stated, the horizontal reaction at B due to a vertical load of one pound at any panel point is equal to the vertical deflection of the panel point loaded, divided by the horizontal deflection of B, both de- flections being due to a horizontal force of one pound acting at B . hand The horizontal reactions on the right, column due to vertical loads at various points of the structure follow; 1 lb. vertical at 3; Hh =|| =Qf^||| = 0.02863 lbs. 1 lb. vertical at 6: Hr = 83^f§3~" 0*03162 lbs. 1 lb. vertical at 4: HTy= ^4 « = 0.03019 lbs. ^ dg 83,483 nps 14 . The horizontal reactions on the left-hand column are equal to the corresponding horizontal reactions on the right-hand column. Table IV gives both horizontal and vertical reactions due to a load of one pound at various panel points of the structure. m " 1- t‘' ■'. ' ' . ■ ;• '’ , ff .|||L , ‘ ^ n'* gr .Cqo ,J^ {\ X ' cfo ?iirs;. t Jt ' xJ^ v'6% iToxC £»^vj . - . ■' • '^ ' ' *\‘' ■' ' " '?}W ’■'B!^''#'' I ' ■ 'K'-- • 'tf'- ' ■ »■ .' *' ■ ‘ ^ i>£i di'^ P'Xtfxty . d3fcwi»fe*/>^ ircc'it • I ^ ■■ 1 .V . r;.'|;e ,'iwM> 4 iOii; X^'^f' -t 'f ■'■• Vl . r ^ "i ..•■If ■id * ' ' "' ^ '^' '* • ^^^' !> ' r^-=* r •» • • . L' ^ ^ ..' ^ ■ ' t 'r ): ■: m:iS i , -I (*. ..v.! jl/iR . ■ '<< " • .'''/A'’ 4't 'v"(^‘: ■.?% < ;,>■ >4::-:-'= : ' ■■ ■•'i- ,-*f J ‘ ' . '* %/: ‘ i. ' t ' "■';. * T , ' 1 ' : . i , if ;' ' ' '■ F. I . fUMi: . ' ■ V '1 Jl •' < ■V ’ I • ‘ ' . t * ’'f'' ' - 7 ' - *A . ’i i.'- 1 > . * 1 *S' . ,’ .’ c " ■ ■■. ... •f.'J. ■ ■ N,,.:Mi'' wl, it ■ ft ' ' \ V. .' . I- ‘ >.U ■.'% V ,...tCiir 'I .., -i^ ■• ’’, -"..v '1?^ ■‘ '".'lA' . . , I '..X^'^ A i ■ty “l1S«)gP’*^ t!l ?>PS L. ' U' ' .< ■ -g . n > j .' *W ' ■« V. ’ ... 6. Application to Class II, Colimins Fixed at Their Bases. This type of structure has six unknowns, viz., the horizon- tal reaction, the vertical reaction, and the resisting moment at the base of each column. There must be a solution for each of the three unknown reactions on one column. Then the three unknown re- actions on the other column follow from the equations of static equilibrium. Case A. Horizontal Reactions. The influence line for the horizontal reaction at B, which is the deflection diagram of the structure v/hen it is subject to a horizontal force of one pound at B, will now be obtained. The meth- od is as follows: Apply a horizontal force of one pound at B (Fig. 9). An equal and opposite force must be applied at D to produce equilibrium. Point D is considered to be fixed and point B is considered to be free to move to Bg, due to the bending of the columns and the de- formation of the truss under the applied forces. Each column, be- ing fixed at its base, must remain vertical at its base. These two horizontal forces cause bending in the columns and develop forces at the points where the columns are connected to the truss. Due to the unknown bending moments at the bases of the columns, these forces are statically indeterminate. They will be denoted by sym- bols for the present, as shown in Fig. 7, and their numerical values will later be found from the condition that, at certain points of the structure, the flexure of the left-hand column equals the de- formation of the truss. These forces acting on the columns are equal and opposite, when referred to the truss. F is the unknown force acting at J. ^ a '•'^ ^F‘T B n ‘ ■ ~v, xi^ ‘?tj ■ V4 a/,i't ■ !• ^tjcitcuxvx X/fwiiTl*;.. 6.i#'^,»ai'/^ ■ — " ■■ 1* .' .t . . . w: ■ "■ '** ■**' M^'rt rfati, -io tmjsMp: ' r' L j. ~ j ! V »*A ^>-...y. t ‘'-^5'' ' ^■ / >» ' 7" ^ if - / if., . ^ > a I 1^ i J l|H| r >£j I ^ . ^ '•*■ " ■f.'’ t'.y-'oq eaj^i. , r\’ ' .\. . ^._ ' ■. .. ■ >«^ ',V'^.'.^' ,T, J 4,ro£ifol •> i-Br .(■■’.^’5;-- / Wttir r’f; . 1-v ^onsav ' * ,^v.: . '.. . .: a'.'-. .. ' f:t.i i# Xfcp.bt .n* tiilft: « 1^‘x iV'^i , 5*0-r4i -^i'6.T^ -1* S-ti-ia£' ii^i0.fta)wt!i,ixiii^ '’’y'* : ^ ■ fi j- ■ MW ' ■■■^•'^^a.i.-H^Ui V.:- i^fcJr-o-qo a.^ton. ,6a . .. !-. os?*# ^ ■’ :-c • • U '■' ' •I '‘-* V u».' ^ ? - ' t J LI ■ .. > . i - : .. . i . * *1 - -b . - aLgC^ :.i- •• as-M rviV' 1'T!’ y. ;.+ fV.: -tia'i/Qv ■^i .•■•.:o.":i- " i' J : : *■•■ -» '^rr ' *:a a’-^x. 1 ' I lc^il V C • i CJ/w . . .... . L? - ■ .i ■z - * '» liy - - -v-i-*, I A « -^ • f* ‘ > ’l "i .1 C.f':, -•) Z^. ‘.''lu "^O il/ti . i:'.. r --■■ ,;v . , . ^ i L I !Oa,> , '• rti : ■ :.■ * s.r c . LC- •■:>. ■ "u •• ;• •■•..'? \^„oo rr /i^>;-! ' I j.''" ' * .:o :* r.L. i n/i t » I -V i-ilx iiv'X 4*>'01*J5** '* L% ~ j' . I- • & 'll. '3 > V ^ ■ : iloU^ . 1 . a , ■-J .T '.'I ‘•^>57 • v^ * • ■ . e-J-5 .j r s ( t > ; J ;;i. *j . ;• " '? .:• .C : • ;,o • • ». ’" 'u; * , 1 lb. vertical at 4: ^ 0.07026 lbs. 0.06632 lbs. The horizontal reactions on the left-hand column are numeri- cally equal to the corresponding horizontal reactions on the right- hand column. The horizontal reactions on both columns, due to a vertical load of one pound at various panel points, are tabulated in Table IV, Class II. 20 Case B, Vertical Reactions. It is now desired to determine the vertical reaction on the right-hand column, due to a vertical load of one pound placed at various panel points of the structure. As it was stated before, the influence line for this vertical reaction is the deflection diagram of the structure when it is subject to a vertical force of one pound at B. An upward vertical force of one pound must be applied at D (Fig. 13) to prevent translation, and moments must be applied at the bases of the columns to keep the columns vertical at their bases. These two vertical forces and moments cause bending in the columns and develop forces at the points where the truss is connected to the columns. Fig. 11. These forces are statically indeterminate be- cause of the unknown moments at the bases of the columns. For the present^ these forces 7/ill be denoted by symbols. F is the unknown force acting at J. C is the unknown force at D or B, acting hori- zontally. These forces are not the same for the two columns (See Fig. 11). The value of the numerical part of the force at each poinl where the truss is connected to the right-hand column, i.e., at K and L, is such that the moment of the couple formed by these two forces is equal to the moment of the couple formed by the vertical force of one pound at the base of each column. This numerical force 1x40 is — g — » 6 2/3 pounds. The numerical value of the symbols will be found later from the condition that, at certain points of the struc- ture, the flexure of the columns is equal to the deformation of the truss. The forces acting on the columns at the points where they are connected to the truss, when referred to the truss, are equal and opposite. 21 Fig. 13 is a greatly exaggerated deflection diagram of the structure under the two unit vertical forces and the moments at the bases of the columns. Point D is considered to be fixed in posi- I tion, and point B is considered to be free to move vertically to Bs, due to the flexure of the left-hand column and the deformation of the truss under the unit vertical forces and the moments at the bases of the columns. The moments at the bases of the columns caus(! the columns to remain vertical at their bases. The solid outline shows the initial position of the structure. The broken outline shows the structure with its original form, but displaced horizon- tally through the distance DD^ = BBq_ due to the flexure of the col- umn DJ, The final position of the structure, which is due to the deformation of the truss under the forces transmitted to it by the columns, is shown in dotted lines. B has moved to its final posi- tion B£ due to the deformation of the truss. j This deflection diagram cannot be drawn to scale until the flexure of the columns and the deformation of the truss have been determined. Flexure of the Columns. Since the forces acting on the two columns are not the same, the flexure of the columns will be different. Column DJ . The horizontal forces acting on this column are the same as in the preceding case (Fig. 8), except that, what was the numerical force before, now becomes the force C. Therefore, what was a nu- merical value in the expression for the flexure of the column be- fore, is now the coefficient of C. -H, = 27,388.37F - 38,305.42C. , J — 5 T ' inches. . 00 ' t/ .■ ■- .< j r, ' ■/. '>••'■ VJ'^ L ■ - ( • ' '!i •; - ■ ' " ■ ' • .. .lO i'*’ ^ ... ■ ..^i ■ . ' 1 -t ., .' "'y' "■ • /•-' ■■■ ' . V 1 /,:^^ . •• V; ’ •' c/r;»,.' : .» ->y« , ^ \il ■" -k"- T •• r ' '.' I r*' /l *, j-V'- .' . :? , ^ 1 - ej j i -;. ^,,1 ", ' ■ ! : : j^: V -* ' « £ i . ^ivL ^ w V< I - 3 ^ . ^'VJi .' •)•' . t/ •• t;.:. •• '••f ... I . ^ ^ i-:. ,. t i-. *'• ,.*V ■ * ' ,n I ' -'•VJ ..’ ■/ ^l< 1 X ••r ■ . i’C! ^ a. ► , f». . . . , f ,*f •:/ 9 ^ *- /- ■ ■') . '. 1 .^3 : „ ■trrs^aiSi ' ! \ f ' ■t ■' .■ a . nX , ’ , ■ ' ’..I 1 . ‘ ■ ' ■■ ■. ‘ • y,' A / f;'”t.C>V 22 -dp * 14,564.55F - 25.940.89C E inches. Column BL: The elastic curve of column BL is shown as Fig. 12. The line BgL^ is tangent to the elastic curve at Bg. The moment diagram is b‘k?b. The displacement of K and ^©spsctive ly^ away from the tangent at Bg are found by the area-moment method. of point K relative to point G, d. 23 , due to the deformation of the truss under the forces shown in Fig. 11, and the deformation of the bottom chord, under the same forces. The deflection and the deformation must be found by the algebraic method, because the nu- merical values of the forces in Fig. 11 have not yet been found. Pg is the stress in any member due to the forces shown in Fig. 11, Pg * SF f QC f T for any member (See Table III), where F is the un- known force acting at Jj ^ is the numerical coefficient of F, and is the same as in Class II, Case A) £ is the unknown force acting at the base of either column, i.e., at D or B^ £ is the numerical • 3^[(72F-)-480) x (36x48+180x162) - 180Cx90xl92j[ £I = (2,223,936F - 3,110,400C + 14,826,240) = 27.388.37F - 38.305.42C + 182,589.16 E d^ = |j72Ff480) X 180 x 90 - 180C = (1,166,400F - 1,9-^,0000 + 7,776,000) » 14.364.53F - 23.940. 89C f 95,765.55 E Deformation of the Truss. It is now desired to determine the horizontal displacement where it/ btiM „ * • rj.\. A i*‘-rrrr ‘i . K'i bitl - ^ -T .01^ VA\t, ^ , .lU'T ■ •* !► (•- 0< . • > Jv’ . i. - *:.: : -.ili; I Q-. j*' :0£l4:t ■: ’{ i ' c • • • i, * . • • ■'• i¥>^"ai u^... M i ^ ' 4. i ' ■ • . M » • V . > • » ■do;; ■ j .; .' . -1 .' . ! V ' , :T,. ij> , :iic. ' ..roi 'i "Oj. . ^ « * ' t > ' ., I- ■< . ■ . '■ ■: ■-"i^ )■ l- -:'^ ' i- • . ,r * . i t- u. r} - :./: 4 a ‘ .»'. '.ifaiy . df lil \ • . ■ .I V , /: . \ r \ ", -« ■•' t ^ : •• 0 Z'lr It ■ . : A , - ■ ’ ' « J nr u ■pl" j ; - • / ir. . k .-0 ;-.-L I-;., - ■ 23 coefficient of C, and is the san.e as the numerical part of the ex- pression for (See Table II); T is the numerical part of the ex- pression for the stress, is the stress in any member due to a horizontal force of one pound applied at K. To keep the truss from moving to the right a horizontal force of one pound must be applied in the opposite direction at point G, for the relative movement of points K and G, due to the deformation of the truss, is desired. Thus, U 3 becomes the stress in any member due to the forces shown in Fig. 14. Z, A, and E are the same as before. The quantity is found for each member, and the sum of these quantities is divided by E to obtain d 33 . ^ 3 *^^ ” ^ f’or the bottom chord members. •AE Values of P^, U 3 , P3U32 for the various members are tabulated in Table III. d 33 = 1.411.97F 4 1.839.55C 4,706.60 E A^JL = 200. 94F 4- 427. 36C 4 670.09 . E The numerical values of F and C will now be determined. From Fig. 13 the following relations are seen: dG+ d 33 or dj^ - do - d33 or “ -43JL = 0 . Substituting the values of these terms given before in the above equations, two simultaneous equations in F and C are obtained 27,317.09F - 49,721.100 -f 91,056.95 = 0 54,575.aiF - 77,038.200 -f 181,919.07 = 0 F = -3.3333 and 0 = 0. Now these values of 0 and F are substituted in the expressioiife for the flexure of the left-hand column and the numerical values of 24 these expressions obtained. dj » 91 . 294.58 inches. dp » 47 . 881 , 77 inches. E ^ E These values of F and C are also substituted in the express- P'tl ions for the strains of the various members, . , and the numierical AE values of these strains obtained. They are tabulated in Table III. From these strains the Williot diagram, Fig. 13, is drawn, assuming G fixed in position and locating J a distance dj-d^ to the right of G and a distance equal to the elongation of GJ above G, Here there is no error in the assximed direction of GJ, and it is not necessary to draw a Mohr rotation diagram. The deflection of any point is measured from point G, and is up or down and to the right or left according as the point is above or below and to the right or left of G . The vertical displacement of the truss at K relative to point G, due to the deformation of the truss, d 35 (Fig. 13), is given by the formula. ^35 “ The values of I AE for the various members are given in Table III; they are the strains in the truss members due to the forces shown in Fig. 11. Ug is the stress in any member due to a unit ver- tical load applied at K. To prevent vertical movement of the whole truss^ an upward vertical force of one pound must be applied at G (Fig. 15). To prevent rotation about G^ horizontal forces must be applied at G and J to form a couple whose moment is equal to the mo- ment of the couple formed by the unit vertical forces. The numerical value of each horizontal force is equal to * 6 2/3 pounds. ^ then becomes the stress in any member due to the forces shown in ■At Tit J: >TKIMg^i.s.>';.iiiic«i r;ttBli:.Xir>”iii^-;^^ .gn-, ?js: r 4''0» , , I ■; . •» , ■■■ « »•-**«' tm . 5 :. ..i\'i«''^ f:r 'tr VV. Jp- **^ ' "'“JT ’* -■ A7 :7.t> criij 'f i.;.Mi^iif iiff feii i ■ ^jhU^ 5»«iH 'ivigi- i63aii?.ir^4d'4t^^^ fr ’V I ^{tveiW t-if* ^>i>4^§i.‘'4i#i;ii i, -o^‘57tv • ’ -V , * ' p ^ l 9l^' X : r, Xv^r ^’% u::;rttt^f^ • . ' • ' ■ if . ■ ' >■ ■ f, ' 'i.'. -\* v^l, 'W ;5^ . A? Kit^yu oi>;iJt;h^,ii 5 ;5i s.7i:at^c^X l>ae C ‘ i '’? '. -;5 - ,j.. V ’ ' , j ., . V , .^''■^ iijf* » • . 3CMk^ ‘ V '!''•’ '*\i |.. ' . ‘.' '*:i2 '■'■ .6^ r ' vXiTrJdXWij - b's ' C k-‘-' ' ' ’ -Xi . ^ , ■ '"^ :m'- '■■ W, lyti' Tiv ^pi^ .of iifc/ poit.^ •.;■’■ pprtSp'^oi. *T 'k ' * / VlBH . . r rtl.*,'. /^ .'. ni' .&^i-hk «4X-. QOuix^x* .’^o.^esfi/Xxsv, -tilt,, f £-' .' , t ', ^ -r'-^'- ..A , , ' 2 ■ - ■ V^'H- ■S'- • ■ . ' '.i ■■' •(»'' 4- ■ # ., ' . • . ' /"•* 4. >!>t{;;l. £ {;* Ci* Ci 7 •>i*x6'4l- 6 Vje.®i IsO' ■ ■' - ^ A'- iv, '/ V . rpi 2 “the value of a small angle in radians being approximately equal to the tangent of the angle. The angular rotation of the whole truss about G due to dj~dG the applied forces is then — ^ ~ . The vertical deflection of point K, or point B, due to the rotation of the truss about G^is then equal to ^(dj-d^) , the chord subtended by a small angle being ap- proximately equal to the subtended arc. The vertical deflection of point B due to the deformation of the truss at K is d 3 g. The total vertical deflection of B is then dg » 6 2/3(dj-d^) f dg5 - 296,899.47 The deflection diagram of the structure (Fig. 13), due to the vertical force of one pound at B, may now be drawn to scale. The vertical deflection of any panel point of the truss is found by scaling from the Williot diagram. Then the vertical reaction at B due to a vertical load of one pound at any panel point is equal to the vertical deflection of the panel point loaded, divided by the vertical deflection of B, both deflections being due to a vertical force of one pound at B. The vertical reactions for unit loads at various panel points are as follows: 1 lb. vertical at 1: Vj^ = ^ “ 0.1532 lbs. dg 296,899.47 ’SS J V , , . :=j5j^i.aaca jiL>ti« 9 r S^Vi' ^■V'v ii " y^'ii . " -'iS^ ^ C;/tri^lfV ^ tJ^ isuyCa"/ .2 C.>t4' i. V: “* ■ *v = as*??*- ■'^.r^lli-1 ^ ■'■■»• ^ -jg - i ^ 1 •, i ' 1 -WJ . - -.-vVtr V - ' •■ ■' ''■'’•r'.V ' ’"^Vf ' P ‘‘i^W fiEfc trlodh to dL'Dx'd^fj'x ,0iPZ I \ ^ ' ■ ' /' ^ ‘ ■‘•"'> ■■ * .■ /..' ' #<;(«»■ 9if^ ‘ . Ij't’ ' ^ T ‘ vitf , ' ,aid^ ..■! X h. tt^.vti (X -tisio^ t : ^' •’ ■ ' '•■V :. i" ih '. .v...": .. ■ ^::i' ■_-'#» 3'^»'3 «2I »., '3 . ■ mm :, ^:v .:’ ■ ■ :V/ _ .. ' ^ '.. ', • ^ »d'T .ftMCta a4 /rariu Ji' l **'-'« \d' ^ ‘t* JPt J^ip^f^v Sr .» .' ...i’ it*^ '•'’^ ■ *■•“.• , ' /V. ./• »tV > ,*^ ja'is'- ..** '■ *. ■• ^ ■ ■ , , " ■* ‘ ” ■ III-" '^. . ^.'.: ■ ... ' ' >4 *: ^ - *^‘ kniv I lAd t 'All*' ^ '•.' f?" v: ' . „'. !6 ■ ■■ ■;■'''* - '■ '-^jt ' * IV ■ ' • ' ^ ^ */ -•• ** ^ ^V| i],"’' !jltft,' tphi^df. fdlm ' . ;’ a i d 0 i f'tfj^ *PW \t- tdgjr^l f ^ • ^ ^ ^ *v; '.4 ,'v ‘ , ' . . 1 . V - ... ■ ■, f- i'f. -.-- — — r '. ■ . iau. i^'f:ii;4 i# a 'w^ ‘ K* ''.li: i -SSSifib, “■.:(, ''^.Si 0:t V -'F #■ ■ r/V.‘ :4wol^o> a^.,%^ ' ,’i; ^ ■ 26 1 lb. vertical at 4: = 0.6897 lbs. The vertical reactions on the left-hand column are found by equating all the vertical forces equal to zero. The vertical re- actions on both columns due to loads at various panel points of the structure are tabulated in Table IV, Class II. Case C, Resisting Moments at the Bases of the Columns. The influence diagram for the resisting moment at the base of the right-hand column will now be determined. As it was stated before, this diagram is the deflection diagram of the structure when subject to a moirient of one inch pound at the base of the right-hand column. Apply a force of ^ pounds at the end of a rigid lever of length k inches, rigidly attached to the column at B (Fig. 19). A moment of one inch pound at B is produced. This moment must be re- sisted by a horizontal force and a vertical force at the base of eacl. column, and a resisting moment at the base D of the left-hand column to produce equilibrium. These forces and moments cause bending in the columns and develop forces at the points where the columns are connected to the truss. These forces (Fig. 17) are statically in- determinate, because of the unknown forces at the bases of the col- umns. For the present, these forces will be denoted by symbols, and their numerical values will later be found from the condition that, at certain points of the structure, the deformation of the truss equals the flexure of the columns. C is the unknown horizontal force at the base of each column and is the unknown vertical force at the base of each column. X unknown force developed at the top of each column. The forces acting on the two columns are not I '"v V- S^.-U\ndr*tti'- ^4 ^ 5«>: 0n^i^n% *«f *" ’’ ■■"' '. 'tt ‘ r / .i ;i^' /^f ^.wtv ■^ '"" * ' ’ " , ’ ' ' ' „ l ' *' ' . 'W'J^Xv ‘ ' iU -'i^t^y,, , *u:i X j 5 *i ,'j!i* 0 ' 2 c^ n <|ft •• r fi^ 44 *i^t' «ii*i ^so ■ ' :■■.■- ■ . ■ ' ■ *: »rv .> ' fi' , r£ ' .-''•'W t* • ^ : • ■ >• • ■ ^ w«lk-; ' , , . •i* 'ii»t .ijw. ■* • i- i Xfi-r ..' ; i v.» ♦<< flo/s Xiir ;jr^fl^^oa^ “'r. .*.v'' ■ .' : SM'^ ./-a ': ■ ■ ^ ■swy. iv ':_ . x‘ tj .-■ . v-4 % V ~ ‘''P i^JJ: v i.ca.i^ » > <■• >\*t ••• .'- ' > ’; * ' ■ ■ •i’*'\l‘?!fi *’ '''*'•• ; .jj '"'Si' •ir-jn.i'i-z < 5 :U ■;«' t>J,i «irto lb ci^ yf^ttCpa:' X '■' ' ^ ir^ " ‘' ■- \il *iSS 3 f^ ’ ' ■ ' 'fXiM . V ;:"' ■- '’!* 4 \ tl in t vXjb ijtt e'‘4 ,-i i: w: ti. d*i» 4041 ^ii?r -‘i i 6al?oq^/(t>4t^«^3A^JAP& o/'-^ -o 5s.- Of D ^ .6t--'*.iAi-’fi^’i4v. •;(« "' E^V'’ ■ '. '^'''* " : '^'r ;^r' 'p- ? ' -' ' -.' . ’■ •■ . 'X. . ■ • • ^^ h^' ‘ ’, * . ■.«'/■■. ' '■ ■■ '■:^ ’ ' ' " ‘ •"' ' ^ '".; •J*‘. i- . p-ii* tin^%q W4f,; Xi'tv, ..t^ <:ioXt>^«?6',- tnaiiC-OO ' ■"■' , C^x . . «&-V' * 9 ^ fe'ti^Oifcfitc«rP -"-rrt K.'.. r-x.r' 0 :rfjr- l,y t :T JS :^. '?»'-JT>:i')b ‘ . „ ■ ■"'*/ >’ '*• ;• V ' *' XN >*,_. “V* >. • ■ d>iOCff.V;* 'Citf' 1 > 4 . \t, ^-,v ©ar.*’ *.io x*? ' 6 ? tlJ:v 'a 4 >jyJi:^Y,'^:x 40 ^ri^^ ■'■ .'■ ."•' ■'•’ ' • ■• ])i %,\'-'^ t' ■! ■'■k’‘' '’r Jti'i •r^ 4rjroj;Aaw, 2'^ dTl#‘iXi4^^ •■-'■'* ' '’• '. ';■ ■» '!■ ' X;-n ’V ’ Ipo'i-' X iiir^.tjTjXx^"V ;wGi),^f(X)r iL' Xr.uIO’i- X«i ^4T* aA^44-^oiol'-i < '- -' '• _.• r ■ ,, ,- • ■' ■ •' -• IS • ^, -' », . T ('-'-'ij.^-^u ■• ,.■ , 5 v V • . ■ ,>v ' ' '''•' - 1 ■,jB .nsuxXco rfsv'^ 1 X jB ^-' *■,.* y % d" 3 n 4 n 3 ^Xcp :pf^ t» jj/c ■•' 'ft w'ft’vorfC*’ ’ | !l ^ni;t o^ ,iM iA •' ii? riqaot^^A v'’^ JL' I ■ / ' ,. ..fj-- '■''^ ■ 0 F;rjc. n ^^»oTo,i i^oji.' ■xo ^at ‘ ■ > ■ --J »• , !■'.'■ , ?'■ .5" 3 r» I .^*7 !,»" ■ la^/s;^ 04 b^Jxjrr.1 » \ ,^^’/yj' ftliJ *c^i* ®t# ^fiBa/Ioo.&tsfrvJ ■"" ■ ' ■■ "^ :; > "? '""^ 'M . *. ■'■■ >>■ ■ ^ pS'3 ' ■■ ^ thy l4^.pf p: ' ! • V '“V - ■ ' -Vi;;. -.„i- ./^•' ^Yy a:it ao/.<^C*XA.r ■ ':Xj>:4^»as 1^ ii'.?5rX.sil ‘ J r*' ' ^ ■ ... '■ .■■; ■ '^, ■' ~Js {itS^iosx oiU ikiAM X-' :<0'iJ ^no ^o^tx^'icm K"' ' . Vv " : •» i :-t ?• l \\iXja4^ '' ' ■' ' i-4-^Xfli XSifif c'iTK^^ rt. ’ ■^ ^ ■ • • • ’■• I ^‘’v,'»\ " .p^ ‘ ’' f •. 1 4 ''A**- _'i^ n^X^iaiilV/ «ir» .. ,-.i -jXiU’t% s.tj lc» •'fesce*^. ^ ■‘ '' : . . h ■■'■■'’ .... " f>fiZ, . i«4v2f<7q;^ap«Y ,«io fcj^ii To.v. : tTj®S’-i;v ^,rix tit v\ osis^ '.'■' ■ , ■'.' ' . •.‘;f‘r.-* f ■ ’ v.'oAi s.'f.o I'j lrt#4ort -liieft^ 'fiiji .ft^^ a!^a^ nmu^to.o::^'^tLMfr^s^i'x ^- ■ • > ' '■'‘' * ^K'*, ■ " ''' "V ~:5X*iip .C.'Ju H 4|t..i>^x‘vnv !i;;^ zt xei, aC'iit4iSffot&fi:;&it3'(5^ My \ rii :X<- XxjO'V I uit fce 7v*c# bM 3fi';/nli^,^;,^^^y ’■ ' ,l;'\ ^ ': : V.' -. ^ '■' > <' ' *^‘' ; ? Tr^-y-~g gTcgrar»MrgT y g!:^^ 28 deterir.ined. Flexure of the Columns. Since the forces acting on the two columns are not the same, the flexure of the two columns will be different. Column DJ : The forces acting on this column are the same as in the pre- vious case (Fig. 11), except that, what was the numerical vertical force before, is now a vertical force R. The flexure of the column, being due to the horizontal forces, is the same as in the previous case (Fig. 8). -dx = 27.386.37F - 38,305.42C E “dft = 14.364.53F - 33.940. 89C E Column BL: The elastic curve of the deflected column is BK2L2 (Fig. 18). Draw TKgU tangent to the elastic curve at Kg. BK*, the deflected po- sition of the lever, at the end of which the force of ^ pounds is applied, is tangent to the elastic curve at B. The moment diagram i£ b'k2b. The deflections of various points of the column away from the tangents TKg and BK' are easily found by the area-moment method. a Iy f(l-180C) x 90 X 120 + 1 X 90 X 60)J - 1_(16,200 - 1,944,000c) El » 199.51 - 33.940.89C E ^ £(1 - 180C) X 36 X 43 ] El ■ (1728 - 311,040C) = 21.38 - 3.850.54C E r\\ ktJpWlCt .V t ’ .< X - J-O': V « ,. 'f ^’IP «T -i. • O'J , ■'Vi»E' COVOlI XxiDi?10< tl vz .. , I - . ;. : ! r O.J ;.i ,\ 1 :J;>0*. . . i » V i / « *s kUf£^:- o : p: L*V' F ..■■ j .;. .IjSk-" - » ■V . v'^'3 <■■ .-'• ) V ^ ‘ •-, -» ,i 'I r.o‘ ' • f ' • ^ '•. ■■ • ' ■•• 7l! » .■ j'\. ’> V% ti%fi , : H'. ' ■- I .-ri A i -l r _ j~~ ux TV ff: p- -'- x:3 -n ~ . t- \ / ™ -.aoRess;^' ] 29 ^4 X 90 X 60 4 1 X 90 X 12o] . 2^(16,200 - 972,0000) El » 199.51 - 11,970.4 40 E 5 =: ~ jjl - 1800) X (36 X 48 ^ 90 X 132) 4 / x 90 x 192)] m 1 - (30,888 - 2,449,4400) El a 380.39 - 30,165.520 E Deformation of the Truss. The displacements of certain points of the truss due to the forces shown in Fig. 17 will now be deternriined . These displacements must be found by the algebraic method, for the numerical values of F, 0,and R have not yet been determined. The displacements desired are d 4 i, the horizontal displacement of J relative to G, due to the deformation of the truss; d^, the horizontal displacement of K relative to G; and ^^JL, the defoririUtion of the bottom chord. ^41 = I P4U1Z/ AE A^Jh for the bottom chord members. is the stress in any truss member due to the forces shown in Fig. 17. = SF 'f' QC TR (See Table III) F, C, and R are the unknown forces explained before. S and Q are the same as the coefficients of F and C in the expression for P in 3 the previous case. T is the numerical coefficient of R and is the same as the numerical part of the expression for P^ in the previous case . U 2 _ is the stress in any member due to the forces shown in Fig. 5. Ujls the stress in any member due to the forces shov/n in P4U3^ Fig. 14. The termi — is the same for each member as the term 30 P5^5^ in the previous case, except that, what was a numerical value AE P/iUt/ before, is now the coefficient of R. Values of .. for all the members are tabulated in Table III. AE (i41 =* 605. 51F + 705. 99C t 865. 90R E d 43 a 1411 . 97F 4- 1 839.35 0 f 4706. 60R E 44 JL a 300. 94F + 427.560 f 670. 09R . E It is nov; possible to set up three equations containing F, £, and R, which may be solved simultaneously to obtain the values of these unknowns. (See Fig. 19) - dQ + dj - do - • d4]_ * 0 ( 1 ) d^ f d 43 or dg - dg - 0 II to (3) The third equation may be obtained by taking the column BL (Fig. 17) as a free body and writing summation moments about B equal to zero. 1800 - 72F - 480R - 1 « 0 p - 1800 - 73F - 1 . (3) 480 The value of dj^ may be expressed in terms of known quanti- ties ( See Fig. 18) . d^ + ki 4 = y(dL t ^5) 7dj^ - ^“^5 " *^^ 4 * Substituting this in (2), 5(dj f 4^JL + ^ 5 ) - 7(d(j + d 43 + A 4 ) » 0 (2a) Substituting the value of R from equation (3) in equations ( 1 ) and ( 2 a) and reducing, +13,499.47F - 13,333.840 - 1.80 = 0 f40,829.88F ^ 64,930.500 - 567.08 = 0 31 Solving these two equations for F and C and substituting the values found in equation (3) to obtain the value of R, it is found F = +0.005404; C = +0.005336; R = -0.0008932. These numerical values are now substituted in the express- ions for the flexure of the columns and the deformation of the truss to obtain the numerical values of these expressions. ■ ^.^^inches. -inches. iii IL The values of the strains in the various members of the trusi , ^4? , are computed and tabulated in the last column of Table III. AE These values are used to construct the Williot-Mohr diagram, shown in Fig. 20. The Williot diagrani, shovm in full lines, is first drawn assuming the point 0 to be fixed in position and the member GJ to be fixed vertically in direction. Then the Mohr rotation diagram, shown in dotted lines, is drawn to correct the error in the assumed direction of GJ, using the condition that the point K does not de- flect vertically away from its initial position as the basis for con- structing the diagram. The deflection of any point of the truss in any direction is the distance from the point on the Mohr diagram to the corresponding point on the Williot diagram in the direction of the desired deflection. Deflections are up or down and to the right or left according as the point on the Williot diagram is above or below and to the right or left of the corresponding point on the Mohr diagram. The values of F, C, and R found above are now substituted in the expressions for the following deformations to obtain the numeri- cal values of these deformations. ■'r fim ■▼ .1 i' ^ '■j, ' n , . i,' y ^ ^ ^ • .^x.ni ^ 2v T" ildii ik^\^dt: AjU^-ci'o • q^' "N A ' L ' ^ - ,i, , _;' ift ■ -'-. f .1.x ttl^zX ^‘4yuy^-‘ s^ofcl fe»4it . " . _, ' ■- "" '. /• ' "' . ‘i;J . j oj ao o 0 1 t, j: e 'p); a 04i X/yvc^a i^' > ■ , , ^ t^- • , ■ . , ,.-■' ' :' '■-(^ ■ ^ A,i iViif.xcr t|ju^ i^iA txf t^.|aoq ^?Jt^’'-i-;facir /ac^ y.3tn dai^-? ;A.;l/;rt.:xn:y,_ wl;x!jt‘- ■ ai’ fcijBii?' vi ,$& -iif £>e#i^. ’ .•■o; de^'V idt '%<^ fSilo^ ^tiV,---- - - ■ - ^ ■■ ■’ ?,ju>‘ s»«5S4^Xifp-crf .3 ito, if ft X ^ '• . • ' ' I " i‘ l.- ’1 bM i#'i|3 *:i'*"''.X i’.>* px.; a.vfK^TC^oXji'SjCI . m - , .V' ] ■ '7*' ••■;.&:^ ' ‘1 >• ■ *i ‘', J L a 4' iJUjT^'^4/:; J'x l.Jx‘'Xr( :t.i> 'SiS */U.6xp,4o4‘'‘4'l .10 :, ■ ri^*-'^a. t'Ml'.'tJt> 2«4iri^T a K f :.:^j ; W', . '■''' ‘ '“f.h' Ji .i^Vi^ib, rt 1 i a ’/^^^ >1 I. .■ ’ I*" ^f- 'v.viiv'^ 'yyik ':^ ,^ « j?, 'f aM|i^lfi!f''-b3t^’-, ..0; ,,'r r« xtiii r*-tpt4:' v-lf it 2 ■‘.f^i''U P'X '.-*>* .?.1X ' tqt ii.«ioial'd--!|tm ■f’f . ■- * ■ ■ jf ' ' \'-’ '^'■X .,M’ *' ’ « fiiik*Mi;t:' la ooyXiv^-Ica'-i j: . .1^ . A'9i iB ,s e^ ^ij^tLJ9MJ if9K b£x^ ' .w^ . ? crr :ay ii'Tr - ' ] r »^,Jtawy^ t ’ . _ I - 1 * ji*S - 1 , u . i' '1 •' ii - A iil i V*' 32 d4i a 6.27 E inches. II 71.76 E inches. ^43 a- 13.24 E • inches. 42 - -0.84 E inches . 4 JL S 2.77 E inches. A4 - 135.63 E • inches The deflection of the tangent is dj^ + ^ = dQ. -f d^2 + ^^4 “ 198.99 inches. E The deflection diagram of the structure due to a moment of one inch pound at B may now he drawn to scale (Fig. 19) . The resist- ing moment at the base of the right-hand column, due to a vertical load of one pound at any panel point of the truss, may now be deter- mined. As before stated, this resisting moment is the vertical de- flection of the panel point loaded, times the length of the lever arm of the force which produces the moment of one inch pound at B, divided by the deflection of the end of the lever due to the moment of one inch pound at B. The vertical deflections of the panel points are obtained by scaling from the Williot-Mohr diagram. The resisting moments at B due to unit vertical loads applied at various panel points of the truss follow: 1 lb. vertical at 3: =- ^q ' I^qq " 5.925 in. lbs. 1 lb. vertical at 7: =■ » 5.292 in. lbs. 1 lb. vertical at 4: M* = z 6.196 in. lbs. ^ 198.99 The resisting moment at the base of the left-hand col'umn for each case is easily obtained by statics. Values of the resisting moments at the bases of both columns^ due to unit vertical loads ap- plied at various panel points of the truss, are tabulated in Table IV. III. APPLICATION TO THE DETERLIINATION OF REACTIONS 7. Deterniinat ion of Reactions Due to Vertical Loads, Since snow loads, roof loads, and other uniform vert- ical loads on bents are usually given in pounds per square foot of horizontal projection of roof surface, the reactions due to a uni- form vertical load of one pound per foot length of truss are here- in determined. The reactions due to any uniform vertical load ar^ then, equal to the distance between bents, times the magnitude of the vertical load in pounds per square foot of horizontal projec •- tion of roof surface, times the reactions due to a vertical load of one pound per foot length of truss. The vertical loads are considered to be applied at the panel points of the upper chord. The load carried to each panel point is one half the load on each adjacent panel. The horizontal or vertical reaction at the base of the right-hand column, due to a vertical load of one pound per foot length of truss, is obtained as follows: Obtain the vertical deflection of edch upper chord panel point of the truss and the deflection of the point B (See Fig. 9 or 15) at the base of the right-hand column in the direct - ion of the desired reaction, both deflections being due to a force of one pound applied at B. Compute the portion of the total load on the tr uss carried to eacn panel point of tne upper chord. Then N = rwd^ B » -o4- where N is the reactioa, horizontal or vertical, as the case may B be, at the base of the right-hand column, due to the load of one iiound per foot length of truss; d^ is the vertical deflection of any upper chord panel point x of the truss; W is the portion of the total vertical load carried at that panel point; and d is the t displacement of B, the base of the right-hand column, in the di- rection of the desired reaction, i.e., horizontal or vertical. The resisting moment at the base of the right-hand column, due to a uniform vertical load of one pound per foot length of truss, is obtained as follows: Obtain the vertical deflection of each upper chord pajiel point of the truss and the deflection at the end of the lever, where the force causing the moment of one inch pound at B is ap — pliedr^^QQ Fig. 18), both deflections being due to the moment of one inch pound acting at B. Compute the portion of the total vertical load carried to each panel point of the upper chord. Then M = k Zwd^ B 2L. , where M is the desired resisting moment, W is the load at any panel point x of the upper chord, d is the vertical deflection of that panel point due to a moment of one inch pound acting at B, k is the lever aim of the force which produces the moment of one inch pound at B, and d^ + is the displacement at the end of the lever due to the moment of one inch pound at B. The vertical deflections of the panel points of the upper chord are due only to the deformation of the truss. These deflections were obtained by scaling their values from the Williot • Ttif I- . ?">i :'^1wm’-.A ■ '»' ' KT.'^/’ • * ''^'' ' - '■'■ ■• rr^ . ''V, .J S • '''i^’-’V^ fWB'CJ ' i .^^ '>J 5 ''^ U *^«l 7 ' to < 5 eAjBf P -« ^ , ''rr "‘ 11 :^ all*^ fci ^t) •«3 jlT;^ >,c.‘ ^oolt pi. ;:4" >• .' .-.j;'"' jiai H'jsi^‘‘ ir X JiT' <# ' tyioJL", 4^ -' - s»' ' ‘ ' ' ' ' ■ '“ ' ’•'•'• t • 'J*! * • ? ' ’• ?•» '»-.», I <'L :3 ■ .>r* JUt.K^li^^ , * .1 eflla Jo <^4403(111 ■tie ^ P'' ‘ ' ■'. - 'I fifi., 4 -J'ttoi,V‘’’e^-' i>.^JfO f'U 'ad^ ''':' 39 HI( 5 ^ ' A ” . ^. •/ -V , ■ ' '- ' \ ^ LL ' + li^ 'i^ “' '■ ^••'*0 t.. ,$®ol jfiaVvtnu i^jpi ,(mUlco^ !► ‘^SOi t^TociV ,‘j,i»*f ! ^0 ..feny ' o i' d fiX* it? 6 ^ ,. :.' .. .: -1 ,v . / ■■'^ . *' X** yi ' ■ a ','Acdv*, 0 'ei«;d; to ,. .,4 ' V '-" i!'--^'^ -'''2!®^' yr'T * ' kH^ M'*’ ■iO' •V'*" • V 04 tHja ttirlatf •^Q f5o5) i'tafc d^icf «{ 6 i! lii -irdl’ f^3 adx/.^oO^ :^u£o^p^ftc;t;pq:xJo^i 1 :^ .'i'. r«^f' f ■■ ■ m^''T *i 4 <(j^ liT' # 70' .4 dl aa , ia w,q , ftp^ 1 . W- , b«i 2 «td,o t^sf,l r ; ■ ■':■■' -V' '^ ■ , . . 7~^~ -^■.' Ik.'*- ft' ** -V*' W ■ - .; -a - - . *• . . !r^,^ ’.V • i Bb ■i,y 'J41 ]0 ’■ ' *■ '' ■'*' ' 1 ' t‘ 'T' t» ' X>?” “■ -,; '" ■ — I ' I Mi ,Mo4» 0 Xo/iPd iS ■•.i 'S'.;:,. .— 'y^, .itijCfi'- li’yei i\4B'':. i£;| 1 '.^. . T8 „ *•', ■ . '- . ' '>. . .■ ... '• '’” ;-;■ . f 1 rt* ^ iyd'iiti: ,S 4 ;d'''^bA:Joa'.dO 0 Jt aorf il . .■» , \ ' ■ ..* 3 " ') 3 ? :,v.S > if 'te ., >nd i’JdX^qq ‘Itiiai; *dT |idlrii'<^T' .wvtAi «>rti to-idG7ioi^*.TOt^ii'!#lF4 ,,W >i.a 6 ax#?Jb«o40‘ :v . X ;! 7*., f-i£ ': ' CS.-.'® X--“.'' 'V -^A« : ■ ■ ‘■■'.sv^i'- ■ ■ ■' . " ' 'lii KLiXiLu^ ^ _ -L Si. ’ ’‘^ * ■ yys i v?? ^B . c 3 i>jpi.i |, i ' ;.|)|«, ' i ‘ i,i,^ j i Stes r t rrg j yr^ i f at .^ -o5- diagrams, drawn by Mr. S. R. Offutt * for his thesis. The vertical and horizontal displacements at the base of the right-hand column^ and the deflection at the end of the lever, where the force caus- ing unit moment at B is applied, were also obtained from his thesis for each bent analyzed. These deflections were originally obtained by computation by the method given on pages to of this thesis. The horizontal reaction, the vertical reaction, and the resisting moment on the left-hand column, due to a uniform vertical load of one pound per foot length of truss, are equal respectively to the horizontal reaction, the vertical reaction, and the resisting moment on the right-hand column for each bent analyzed; for the load is symmetrical on the two halves of the truss. The distance from the base of either column to the point of contraf lexure of the column is found from the equation, M M - HY = 0 or Y = » where M is the resisting moment at the base of the column, H is the horizontal reaction at the base of the column, and Y is the distance from the base of the column to the point of contraf lexure of the column. Concentrated loads due to cranes sometimes come at the panel points of the lower chord of a truss. For this reason, some panel point was chosen, and the reactions at the base of each column, du. e to a concentrated load of one pound at that panel point, were computed. This was done for each lower chord panel point of the truss. These panel points are points 1 and 4 % See footnote, page 3 . '.HI ^ \r.,i .; • ^.<»t' ' > • i^rv f't 49'i^'•’Jo’VA>•.4' ^ ii^'- ^. .iu .toti '-^t^' . ' '■ ■ '/■'■ "W ' r ;.* '-■%:>i 4/-' .i^mu ■ ;U0 ‘jayi ^ * ' ^ V«».' * 1 * * '* ♦ , ■•■ ^ , , '•’ - '• ,*i , . , • ■ ,, ,'::'ii. ■»'!•:•. ^ --V ... v;. - ■■, ^ A •'» [ 4 , OS>\:^-- wu;;/ f ■ -r .!• '• /. i.**' . “■ ' • p^ V' s Ai '®. •p;r' ,y> 4 .>*/ Xj.- l^nev '' ., J * , ' f ■ ^' ‘ ^ * ' ” ix a,«CH s-ii ' =^:av To, 1 »oX' •'» ' ‘^r:.-/tK s ' . ■»■ ‘ ' ^ -i*yy ^ , : : r. 'Pili To ctii u oia^&58MTfj| ri i?-‘rij'. sY-'^ ^ » , ■ • „ i» . - •’ ■ ‘ *■ ^v’."iiii 5 7 /% '^ioo 1 -'MH ' si; r , - ...rjo a%.Y«‘-K''l :V- ' ■■'.'■Al -ii6- for the bents having 30-f t. , ^0-f t,50-f t, , and 60-ft. spans (See respectively Figs. 23,24,25, and 26 ), For the bent having the 20 -ft. span the point is point T (See Fig. 22). The reactions on the right-hand column are obtained in the same manner as the reactions due to the uniform vertical load, and the reactions on the left-hand column are then found by the equations of statics, j The points of contraf lexure for both columns are obtained in the j same way as the points of contraf lexure due to the uniform vertical! load. The reactions, due to any concentrated load at any panel point of the lower chord^ are directly proportional to the reactions due to a one pound load placed at the panel point. 8. Determination of Reactions Due to Combined Wind and Vertical Loads. The horizontal react ion, in the case of the hinged columns, and the horizontal reaction, resisting moment, and point of contraf lexure, in the ca^e of the fixed columns, due to a wind load of 20 pounds per square foot of vertical projection of bent combined with various uniform vertical loads, varying from 30 lbs. to 100 lbs. per square foot of horizontal projection of roof sur- face^ acting on the bent^were obtained each column. A wind pressure of 20 lbs. per square foot is a commonly assumed value in this region. It is the value for which the bents were designed. The wind was considered to come from the left, thus, making the right-hand column the leeward column and the left-hand column the windward column. * t a ^ M t. ' X—” , ■ ■■'■ ■ , l!.’s > ' * J C" '•■ .'t : -J. <■ ' ^ »'2S i ^'t;' . *, y . -.•..I ■ ■' ' ■ ! ' j ‘..fr3S •.i^;,^rl>i'''j'it . ("^j:'’ -V'^^ fc sE) T jiiictfl •* jiiiofl A*t.^ -.AltV 0^ .' • ■••™ • ■ ' i ^1 « f fU 4,t bi*iiJi;i.’j?tJv di* t t? i‘ t\Oi j f Id V'lt Jtte rf#' *»xa ^ ■ -■ . % ' V. ■ . ■ ■ if* fti iw fcosuii^o- pH’ ‘ *Ti>r*tni/ ■• ■■■* !^o -■ua' ^ r '■ , ^ • • *Si lyti^ ■!iiinti%:.hji’Ci:l batjo^nao. M^qi ‘ ^ >. r >*{(«'•* 'Jr.;- >' . ■•« .' ' 1JH'...'- % wifi to »fip ff? .fiOiAWi' ■ * T r‘i<5 «'2d ^fi4 il.E»i.^j|r*’X0'i' .iiAt2^7 Ml IN: ij .,, if. ,s) * '‘Att .^1 ^>' . ■ '“ ■''* pv^ e„dT ii. • *•. ■ ' ' . , iKf'4 ■ ^^ Mcsa - 37 - Values of the horizontal reaction and moment at the base of each column, due to a wind load of one pound per linear foot act- ing on the windward column and 0.7455 pound per linear foot acting on the windward slope of the roof, were obtained from Offutt' s thesis.* This horizontal reaction or moment acting on either columr^ times 20, times the distance between bents (15 ft. for all of the bents analyzed), gives the horizontal reaction or moment at the base of the column, due to a wind pressure of 20 lbs. per square foot. The horizontal reaction or moment at the base of either column, due to a uniform vertical load of one pound per foot length of truss, times the intensity of the vertical load in pounds per square foot of horizontal projection of roof surface, times the distance between bents, gives the horizontal reaction or moment at the column base The directions of the horizontal reactions and moments at the bases of both Inward and windward columns are shown in the above figure. is the horizontal reaction due 'to the vertical loads, H is the horizontal reaction due to the wind loads, M is the moment due to the vertical loads, and M is the moment due to the * See footnote on p-ge^. ^vr^„f^ • " '.‘l'.v. ■. • ...ii ■ J .vr,i. ' :>:'l^- 'TfOu. ..■' J?4 } Pttu.^f,.r^ M. 1^ >:^a4ia,-Xi^ ari^ ^ '' ; . 1 ' ,*T^ -V '• \C .if'"- :® >* •‘ 4 . Av^ L '^j '5^ - ^34 < j>,.’^.' o’ti^"- ' « tt.i» jcri oiCi.;|J4C *' f '' ■'’^ /■ ’ ■' ' 7^ 1 ■/*.;' |: :■; ^ *«■ ^ l.^-^f. i-.»x ft c ' 10 • ' *- 4 '." • ' *ti '•A- F ^l . > ’ J .to’ 7 7 ij ' I -ttK •* I:i^ ^.:-rA ■ t _i i. \r-^m^ ■atjR'14't' o*i'^'; .‘^yW ,iycU/4^^'X’ '/.if aai ~". ( w "?/' Ilk ‘ 4 /sJ^i' > ;wo.y.j^jAga * Mimv Ji 4 '<. //// 'l! ^Sk, fli ‘ I'k 4^ ^,- '‘'v i £-il - tM '• J' ; 4- er’V ;.a ilv, V ';,i.:^', -: V- rm,. ;:. , V Uj ^ 7 .. 4 :-.." *ff /vi*-. ' K- '4 . ■ rV, ■ ■• ;■ ■'. . .. , ' ■ 1t^V' ' 'fr^0i:: «;t?' ^'.ij _ , _ , i *' ' ' . ‘ ' ■* ' ' '‘lB^>f' ■‘/^1,-1 ,i. / *^’v, ' .s'‘.'’Sb f ij , |«oui^jir^4l - '’^9 ttfn ^ >4 ■ » i^i* Philip -ttcf fi^Wi'i 'V-.ft y: . . « j:. if>' i 4 \i SI V ;: -xSit pi ■ ^, ,'r ■ rt.-f' , tw#«"?'iiigif'' * * y s^qtai^apitera -o8- wind loads. The combined horizontal reaction on the leeward column is the sum of the wind load reaction and the vertical load reaction on that column. The combined reaction on the windward column is the difference of the wind load reaction and the ver - tical load reaction on that column. This same statement is true for the combined moment on each column. Reactions to the left and counterclockwise moments are considered to be positive; re- actions to the right and clockwise moments are considered to be negative. The combined reaction on the leeward column is always positive, and the combined reaction on the windward column is positive or negative, depending upon whether the reaction due to wind loads or the reaction due to vertical loads is larger. The location of the point of contraf lesure on the leeward column is obtained from the equation. where Y is the distance from the base of the leeward column to R the point of contraf lexure in inches, M is combined wind R load and vertical load resisting moment at the base of the column in inch pounds, and H is the combined wind load and vertical R load horizontar' react ion at. the base of the column in pounds. For the windward column. LL ~ ^ Y = e or Y. R R R R or 12 va 'i'l , wf^S/lf^ * * * P.\ '{'l4']L'<>J;^ „.ti^ W;.ia -iSCi ti^O £» ,|'!tl,tiVvo •m <*oros . ' , ■.•'■* 3 b.. ^it --.^.,-' U^. • 'j 4 'i' s: • i. 4©-* l^ai' « Ik: w '>iis *T’ft flf- ♦ i fi-Ht 4;4- ••; rt- . ■ . ,‘V. ' .te. ■ .,: ; ■ y JW i >iwij ' ■ .** v'' i^BV5 •./■_ *. B f .4,^ v^>*i » tifci .'^ *!■ '' rtJr.t, "♦ 'jJsfikfc'' ^ » i-/ '. 'Sfl ST * , jsl: .-’. * ar 'J « ■ 4c p '.;>-■ % 5J ..‘•as? ; *^:,'*; V4V' - fUK ■'^^1 ^ y » -, U ' far Us ■ 'i?' :‘«o jrx> ?>■ 0- 6k' '^>:C>XVi ■ , • ^ '^y ' “ * , - ■ '^‘■'^> ^ .'1''^ ,. •.‘^^4ii™i . i!:"?! ff t 4 it>^ *. . . « ^ . --i ,. . ^ jf‘it\.,a v*5' ‘ r: it ’■ ' ' ’ i iiv' ' k Hi - ' -' '• ’ Ti'/ iK ■a* . " f ^ lif , ,^» «+\c*ii^ aV oi^^t ^ ' ■ ff ‘ ■ '' :' 'i. ■ F' ic *n\H;} . ^/j .+.• ir< .-)g^« . •><>/, u^.«i-Jr f^ k*..' .; JA’ 'X- /ivi' .ftiv' Xs- .M. ■■ ^. ^ n« -#'4 • ,'Vv ^v,r€ niiifjti^piH * K.''‘*l '*' '/a ':^ .-i ' ^ ’ 1. ► ^ 7T)-V.t Jf#< Pliri.* !^ ;.-■: MP^rref^ l^r. v*'£ • ri. f.-.t. C •, i.ii .. X : : ■ .;’ '’ ;■ : x :. - to Jil. . . V L.T' .. ■ihki -i:':' ■’( ■ ;'^-i . ■•■ ' i: .i>. - ^ ^ lV- ^ fi ' ' * oV-‘.-> . '*. I 1 ‘ .' '• ^ '.'/iv;'* ' !'.' t ^ 6 '\] ^4 ■ ■ f f il ,y f**' ' y ’lJO,’ • :• ..■ fr.-:. ..'■ . ,Z' ' ’■-■ . ■ I >' . . , r '\ vXu V' •.Oi'.'"' ; .IOC ■ • V • : . ■ .. .• c ;,.: U; .,vC^ on ; „ ' • , ;'• '), .. ■ if ;:i- ■ ; •■: t >t« ■• - 4 ? /. ) ' ' :.:a ■ .W ■ 9 k-y, .. ./ . ',t- 'i;- -u .... •-,1 . - . .. i: ■•>. 1,3 . . .;■■ ■“• ' ni.-v.' > I .i '■■•• .'•. 1 .; .' O'-; .! .»'• 'i ’A oi,:^ •;•-’*• 1'-^ ,1 p., !i)9f« • ' > ,1 ‘ * - 'V. ' '■ :• • iyzy.m. .> J ■Ml. .< JL.»uV' •'■ A ■^ /i','& -ft' f ■ >r. Ci. - ; •• 1. 'T:rye»&. - 40 - pointy one third the distance from the base of the column to the foot of the knee brace. The columns which are 31 feet high eere made considerably heavier than necessary to see what effect stiff columns would have on the results. The columns of the twenty-foot span, which are 10-ft,, 12-ft., and 16-ft. high, have the same section, which is hfeavier than required. The 19-ft. column has a lighter section. The bents having the 2©-ft. and 30-ft. spans have a shorter distance between the foot of the knee brace and the top of the column than the other bents. This distance was selected to see what effect the position of the knee brace waald have on the results. Figures 22 to 26 are line diagrams of one bent of each span length. Points J, G, F and E are the girt points of the column DJ. Tables 1,5, 6, 7, and 8 give the designs of the bents. 10. Results. A complete analysis of one bent is given on pages >f ■ ■ ..Y'tt;- f'.'j \ ..'7 < ' ' 'u ^ ■'■'Vy .,. • ■. i’-'-JOc vi.: > V'-^ ■■ .:iv *!*^ . . ' ' 'pi/ c *i' , ■ r. t ■:..i,Xb-^.t‘.i. :> ^ri:t #.>i t - 41 - horlzantal reaction at the base of the left-hand column, where H is the reaction and t is the total height of the bent. L Table 0 gives the comparison factor for each reaction, and also the ratio X/ d for each column, where Y is the distance from the base of the column to the point of contraf lexure and d is the distance from the base of the column to the foot of the knee brace, due to a uniform vertical load of one pound per foot length of truss. Tables 10 and 11 give the same quantities due to a con- centrated load of one pound at same panel point of the lower chord of the truss. Values of the horizontal reactions^ due to a wind load of 20 lbs. per square foot of vertical projection of bent, com - bined with various uniform vertical loads, are given in tables 12 and 14 for hinged and fixed columns respectively. Tables 13 and 15 give the values of the ratio R for hinged and fixed columns respectively, where H is the horizontal reaction at the base of the right-hand or the leeward column die to the combined wind and vertical loading, and H is the total horizontal component of the wind load of 20 lbs. per square foot acting on the bent. Table 16 gives the values of the resisting mom*ents at the bases of both columns, and table 17 gives the location of the points of contra- flexure on both columns, due to the combined wind and vertical loads Graphs No. 1 to 6 shovir the effect of variations in different characteristics of the bents on reactions due to vertical loads. Graphs No. 7 to 36 show the effect of these variations on the ratio and the location of the point of contraf lexure on R H each column. These graphs will be discussed later. W '^' 4 rit :\!,i , f ? kx *£ v *^ . 'TrtK V ' J ’ I ^ 0 ' j > if ' jBv . ^ Q , l : Jf j 6>#«'^ JC 4». i ' • j rrv'raa t wTyj -I-., •®1 ,» ■ V*' i ', /^*J » - :':ifa ,t* -.r<)u#l /iOk5ttC(inoo ■ etbi^ijj.'V ■ ' ' ••• . . ■ . i ' ^ "'iQ Ix L "!!: "t '.- lo . f jJor *.. tot .'::> t " r ^¥’4 jH ? til ; ;• 'f'Ur t X ■'v^ : ». 0 jc/.o<3 .r , . •! •' ;'• ' ■' : , ^ ;a ' .- ;^ ■n^iod '\o . *1 I Jr .« ,»«*Cai .} “to v * c - r , f Ott V>? 4 Wi . 31*1' ,l^-,r..?t5i . w*i ‘iv LV C*a,oi o wdi •-C "4 O ', i " * v . . - jff '0 i ! *® S'M .■'.1 /. l >4 Xl h $ . . it ^40. tcr ' l » Ci)t lit |/<0 :;■ i '^0<^ ' \‘ ,,-™ - *'> c , ;' r .*^ iJ : . .' t !» tp * rt : l .^ riii;"iiV 1;? Pfel . ' yt . c ; ,! i « ii '^ II j V ;^ , ui • , tt A ^ 14 ^l 0 '>r < im ’ ^ IK <» ,^- i : - .. *‘‘ j'l ,.-. : I n .4' A ^, i 1»» '4^ ■ Ji '- V - if .' ■ ■ -■' . C . • ' f ' ^ i , ftfiVifiot I' - * . . Ci.\' l-sr '»irV ■ i •- j| OtW't #i!V^ t . Tkv > fi, ■ ' ' . '^^ 3' ^ A'IBKi ' • A>-:,r Jii ^10^.^^.4''^ Jl # ^iv i't, « .^DUl-iUt'^ 'TP... '1 , .;' ■ ' . ■•*' *. - t - •‘<*4,; J i'i IS t AO jri^bfi. ^4'V> *•'.4 ■*’e^'^X. 0Vc’.;?' Vo ,T1- :.;i ^l^»|if: er. 'vV F ' . ■- V' ;‘ •'■ ^ ' I, /'''.' '.'.'^n .•, ?v , ■ ■• ■ '■ ■' ■ ■ - Va ' '.1 . ..'< iJV ;,-«»\ . flt . . . ^ ' ... i - . *\' ,-,■ A -♦ ■>4 jB J . ' '' ■ Jj[\ .1 •!,^~‘ S ' ■ Sst : '•. f • .J' -42- IV. TEST OF PAPER MODEL OF BENT. 11. Description of Test. In order to check the reactions computed according to the theory presented herein, a paper model of a bent was con- structed, and the reactions were obtained from measured deflections of the model and compared with the calculated values. A model of the bent having the 40 - ft. span and 21- ft, column height was made of heavy paper. The span of the model was 15 inches. The truss and columns were so proportioned that the moment of inertia of the truss of the model _ the moment of in - mOrnent of inertia of the steel truss the moment of in- ertia of the column of the model . This method of proportioning ertia of the steel column the truss and columnswas selected, because the truss acts as a beam supported on the two col\imns. In getting the moment of inertia of the steel truss or the truss of the model, a section was cut by a plane normal to the lower chord of the truss at the quarter point of the truss. Any other point might have been chosen, provided the same point were taken on both model and steel truss. The weA members were neglected, and the areas of the chord members were con- sidered as concentrated at the centers of gravity of the chords. The gravity axis of the section in a plane at right angles to the plane of the truss was found. Then, the moment of inertia of the section of the truss was the area of each chord member times the square of the distance from the center of gravity of the chord to the gravity axis. The truss members, themselves, vvere proportioned according to the areas of the sections of the members. A drawing of the model is shown in Fig. 27. -4o- The model was floated on small steel bearings placed on a drawing board. A small weight was placed on the model over each bearing to prevent buckling of the model. In getting the deflections of the model, the left-hand column was fastened at its base to the drawing board by needles, and deflections were applied at the base of the right-hand column by a micrometer. In the case of the hinged column, a single needle was pressed through the base of the ISft-hand column as a hinge; in the case of the fixed column^ three needles were used to prevent rotation of the coluJfin, at its base when the deflections were applied. To obtain the deflections of the panel points of the model, a microscope, containing cross- hairs in the eyepiece and mounted on a micrometer, was used. It was focussed on needle points, pressed through the model at the panel points. The method of obtaining the horizontal reaction at the base of the right-hand column, die to a load applied at any panel point of the truss in any direction, will no?/ be described .♦ | The microscope was placed over the panel point where the deflection j j was to be measured and oriented so that the motion of the micrometer was parallel to the direction of the desired deflection. * For a full discussion of the method used see a paper presented by Mr. George E. Beggs before the American Concrete Institute at Cleveland, Ohio, Feb. , 1922, on "An Accurate Meehan - ical Solution of Statically Indeterminate Structures by Use of Paper Models and Special Gages." I Hi'rAtr - Ki 7 »' vl:*' 'ln'-ii-'i ■ V'l »J« MiJSii ac'* ii«SStfl'C'iT i ♦ 4 v^ ju -.';/(? c €jib^ ‘ro\».^jiJvf ■^ • 'J. ■ ' ‘>1 vim ^ .iU oXoti to ;’] ? . V,.' ■ ■ ,.,‘' ;^v''^ ' ";'S 'er 'c( .1 ■ ): ’. 1 J ffl ^ '■' ’ '■ ■ ^ ||.r;p/; -. i.7->, «- : • f ■ . ■ ;f - ■ ■: .. '.• ;; 4' Hi - ' fiJ %hv «li. .5- .*, faO. t'tJt-fl'Ua». ^4||^^^: f ' •sr'^" if. r • ■ \ >' ■ f, ' r • ' ' . i§M*i mi Si, ^'T. .X'tiT -.V y i rr.n . .* * tirtf ,v ... *• ., ¥i i. E ■ , bb^t^sci^b‘ ^ tTU' ^.floXioifetXft' Vfu itI-' 0 . ^ flC“ L'. I 'l. ^ H UHifi ’; ,a^i ‘'4 a»'/o -"OoqiaX^' i=. ' y^u^d ^ -j'V' ,• r '.^w“ " ■ l«i E*'Sr' ^ .' ^ ^ a '• • ‘>^' " * r A tf) ... 'jb :* ■ ^^■•' rip,0, /^-,;4 X'Xc'»aA VI h:.'^^c9iJ .nKr'X;v.'’tot^,l^®‘^;-'^ ■'■ * •; 'P'h 'X«.XoG 4 C - ’ ..... V'^ i',,.Kj " ^ V'- , ' ■ '■ ,'' 3 « ' ‘^■•'iao ' * ' St 'p j g t ' iTir d W-o . rjr i*r. ‘ : !te *j^Uf m t r*r 44 A cross-hair was brought over the needle point by the micrometer and a reading of the micrometer was taken. A horizontal deflec- tion was applied at the base of the right-hand column by another micrometer, which caused the needle point at the panel point to move away from the cross-hair of the microscope. The cross-hair was again brought over the needle point and the micrometer read. The difference of the two readings of the micrometer was the de- flection of the panel point of the truss. Then, from Maxwell's theorem, it follows that Hr = ^ , where is the horizontal reaction at the base of the right-hand column, due to a unit load applied at the panel point x of the model where the deflection was measured and applied in the di- rection of the measured deflection; is the measured deflection of the panel point; and dg is the horizontal displacement applied at the base of the right-hand column. The vertical deflection at the base of the right-hand column was found by the same method. The resisting moment, due to a unit force acting at any panel point and in any direction, was obtained by applying a deflection at the end of a short lever, which was an extension of the base of the column, and by reading the deflection of the panel point of the model as before. Then, from Maxwell's theorem. where Mr is the resisting moment at the base of the right-hand column, due to a unit load applied at the panel point x of the model where the deflection was measured and applied in the dir- ’■' . *, 'r ' , . '■ V i '^f-r i. ^.-vl i. '1' ■ -'p^'i -i4srfqRy»)':a... /. «u5«f "UyaiflMort’ a..4-ws f” * o; oc,t.3* asfliox diSven i]tOi!uff^ ‘ii$#©.‘tid*'.oi.'ir ■ - ■' ‘ ■ ■ '•■ ■ '■ - ' ' ' ''ll K0\ ii>^ ■ nJL0^' '*] ■ #•■ (■< . ',W ISJ. ^ ' • ''.'<1 «...-■ ■ ■ ,' Jl /■''-'‘‘/■‘..r- -^' 'v-' • *,4 ‘B 4 r/ir f r’l'ft* i ( ^4lJ ijgl* -4 ’ ' ■ mHP'-- '*'• . ' i - ' ' ■ ,, J-'''\ti : . ii ‘ ■ !» \ ^ . ’ • -^ IZ ', • ■ • ' '. 'iki , ^ ‘ ^ 'Wr' -j ■ ^ ^ fr' V ^ ^ v’ K V ' T x ;ttii’. .[ Xe,-:>. ■• ,6id* ..-.U 'teXtJroj KeciX I : >•'. y.7v*)fc;ioni aMWv'mxi^:;i 'iXlt "‘ •-■ ' ’ *• . •' ^ > '7A#3 > - i. ii'-*'- vM ■ CT«ii aoio^i:tlSb' ^’'W * '!M’A »», ; , ■n« i rtiV '■ V ti£>. ''Ui^ etJ;f iJs- >. - 1 c^.'''fcifiicf J'j* ■ I# frKf-; :J\V' - ■-•,'* .’ SK .' ; :; ^ 1 r-‘ b;‘ ; ax ^' 4:5 ■»r^^ ' t ;; ' ^ i i 1 i. a^ •fc: ■ / 4 f‘'f' , . ,|#x ? #X.BIirt - *;CJ 8 '^.;tO i fJ 3 «^ t - 3 SlO i i V. . ■wJc<'t' '■ ’ ''" "' '■ '•'"■ri t-’V ’ ■ #. l-.:-iC-‘. ihz^.^c fc;,'; !'<♦.> I "K ' •■< - r» ''^.j f*.* • ./ ■' s ■ -, ■- ■ }'.r • ./ *’ , ' J^r ••**■-» K. . V -V' - • ^ , /T \ V:., ■ ■ •• :. ' "" y | 4 S,J J . ; rtoX 5 (jsf>»’J 'ir. .•r';^XA'' '. ' I ^.Vi‘ . r‘; :•■*•'•,» '■ . ■• '^\ 4 tj 0 L 1 5 ?k’ pXf: I V' * '* ' wo;."' -V ••.^'./•■> •: ^v ’'^A',' .;■ '■' . ' I l»'; 3 pVf"-‘ ' • .'•‘**v» "0 ''V 'u' ' •.!••''' '■ • *' . ;f :is' V * :■ ■" I ' '.'I. , ',.. ,■; •■’iiijifoT^ '■ ' •''■ ‘ ’■' ' '"'’a ^’V:V. ,v. v-uxv'' t*!t'?X3t\;r-A;'v.o y .v^Ai'jeXr-.oi t-t?- •* ^ ' f»:«p .iptjb^QXp: } ' ■ , y ) '. ' . . " ■ . ," ■ ' ^ V -f^ ',"•' }j J di 7. c r, it , , . - n i^'rrP^ “to- . tlC' ._ . ij ^ • * .*• . ► > . ^ I . ^ *X': isi ■’ '"'' 'it' ■' '‘' "! .' ' , ' ■i''" j . <■' '', - ,■ li ^ J. < /V >- ^ ;• . .- - _3S/>., ’ pstkd'^^i fil ' isLjfrdz ”, - . ‘l ’ ,' J^- ‘A',,;' ’'I'lfjL'V' ^ ^'2 V ■ •!■ ' '. Jii«yi,c c . ei^-3 tP o' ajt *;*«««> i .i IkWPaXlP^v -(• : t • 16 M? . .c*.«&.,4,ir, tpiT/ riijpieQ' ': e*^.' artsssriawi?^:: ■-:--r=r- 3 C^ - 46 - fact is probably due to a slight horizontal movement of the pin use . as a hinge at the base of the left-hand column. In the method of calculation presented in this thesis the truss is assumed to be pin-connected at both points where it is connected to the columns, and the truss members are assumed to be pin-connected. The model had rigidly connected members, and both flexure and direct stress could be transmitted to the truss members at the joints. The condition represented by the model more closely resembles the actual condition of the joints of the truss. However, the close agreement of the calculated and measured reactions shows that the difference in the condition of the joints does not materially affect the results. The test as a whole support the theory upon which the method of calculating the react! onspre - seated in this thesis is based. '■ ■ ’.>"' » '} ® 'It. r -■ " i ?T' ,.v, m 5-^ 4;^ X '■'.'■■K '’v ' V r / ‘ v^S ■ ■ ■' i.^ '. .« ./•■IF,' .^1 >tL'‘. . n. . rj'^'ir *• ^i?A* • ^ '\ l-v ' V • ■/ ■...,. • ;-t '•'ii\: ,-iv -0«»- '6itt ’5#-' r - ■ - ■ * .,. f* *': 'If ' ^- ‘ ■ , ■ '% «.a • ■ ♦ ■' iL... , .--r- ...',.-^._i2 . . ^ . -r.' ‘..■' t fr'a ■? ^ “i> . ti'.^ ; _4 . ..-I* 1 * . » . • r > ' ' <■ ■' • t ^ ’■'a " 1’ I. ■ -4^ ;i^4.;.pcl^.: ^a--^ .rr ♦a'w JrV|‘V .sXkf^-A: |-, J >'v? 1 ->:i .teiti ” iM / • • . • 'i *. i fm ' "'^ T" •' t '■ •■,. ' f. ^ I .'.'Sf ■’.■ ■'^i ,/?., Cr-‘’4.'. .;V ”''^~ •• r% rr .’ 4 ^^'' ^ 47 V. DISCUSSION Am> CONCLUSIONS. 13. Effect of Variations in Different Cliaracteri sties of the Bent on Reactions Due to Vertical Loads .- Horizontal Reactions Graph 1 shows the reaction factors for horizontal reactions^ H/t, due to a uniform vertical load of one pound per foot length of truss, plotted with column heights as abscissas. Graph 2 shows the same reaction factors plotted with span lengths as abscissas. The effect of variations in column height on the horizontal reactions is shown by Graph 1. In general, as the coltcm height decreases, the horizontal reactions increase. Between the 21-ft^ and 16-ft. column heights the increase is very rapid, in the case of both hinged and fixed columns; the reactions more than double in value. The effect of the size of the column on the horizontal re- actions is shown by the bend in the lower portion of each curve of Graph 1, except the curve for the bent having the 20-ft. span. A much heavier column section was used for the bents having a 31-ft. column height than for the bents having other column heights , The effect is to increase considerably the horizontal reactions for all bents which have the stiffer columns. This effect is also shown by Graph 2, where the curve for the 31-ft. column height liesi above the curve for the 26-ft. column height. The effect of the position of the knee brace is shown by the broken lines on Graph 2. The distance between the foot of the knee brace and the top of the column was made four feet for the ’ I IJ fc>- t. .. 9 riJ I ■; f " ■ llQ X I M r; .+ I jjV -■ C' ■• ;j j V '■ cn ^ U o' ■ -* : . O' i.' j ) ) • 'i 1 '. /;» -'O ,j ./■ I' ■ ff- • 4. ^ - I. . c .nrt .> J .. . U ' i T ■'» t ^.. t . ■ ■; . ' •('.{>“ : : orcoi 0 . . ' ■ , . :i ’•- ' j 0 , ^ ^ rf - ' \j . , O”. , •■: : ■ .: -T.i'jv , • ' e* j'.) '.:r ©oiu o'' 'id iv->L ' r..o' ) nv'T' ; oj:'/ rf.r.ro-- i , : i:/ a'l"' \:l t> ,' . r> 1 rr ■■Tsrwessr^- - 4 s- 48 bents having the 20-foot and 30-foot spans, while it was made six feet for the bents having other spans. The points for bents having the distance of six feet between the foot of the knee brace and the top of the column fall on a smooth curve for each column height. The points for bents having the 20-and 30-foot spans fall consider- ably below these curves. The broken lines are drawn through these points. Thus, the effect of placing the foot of the knee brace higher up on the column is to decrease the horizontal reactions. The effect of variations in span length on the horizontal reactions is shown by the smooth portion of the curves on Graph 2. All of these curves^ except the curve for the fixed column having a height of 16 feet, are nearly horizontal. This shows that vari- ations in span length have only a small effect on the horizontal reactions. The degree of fixity of the colmin bases has a large effect on the horizontal reactions. This is shown on Graphs 1 and 2, wher< the curves for the fixed columns are always above the curves for th( hinged columns. In general, the values of the horizontal reactions for any bent of the type analyzed, having fixed columns, are some- what more than twice the values of the reactions for the bent, when the columns are hinged. Refer to the seventh column of Table XII. The value of the horizontal reaction due to a vertical load of one pound per foot length of truss is 1.514 lbs., in the case of the bent having the 20-foot span and 10-foot column height. This value is about 7.5^ of the total vertical load on the truss. In the case of the bent having the 60-foot span and 26-foot column height the horizontal re- action is slightly greater than 1^ of the total vertical load on the truss. These two values are respectively the largest and 49 smallest values of this percentage, in the case of the hinged col- umns. In the case of the fixed columns (see the seventh column of Table XIV), the largest value of this percentage is 15.5^^, this value being for the bent having the 20-foot span and IC-foot column height. The smallest value of the percentage is for the bent hav- ing the 30- foot span and 26-foot column height and is about 2.5'J^. In at least the case of the shorter column heights, the horizontal reactions are sufficiently large to cause a »iaterial compressive stress in the knee braces and to affect materially the stresses in the members of the truss due to vertical loads. The stresses in the mem^bers of the truss due to vertical loads would be somewhat de- creased by taking into account the deformation of the truss. Resisting Moments . Graphs No. 3 and 4 show the reaction factors for resisting moments, due to a uniform vertical load of one pound per foot lengtl of truss, plotted with columm heights and span lengths respectively as abscissas. Graph 3 shows that the effect of stiff columns on the resisting moments is even greater than the effect on the hori- zontal reactions. This effect is shown by the sharper bends in the curves for moments between the points for the 26-foot and 31-foot column heights. The moments, in general, increase rapidly as the column height decreases. The position of the knee brace greatly affects the resisting moments, which effect is shown in Graph 4 by the portion of the curves drawn in broken lines. A shortening of the distance between the foot of the knee brace and the top of the column decreases the resisting moments. As the span leng-th in- creases the values of the resisting moments increase. ri!' * V . '' r ■ <-w« (y^H r"«, vVI -s:™ .■v” ■ i-'.j ,0 c iJ kI < > I . "'., '■« 0\ L '• * *\ t‘ i. V jt ^ y. !?■ ^ ft i. U >' . V li ■:j f 1, r t,.: -V • . ; i .0 , ■ ‘ ■! r" * ' , V . ^ t ^ ^ \ i - - i- J-.‘ ^■■' *6':^ y* 'V^o:. , . V r '*e ' -^’Z or' ■ . V/ i ;.i_i.'.^ ,. ^ . . i '.u'o . t! <( ■ov rf t I ' r o '.) • i' . vj; ; '^''^1... ’■ *■ - V k r „ ;. ..... ...^ .- l.^tL , • r£!s%.c .:.’• iiX ’i 1 I'rS-’j /■\:r tyior :v:v a-*.. ,..^. ..' *t i ■• V -s. 50 The horizontal reactions and moments are thus affected by- variations in span length, in column height, in size of column^ and in position of knee brace. Because of the effect of variations in so many characteristics of the bent on the reactions due to verti- cal loads, it was impossible to present empirical curves for ob- taining the reactions due to combined wind and vertical loads, as the writer had hoped to do. Vertical Rea c tions . Graphs 5 and 6 give the reaction factors for vertical re- actions, due to a concentrated load of one pound at panel point 4 of the lower chord, plotted with column heights and span lengths respectively as abscissas. These curves are approximately parallel and consistently spaced, showing that column height and span length are the only characteristics of the bent materially affecting the reaction factors due to vertical loads. The vertical reactions, themselves, are dependent only on the span length. The degree of fixity of the column bases has little effect on the vertical reac- tions. This fact is shov/n by the approximate coincidence of the curves for hinged and fixed columns on Graphs 5 and 6. 14. Errors in Comii'.on Assumiptions for Various Combinations of Wind and Vertical Loads . AssuiBption of Equal Horizontal Reactions . The ratio, 5^: , is the ratio of the horizontal reaction on H the leeward column to the tots.l horizontal component of the wind acting on the bent. Graphs 7 to 11 show the ratios — ^ due to va- H rious combinations of wind and vertical loads, plotted with column heights as abscissas. Graphs 12 to 16 show the same ratios plotted with span lengths as abscissas. Graphs 17 to 21 show these ratios I I g — _ _ ■ ■ ■ ■ . . — — 1| O- 51 plotted with the ratios ag abscissas. span length Hr The first set of graphs shows that the ratio — £1 is large/' H for the shorter columns having the 16-foot and 21- foot heights than for the longer columns having the 26-foot and 31-foot heights. Any vertical load tends to increase the value of the horizontal re- action on the leeward column over what the value of the reaction would be due to wind loads alone, and to decrease the value of the horizontal reaction on the windward column. The larger the value of the vertical load is^the larger will be the value of the ratio _H . In the case of the hinged columns, this ratio becomes nearly H equal to unity, in the case of the 16-foot column height, but never exceeds one (See Graph Ro.ll). In the case of the fixed columns, under the vertical loads of 60, 75, and 100 lbs. per square foot, the horizontal reaction on the windward coluiian for the short column heights changes direction. It acts to the right instead of to the left, and the ratio & becomes greater than unity. This change is H shown on Graphs 9, 10, and 11. Hr Graphs 17 to 21 show that the ratioj||i^ decrease as the ra- tios - Hpi - Sy increase. The ratio the case of the span length hinged columns, ranges from a minimum value of 0.36, when the ver- tical load is 30 lbs. per square foot and the ratio of^ is equal to 1.033, to a maximum value of 0.95, when the vertical load is 100 lbs. per square foot and the ratio of?L is equal to 0.500. In the case of the fixed columns, this ratio ranges from a minimum value of 0.33, when the vertical load is 30 lbs. per square foot and the ratio cfTL is equal to 1.033, to a maximum value of 1.62, when the vertical load is 100 lbs. per square foot and the ratio ofl^L is equal to 0.267. The values of 2 h are, in general, larger for the H f >£.tX A ;,iv. V *v r> ', • 9 '- ■V' .•*■/ t>ii ■\ . ' . ^ : V ,, ' 'J 'F’’;^;.'..' - : ! 'U t)i. '.. '■■ s.i:, f.- : .VI 'Cl ...: :>Ls- i f. . .''If-. ' •V • .V » , ■ •‘•Jtr V. y .. ■ ft' , I uti U t . iA .*■ vj r ' •. ■• i , ■ -n.' ■-.' ■' ■ -Ji :'A r • . .r' ' .......tltl . . r V* i*,l ; ■ . i?’’ 15 ' I , I *. r <> I r .^ . r ■ ' . -V -> ' a .1 I". ...■%.• J V I ■- I ,; ■ *v ‘r*. ‘>jy ; r ■ '.liYi,: ; $ wd-J ’^a!. a t; rf’.lox ■; ci&’i. ♦ * s^:” ' i, ‘ vvri':.: a .;• iaj^.vr- I r'-.- 0.1'.:.'; ..'H-C •V, ^ ‘ Ai. C . . d ma-’ ’ r‘* ‘‘ V r *►.4 M ^ •• Vi T. X «u*4 . . < . :0. ... . A . J ■. C* ff ^ ' •* ' . ”1 ■ • \ • .-- .v!. ■ . ,c a^, ' •I' .-•• ■ too? >-••. -ii*; .'•‘i.. ’ 3 ,, . ’• 1 - , :, .n a. .r ::4.: ."u- \.1‘ z\lQ.zi-iyyi r: W .' ' ■ -.srrz.'.r. 52 fixed columns than for the hinged columns, and, in the case of the short columns, the values of the ratio for the fixed columns are much larger. Due to the great variations in the values of the ratio ^ , it is impossible to give an average value of the ratio that will fit any great number of cases. In general, the values of the ra- tio are smaller than the commonly assumed value 0.5, in the case of the bents having longer columns and subject to the smaller vertical loads; in the case of the bents having shorter columns and subject to the larger vertical loads, the values of the ratio are consider- ably larger than the value 0.5. Points of Contraf lexure « The ratio y/d is the ratio of d. i s t an c e f r pm _ b a s e __ p f_ c _ o _ lumn distance from base of colirmn to point of contraf lexure to foot of knee brace Graphs 22 to 26 shov/ the ratios Y/d for both columns, due to various combinations of wind and vertical loads, plotted with the column heights as abscissas. The ratio Y.^d for the leev/ard column, in general, decreases slightly as the column height increases, due to all values of vertical load applied. There are a few cases, hovifever, where an increase occurs. Graphs 27 to 31 give the ratios Y/d for both columns, due to various combinations of vertical and v/ind loads, plotted with span lengths as abscissas. These graphs show that variations in span length produce no regular variations in the ratio V/d. The location of the point of contraf lexure is best studied from Graphs 32 to 36, which show the ratios Y/d for both col'amns , column height plotted with ratios l~engt h — abscissas. 53 The ratio for the leeward column does not vary much for all values of the ratio . This shows that the locatidi s^an xen^xn of the point of contraf lexure on the leeward column is practically independent of all characteristics of the bent. The lowest value of is 0.497 for the bent having the 50-foot span and 31- foot column height, with a vertical load of 100 pounds per square foot. The highest value is 0.592 for the bent having the 20-foot span and 10-foot column height, with a vertical load of 30 pounds per square foot. The average value of the ratio, when there is a vertical load of 30 pounds per square foot (See Graph 32), is 0.55. As the magnitude of the vertical load increases, the point of contra- flexure of the leeward column is slightly lowered. When there is a vertical load of 100 pounds per square foot, the value of the ratio becomes about 0.51. These values are very close to the corrmion- d ly assumed value of 0.50. In the case of the windward column, certain combinations of vertical loads and wind loads gave no point of contraf lexure and other combinations gave two points of contraf lexure (See Table XVII), In case there were t\?o points, the point that was nearer the average value was plotted. Graphs 32 to 36 show that there is a much wider variation in the values of the ratio y/d for the windward column I than for the leeward column. This variation becomes greater as the vertical load becomes larger. An average value of the ratio ^/d for the windward column is 0.46, due to all combinations of verti- cal loads and wind loads applied. 15. Conclusions. As a result of this investigation the following conclusions may be drawn: 8£w sx/Ijav it ni aott&lrBV 54 1. The effect of the deformation of the truss under vertical loads on the reactions is too large to he neglected. The values of the horizontal reactions and moments due to vertical loads are suf- ficiently large to materially affect the stresses in the knee braces and truss members. The additional compressive stresses in the knee braces, due to vertical loads, would necessitate the use of a larger section for the knee braces. The stresses in the mem- bers of the truss, due to vertical loads, are, in general, somewhat decreased, when the deformation of the truss is considered. 3, The assumption that the horizontal reactions are equal is largely in error. 3. The common assumption^ which places the point of contra- flexure midway between the base of the column and the foot of the knee brace, is not materially in error, provided, the columns are rigidly fixed at their bases. In general, it is slightly higher than this mid-point for the lee7,^ard column and slightly lower for the windward col'omn. 4, The resisting moments, computed with the assumption that the horizontal reactions are equal^would be materially in error, and they had best be determined by an analysis of the bent. The above statements apply only to bents having one quarter pitch Fink trusses. Bents with other types of trusses might now be profitably investigated. 55 VERTICAL DEFLECTIONS OF THE VARIOUS PANEL POINTS AND COL'IPUTATION OF THE REACTIONS AT THE BASE OF EACH COLUMN In the following results, t, total height of structure, = height of column + 1/4 of span length; d, with a subscript, in- dicates the vertical deflection of the panel point whose number ap- or the displacement at the base of the right-hand column pears as the subscrip"^’ H, V^ and M, with a subscript, but no ex- ponent, indicate the horizontal reaction, vertical reaction, and resisting moment , respectively , at the base of the column whose let- ter appears as the subscript, due to a uniform vertical load of one pound per foot length of truss. Thus Hj;^ is the horizontal reaction at the base of the right column due to the uniform ver- tical load. H, V, and M, v/ith both a subscript and an exponent, indicate the horizontal reaction, vertical reaction, and resist- ing moment^ respect ivel3r at the base of the column whose letter appears a s the subscript, due to a single vertical load of one pound placed at the oanel point whose number appears as the ex- ponent. Thus, Hj^"^ is the horizontal reaction at the base of the right column due to a load of one pound at panel point Numiber 4. d is the distance from base of column to foot of knee brace. Y is the distance from base of column to point of contra- flexure . Span 20' — - Column height 10' Total vertical load = 20 lbs. Total height of structure = t = 10 + 5 = 15 ft. I Class I Case A: d „ r= 6,241 Z> ^ "3 = 10 3^ 10 X 2.5 = 25 d = 0 615 0 : 615 X 5 = 3075 p. CO II 650 s: 650 X 5 = 3250 II O' 615 Q: 615 X 5 = 3075 10 L: 10 X 2.5 = 25 9^450 ^ = 9450 - = 1.514 lbs. L R 6241 «L - 1.514 = 0.1009 t t 15 d T = 755; H N t t 755 6241 0. 13C98 15 = 0.12098 lbs. 0.008065 56 Case 3: 10 Tr T 1 X 75 '^R ‘ 240 lb s . ~ = 0.3125 lbs. V L T 1 X 165 240 V 0.6875 lbs. t 10 - 15 “ 0.5125 15 0.6875 15 0.6667 • = 0.02083 • = 0.04583 Case A: d. = 5 3 =413 d„ =405 s Class II dg = 1972 j: 5x2.5= 12.5 o: 413 X 5 = 2065.0 s; 405 X 5 = 2025.0 57 dQ = 413 dL = 5 t t = 465 H. R H Q: 413 X 5 = 2065.0 L: 5x2.5= 12.5 6180.0 6180 1372 3.1339 = 3.1339 lbs. IF = 0.2089 H. T = H T R -iSi- = 0.23560 lbs. 1972 . = 0.01572 L- 15 Case B: dp = 20,374 = 10 lbs. d^= 6,170.3 y T ^ 6,170.3 _ R 20,374 =_k = 10 _ C.667 t t 15 0.3029 lbs. 0.3029 15 0.02019 = 1 - 0.3029 = 0.6971 lbs. L t Case C: _ 0.6371 15 = 0.04647 ^ 35.55 = 0.039 j: 0 . 039 X 2 . 5 = 0.10 <^0 = 3.12 o: 3.13 X 5 = 15.60 d s o CO • 03 II s : 3.80 X 5 = 14.00 ^0, = 2.42 0: 2.42 X 5 = 12.10 = 0.034 L: 0.024 X 2. 5= .06 41.86 *' 1 ' \ . lU'* • * < y * " 1% r t « 59 Case A: = 11,015 j: 12 X 2.5 = 30 0 : 800 X 5 = 4,000 s : 760 X 5 = 3,800 Q: 800 X 5 = 4,000 L: 12 X 2.5 = 30 11,860 H = H = 1.077 lbs. R L d = 905 T H = H ^ = 0.0821S lbs. L R Case B; =' 10 lbs, ^ 0.3125 lbs. \ " = 0.6875 lbs. Case A: 2 0 8 Q L Case 3: B 15.0 2375.0 2325.0 2375.0 8 X 2.5 = 15.0 Class II d_ = 3,215 6x2.5 475 X 5 465 X 5 475 X 5 ^ 2.2100 lbs. = 532 = 0.16548 lbs *^105.0 = 10 lbs. = 0.3045 lbs. dg = 25,494 d^ = 7,762.5 = 0.6355 lbs. Cass C; d +A ^ 39.82 K 4 60 j: 0.041 X 2.5 = 0.10 0 : 3.18 X 5 15.90 s : 2.72 X 5 13.60 Q,: 2.51 X 5 12.55 L: 0.027 X 2.5 = 0.07 42.22 M = M- = - 1 - ^ - ^ - — = 101.79 in. lbs. ^ L 39.82 = 3.58 = 8.631 in. lbs. = 6.711 in. lbs. Yp^ = Y, = .. lp , l « - 7 - g - := 46.06 in. ^ 2.21 Y = 52.16 in. Y, = 40.55 in. R L Span 20' Column Height 16' t = 21« Class I Cass A; = 27,387 15 X 2.5 LO • to II = 0.567 lbs. 0 : 1050 X 5 = 5,250.0 = 1,182 s: 990 X 5 = 4,950.0 T Q: 1050 X 5 = 5,250.0 = 0.04316 lbs. L: Case B: 15 X 2.5 = 37.5 15,525.0 = 10 lbs. = 0.3125 lbs. = 0.6875 lbs. f \ / H H I i II I I A t : • V M I \ 61 Class II A; S = 7,410 j: 8 X 2.5 = 20.0 It 0 : 595 X 5 = 2975.0 dm = 8 : 580 X 5 = 2900.0 T Q: 595 X 5 = 2975.0 H T L: 8 X 2.6 = 20.0 8,890.0 R = H = 0.09042 lbs. L B: d = 35,734 B V ^ R = = 10 lbs. d^ = 10,962.5 = 0.3068 lbs. V = 0.6932 lbs. L C: d^ = 78.133 2 ' 0.041 X 2.5 = 0.10 0 : 3.18 X 5 = 15.90 s: 2.82 X 5 = 14.10 Q: 2 . 62 X 5 — 13.10 L: 0.029 X 2.5 = 0.07 43.27 . M . 43^27„x 14£_ ^ , 0.75 in. lbs. 78.133 R "L dip s 3 . 56 T = 6.561 in. lbs. Y = = ■J ! 9. .7 L- = 66.47 in. R L 1.1S97 M/ = 5.195 in. lbs Ij Y^^ = 72.56 in. Yj^^ = 57.43 in. 62 Span 20’ Column Height 19' t = 24 feet Class I Case A J 0 S Q L = 71 , 18 X 2.5 = 45 1170 X 5 = 5850 1160 X 5 = 5800 1170 X 5 = 5850 18 X 2.5 = 45 17,590 419 = 0.246 lbs. = 1,380 H = H 1 r: 0.01932 lbs. R L Case B: = Vl = 10 lbs. V = 0.3125 lbs. = 0.6875 lbs. ^ L Case A: J: 9 X 2.5 0: 695 X 5 S: 665 X 5 Q. : 695 X 5 L: 9x3.5 Case B: Class II dg = 13,450 33.5 Hp^ = \ = 0.829 lbs. 3475.0 J 3325.0 T = 790 3475.0 H T = H T = 0.05345 lbs. 32.5 ^ ^ 10.320.0 d = 43,414 •D \ ^ \ lbs, d = 13,362.5 1 V 1 = 0.3078 lbs. V I" = 0.6922 lbs. R L • 5 . * w A !B 63 C: d K + It J: 0.042 X 2 . 5 = 0.11 0: 3.09 X 5 = 15.45 S: 2.65 X 5 = 13.25 Q: 2.60 X 5 = 13.00 L: 0.029 X 2. 5 = .07 41.88 IvL = M_ = X 00 00 1 — 1 180 _ ; 115.687 diji — 3 . 43 = 5.337 in. lbs. M T == 4.209 in. lbs. ^ L Y_ ^ V - 65.16 “ - 78. oO in. 0,829 = 84.11 ^ 66.34 in. S4 Span 30' Column Height 16' Total vertical load on truss = 30 lbs. t = 23.5 feet. Class I Case A: d_ = 45,914 3 J: 23 X 1.875 = 43.13 II Hj. = 1.062 lbs. 2: 1440 X 3.75 5,400.00 "^4 = 2,035 3: 1925 X 3.75 = 7,218.75 6; 2160 X 3.75 = 8,100.00 H 4 R = H ^ == 0.04432 lbs. L 7: 1925 X 3.75 = 7,218.75 ^1 = 1,650 10: 2160 X 3.75 s 8,100.00 = H. ^ = 0.03594 lbs. 11: 1925 X 3.75 = 7,218.75 R L 13: 1440 X 3.75 = 5,400.00 L: 23 X 1.875 J2; 43.13 48,742. 51 Case B: = 15 lbs. V 4 _ lx 113.5 R 360 0.3125 lbs. 1 X 247.5 360 1 X 58.35 360 1 X 305.75 0.6875 lbs. 0.15625 lbs. : 0.84375 lbs. 360 JlIV Case A: Class II d_ = 12,530 65 J: 12 X 1.875 = 22.50 II 2; 830 X 3.75 = 3,112.50 d4 = 3: 1085 X 3.75 = 4,068.75 6: 1230 X 3.75 = 4,612.50 7: 1100 X 3.75 = 4,125.00 10: 1230 X 3.75 = 4,612.50 H 1 11: 1085 X 3.75 = 4,068.75 R 13: 830 X 3.75 = 3,112.50 L: 13 X 1.875 = 22.50 27,757.50 i B: dg = 133,714 ,07662 lbs. R Case C: = 15 lbs. u L = 41,243.9 0.3084 lbs. V. 4 _ = 0.6916 lbs. d^ = 20,195.9 V ■*• = 0.1510 lbs. R = 0.8490 lbs. + = 130.325 J: 0.072 X 1.875 = 0.135 2: 4.60 X 3.75 = 17.250 3: 5. 75 X 3.75 = 21.563 6: 6.27 X 3,75 = 23.512 7: 5.10 X 3.75 - 19.125 10: 5.28 X 3.75 19.800 • • f — 1 rH 4.39 X 3.75 = 16.463 13: 3.08 X 3.75 = 11.550 L: 0.03? X 1.875 mZ 0.069 129.467 66 ^ 139 » 467 X 144 \ 130.225 = 143.15 in. lbs. ^4 = 6 >.09 <^1 = 5.33 1 in. Mr4 = 6.734 in. lbs. » II < CD 00 in Ibi II 5.258 in. lbs. M 1 = L 4.004 in a Y = . 143.16 = 64.62 in. R 2.2153 T II >* 73.37 in. 76. 93 in 57.29 in. V = 52. 26 in Span 30' -Col. Height 21' t = 28.5 ft. Class I Case A: d_ = 108^344 5 J 30 X 1.875 = 56.25 2 1840 X 3.75 = 6,900.00 R 3 2380 X 3.75 = 8,935.00 6 3670 X 3.75 = 10,012.50 7 2320 X 3.75 = 8,700.00 10 2670 X 3.75 = 10,012.50 R 11 2380 X 3.75 = 8,925.00 d, = 13 1840 X 3.75 = 6,900.00 1 L 30 X 1.875 = 56.25 = 60,487.50 R 4 _ Case B: V = V = 15 lbs. it L = 0.3125 lbs. = 0.6875 lbs. V_^ == 0.15625 lbs. R V_^ = 0.84375 lbs. L 5583 lbs. 0.02358 lbs. 0.01957 lbs. r » \ / Class II 67 Case A: dg = 28^372 J 18 X 1.875 =S 33.75 2 1070 X 3.75 = 4012.50 3 1370 X 3.75 5137.50 6 1500 X 3.75 s 5625.00 7 1300 X 3.75 4875.00 10 1500 X 3.75 5625.00 11 1370 X 3.75 =S 5137.50 13 1070 X 3.75 4012.50 L 18 X 1.875 33.75 34,492.50 = Ht = 1.2157 lbs. = 1420 = 0.05005 lbs. d^ = 1200 Hi = H 1 = 0.04230 lbs. R L Case B: (ig = 181,595 ^R = V = 15 lbs. L II 56,206.3 d^ = 27,677.; I V = 0.3095 lbs. = 0.1524 lbs. =* 0.6905 lbs. = 0.84 76 lbs. Case C: ^ +^4 = 241.020 J: 0.071 X 1.875 = 0.133 2: 4.45 X 3.75 = 16.688 3: 5.55 X 3.75 = 20.812 6: 6.08 X 3.75 = 22.800 7: 4.92 X 3.75 = 18.450 10: 5.00 X 3.75 = 18.750 11: 4.30 X 3.75 = 16.125 13: 3.05 X 3.75 = 11.438 L: 0.038 X 1.875 = 0.071 125.267 Mt = „ - 125.267 X 204 = 106.03 in. lbs. L 241.020 II 5.93 = 5.18 V - 5.019 in. lbs. ^ 4.384 in. lbs. 3.939 in. lbs. Mgl = 2.998 in. lbs. 1. R H - 1.2157 - = 87.22 in. Y 4 R = 100.28 in. = 103.64 in. Y 4 L == 78.70 in. == 70.87 L in. I V- • .i V '/ ' Y*' .A i i 68 Span 30' Column Height 26' t = 33.5 ft. Class I Case A: dg = 212,139 J 37 X 1.875 = 69.38 Hp = H^:= 0.3560 lbs. 2 2250 X 3.75 = 8,437.50 3 6 3000 X 3.75 = 11,250.00 3325 X 3.75 = 12,468.75 d^ = 3150 7 2950 X 3.75 = 11,062.50 10 3325 X 3.75 = 12,468.75 K^4 » N 4 ^ 0.01485 lbs. 11 3000 X 3.75 = 11,250.00 irt L 13 2250 X 3.75 = 8,437.50 d^ = 2560 L 37 X 1.875:= 69,38 1 75,513.76 = H ^ = 0.01207 lbs. R L Case B: = 15 lbs. = 0.3125 lbs. = 0.15625 lbs. V_^ = 0.6875 lbs. i) Jj = 0.84375 lbs. Class II Case A d^ = 54,557 J 19 X 1.875 = 35.63 = H, = 0.7852 lbs. R 2 1280 X 3.75 = 4800.00 3 1640 X 3.75 = 6150.00 d. = 1720 6 1830 X 3.75 = 6862.50 4 7 10 1590 X 3.75 = 5962.50 1830 X 3,75 = 6862.50 = 0.03153 lbs. 11 1640 X 3.75 = 6150.00 d^_ = 1460 13 1595 X 3.75 = 5981.25 L 19 X 1.875 = 35.63 = 0.02676 lbs. 42,840.01 R L Case B : dg = 229,477 = 15 lbs. d, ^ = 71,168.9 d^_ = 35,158.4 ^ 0.3101 lbs. = 0.1532 lbs. = 0.6899 lbs. h Jj = 0.8488 lbs. t f ♦ r '.T If ■i I . • X > y '■ . ^ i ’ /. w % ^ X CC X Ov- ■r, V< M /. I 1 I j. .jf L. ■ I t J / ^ t I I 4 ; ^ 'uX \ . '. . . I'f. .-i* /J 3 ;.. e-a^r. a . ri^ ' f .. : ., )i i . - -f l ■aae aEeCT ? . - Case C: + ■A 4 = 383.494 J: 0.070 X 1.875 = 0,131 2 : 4.30 X 3.75 = 16.125 3: 5.35 X 3.75 = 20.063 6 : 5.65 X 3.75 = 21.187 7: 4.40 X 3.75 = 16.500 10 : 4.58 X 3.75 •= 17.175 11 : 3.98 X 3.75 = 14.925 13: 2.75 X 3.75 = 10.313 L: 0.037 X 1.875 = 0.069 116.488 = M. = 264 X 116. = 79.16 in. lbs. R L 388.494 d4 = = 5.58 d^ = 4.82 V = 3.792 in. lbs. = 3.275 in. lbs = 2.928 in. lbs. IL^ 2.177 in. lbs = Yt = _Z£il 6 ^ 100.82 in. 0.7852 Y„^ = 130.27 in. Y ^ 122.38 in. R R Yj^^ = 92.86 in. Y^^^ = 81.35 in. Span 30’ — - Column Height 31' t = 38.5 ft. Class I Case A: d_ = 164,493 3 J: 44 X 1.875 82.50 II 0.5322 lbs. 2: 2645 X 3.75 = 9,918.75 Jj 3: 3480 X 3.75 =: 13,050,00 d^ = 3700 6: 3875 X 3.75 = 14,531.25 4 7: 3300 X 3.75 35; 12,375.00 tr 4 TT 4 = 0.02249 lbs 10: 3875 X 3.75 s 14,531.25 % = 11: 3480 X 3.75 = 13,050.00 dn — 3025 13: 2645 X 3.75 =£ • 9,918.75 ^1 L: 44 X 1.875 = 82.50 = H 1 = 0.01839 lbs 87,540.00 70 Case B: ^ = 15 lbs. Vr^ = 0.15625 lbs V = 0.3125 lbs. = 0.84375 lbs = 0.6875 lbs, Class II Case A: = 42,340 J: 18 X 1.875 = 33.75 Hr = % = 1. 2: 1370 X 3.75 == 5,137.50 3: 1800 X 3.75 = 6,750.00 d. = 1900 6: 2000 X 3.75 = 7,500.00 fr 7: 10: 1760 X 3.75 = 2000 X 3.75 = 6,600.00 7,500.00 Hr^ = = 11: 1800 X 3.75 = 6,750.00 = 1580 13: 1370 X 3.75 = 5,137.50 L: 18 X 1.875 = 33.75 _ = H 1 = R L 45,442.50 Case B: S = = 105,261 II = V_ =15 lbs. L p* II = 32,354.1 di = = 15,751.0 = 0.3074 lbs. V 1 R = 0.1496 lbs. = 0.6926 lbs. = 0.8504 lbs. Case C: ^4 "" 228.323 J: 0.047 X 1.875 r= 0.088 2: 3.57 X 3.75 sz. 13.388 3: 4.36 X 3.75 16.350 6: 4.65 X 3.75 =: 17.437 7: 3. 73 X 3.75 s 13.988 10: 3.77 X 3.75 sr 14.137 11: 3.20 X 3,75 S£ 12.000 13: 2.37 X 3.75 s= 8.888 L: 0.013 X 1.875 0.024 96.300 = g34_x._9J.._5p_Q ^ 140.03 dn. lbs. R L 222.822 0733 lbs. 0.04487 lbs. 0.03732 lbs. 71 = 4.51 = 4.06 = 6.558 in. lbs. H 4.722 in. lbs. V = M 1 r= L = 140.05 ^ 1.0733 \ = 146.16 in. Y/^ = 105.24 in. L 130.47 in. L 5.904 in. 3.510 in. 158.20 in 94.05 in lbs. lbs. 72 Case A: Span 40* Column Height 16* Total vertical load on truss = 40 lbs. t = 26 feet. Class I dg = 32,309 = 1.897 lbs. J: 25 X 2. 5 = 62,5 2: 1410 X 5 = 7,050.0 3: 1840 X 5 = 9,200.0 6: 2025 x 5 =10,125.0 7: 1800 X 5 = 9,000.0 10: 2025 X 5 =10,125.0 11:- 1840 X 5 = 9,300.0 13: 1410 X 5 = 7,050.0 L: 25x2.5= 62,5 61,875.0 d4 = 1970 = 0.06041 lbs. d^ = 1590 H ^ = H ^ = 0.04876 lbs. R L Case B: V. R V_ = = 20 lbs, K L = 0.3125 lbs. vj- = 1 X 75 ^ R Case A: 40 = l-X, 27.. 5 0.6875 lbs. 40 Class II dg = 9,445 = 0.15625 lbs. 480 V, 1 = 1 X 405 , ' 480 0.84375 lbs J: 15 X 2.5 =: 37.5 2: 910 X 5 4550.0 3: 1190 X 5 = 5950.0 6: 1320 X 5 s: 6600.0 7: 1150 X 5 ss 5750.0 10: 1320 X 5 = 6600.0 11: 1190 X 5 5950.0 13: 910 X 5 = 4550.0 L: 15 X 2.5 = 37.5 4C >,025.0 d4 = 1260 = 0.13340 lbs. = 1040 H 1 = H ^ = 0.11012 lbs. R L Case B: V. dg = 211,776 R = = 30 lbs. d^ = 65,530,5 d^ = 32,169.4 V. R = 0.3094 lbs. V. R = 0.1519 lbs. V, = 0.6906 lbs. V 1 = 0.8481 lbs. L 73 Case C: . + A _ 4 “ 101. 27 J; 0.098 X 2.5 = 0 .245 2: 5.55 X 5 = 27 .750 3: 6.95 X 5 = 34 .750 6: 7.48 X 5 = 37 .400 7: 6.20 X 5 31 .000 10: 6.28 X 5 = 31 .400 11: 5.50 X 5 = 27 .500 13: 3.78 ; X 5 = 18 .900 L: 0.056 X 2.5 = 0 .140 209 .085 \ - = M = L 209.085 X 101.27 120 = 247. .76 in. lbs. - = 7.35 = ( 5.24 V = 8.709 in. lbs. II 7.394 in, lbs. = 7.221 in. lbs. M^l = 5.306 in. lbs. = Y = = 58.47 in. ^ ^ 4.3377 Yj^^ = 65.28 in. Y_^ = 67.14 in. K Y^^ = 54.13 in. Y = 48.18 in. L Span 40* Column Height 21’ t = 31 feet. Class I Case A: d = 83,483 jD J 30 X 2.5 = 75 2 1790 X 5 8,350 3 2390 X 5 = 11,950 6 2640 X 5 = 13,200 7 2240 X 5 r= 11,200 10 2640 X 5 = 13,200 11 2390 X 5 = 11,950 13 1790 X 5 r: 8,950 L 30 X 2.5 =: 75 79,550 0.9523 Ihs. = 2520 = 0.03018 lbs. d^ = 2000 = 0.02396 lbs. r« 74 Case B: = 20 lbs. V_^ = 0.3125 lbs. K = 0.6S75 lbs. = 0.15625 lbs. V ^ = 0.84375 lbs. L Case A: Class II A = 22,346 D J: 19 X 2.5 S= 47.5 2; 1070 X 5 = 5350.0 3; 1400 X 5 7000.0 6: 1570 X 5 7850.0 7: 1380 X 5 =: 6900.0 10: 1570 X 5 7850.0 11: 1400 X 5 =: 7000.0 13: 1070 X 5 5350.0 L: 19 X 2.5 47.5 H R = = 0.06632 lbs. 47,395.0 = 1240 H ^ = Hi = 0.05549 lbs. K L Case B: Vj^ = = 20 lbs. d4 = 92,132.7 d = 296,899 B R = 0.3103 lbs. = 0.6897 lbs. L d^ *= 45,470.5 V_^ = 0.1532 lbs. K = 0.8468 lbs. Case C: dx^ +A^ = 198.99 J: 0.091 X 2.5 == 0.228 2: 5.05 X 5 = 25.250 3: 6.55 X 5 = 32.750 6: 7.00 X 5 = 35.000 7: 5.85 X 5 = 29.250 10: 6.12 X 5 = 30.600 11: 5.45 X 5 = 27.250 13: 4.21 X 5 = 21.050 L: 0.051 X 2.5 = 0.127 201.505 “r = M = 180 X 201.505 . L 198.99 <^4 = 3.85 1 = 5.78 I aagjjjjjJ or, . r I •>c •* ^ 4 - \ /. r I 1 •j ^ ■ ..I i ^ • / *> * * •. I V- * • • 4 . 7 . ': r 3 i' c i - A . 1 u ■ . ' ' ^ •'i UftJ A najiTi rjinr I ~ii''~ 75 II 6. 196 in. lbs, V = 5.228 in. lbs II 5. 140 in. lbs. = 3.764 in. lbs Y “T) 182.37 _ 85.94 in. n L 2.121 II 93 .43 in> V = 94.22 in. v = 77 .50 in. = 67.83 in. - 76 - Span 40' - Col. Height 26' t = 06 ft. Class I. Case A: cl = 172,255 B J; 30 X 2 1/2 75 2 : 2250 X 5 = 11,250 3: 2925 X 5 = 14,625 6 : 3350 X 5 = 16,250 7: 2825 X 5 = 14,125 10 : 3250 X 5 = 16,250 11 : 2925 X 5 = 14,625 13: 2250 X 5 sr 11,250 L: 30 X 3.1/3 zz 75 98,525 Case B; V R r4 V’ = 20 lbs. L = 0.3125 lbs. R Vr^ = 0.6875 lbs. H = H = 0.5720 lbs. R L d = 3,100 = 0.01600 lbs. d. = 3,575 i T-1 K = H = 0.014S5 lbs. R L V = 0.15625 lbs. R = 0.84375 lbs. L Class II. Case A d = 44,707 B JJ 23 X 2. 5 = 57.5 2 : 1320 X 5 =6,600.0 3 : 1750 X 5 = 8,750.0 6: 1910 X 5 = 9,550.0 7: 1700 X 5 = 8,500.0 10 : 1910 X 5 = 9,550.0 11 : 1750 X 5 = 8,750.0 13: 1320 X 5 = 6,600.0 L : 23 x2.5 = 57.5 58,415.0 = H, = 1 . 3066 lbs . R d, = = h; = R L d = 1 1 1 H = H = R L i,S20 0.04071 lbs. 3,500 0.03355 lbs. " ■ I .■( i‘‘ I > ' . ■ ■ -V r .1 -77 Case B: = V d V V d = 382,02^ B = 20 lbs. = 118,732.7 = 0.3108 lbs. = 0.6892 lbs. d^ = 58,770.5 = 0.1538 lbs. R L = 0.6462 lbs. Case C: + ^4 = 332.11 J: O.OSO X 2.5 = 0.23 2 4.90 X 5 = 24.50 3 6.22 X 5 = 31.10 r> D 6 . 60 X 5 = 33.00 7 5 . -10 X 5 = 27.00 10 5.56 X 5 27.75 11 4.85 X 5 24,25 13 3.45 X 5 = 17.25 L 0.052 X 2.5 = 0.13 185.21 R Ti/r-i M = M = 240 X 185, <=j 1 = 84 in Ih R . R L S32.il = 6.35 dj =5.62 = 4.589 in lbs. Mr = 4.061 in lbs. = 3.773 in. lbs. L = 2.885 in lbs Y = Y = 133.84 = R L T73U5^ 102.43 in. = 113.72 in. 1 Y R = 121.04 in. = 92.68 in. = 85,99 in. L Case Case Case Span 40' - Col. He t = 41 ft. Class I. A: d = 135 , 181 B J:- 40 X 2 1/2 100 2: 2600 X 5 = 15,900 3450 X 5 = 17,250 6: 3825 X 5 = 19,125 7: 3350 X 5 = 16 , 750 10: 3825 X 5 = 19,125 11: 3450 X 5 = 17,250 13: 2600 X 5 = 13,000 L: 40 X 2 1/3 = 100 115,700 B: V = V =20 lbs. R L = 0.3125 I'os. = 0.6875 lbs, L Class II. -78 ght 31' H = H = 0.8559 R L d = 3610 H^= = 0.02670 Ids. R L d;j_= 3000 = 0.02219 lbs. R L = 0. 15325 lbs. R = 0.84375 lbs. L A: d = 35,685 B 21 X 2.5 = 52.5 2: 1425 X 5 = 7,125.0 o : 1890 X 5 = 9,450.0 6: 2100 X 5 =10,500.0 7: 1890 X 5 = 9,450.0 10: 2100 X 5 =10,500.0 11: 1890 X 5 = 9,450.0 13: 1425 X 5 = 7,125.0 L: 21 X 2.5 = 52.5 63,705.0 H = H = 1.7852 lbs. R L d = 2010 4 = 0.05633 lbs. R L d^ ^ 1650 1 1 K = = 0.04624 lbs. - 79 - Case B: d = 175,594 B >' II > 20 lbs. R L d = 54,228.3 d = 26,518.3 4 1 R = 0 . 3088 lbs. = 0.1510 lbs '^R 1 = 0,6912 lbs. V L L = 0.8490 lbs Case C: d. + A = 196.29 K 4 J :*• 0 . 063 X 2 . 5 = 0.16 2: 3 . 98 X 5 = 19.90 3 : 5.00 X 5 =25.00 6: 5 . oO X 5 = 26..50 7: 4.27 X 5 =21.35 10: 4. 45 X 5 =22.25 11: 3.75 X 5 =18.75 13: 2 . X 5 = 12.75 L: 0.030 x2 « 5 = , C 8 r4S . TT' 1 = M = 146.74 X 300 = 224.27 in. lbs. R L 196 . 29 d 4 = 5 , 22 d^ = 4.52 = 7.978 in lbs. = 6.202 in. lbs. L 'aI = 6.908 in, lbs. R 1 M = 4.5S8 in. lbs. L Y = Y = 224.37 ^ ^ 1.7852 Y"" = 141.6b in. R 125. So in. 1 Y = 149.39 in. R Y^ = 110.10 in. L Y = 94.90 in. L -80- Span 50' - Col. Height 16 * Total Vertical Load on Truss = 50 lbs. t = 28.5 ft. Class I. Case A: = 29,654 J« • 20 X 3. 125 = 62.50 2 1170 X 6.25 = 7,312.50 3 15 30 X 6.25 = 9,562.50 6 1695 X 6.25 10, 593. 75 7 1455 X 6.25 = 9,093.75 10 1695 X 6.25 = 10,593.75 11 15 30 X 6.25 = 9,562.50 13 1170 X 6.25 = 7,312.50 L; 20 X 3. 125 = 62.50 64,156. 25 H R H I R H = 2.1635 lbs. L , = 1641 = 0.055 34 lbs. L d = 1344 = 0.045 32 lbs. L Case B: V = V =25 lbs. R L = 1 X 187.5 = 0.3125 lbs. ^ 600 = 1 X 93.75 = 0.15625 lbs = 1 X 506.25 = 0.84375 lbs ^ 600 Case A: JL - X A^S.5 ^ 0.6875 lbs. 600 Class II =8,369 H = H = 5.1694 lbs. 12 X 3. 125 =r 37.50 R L 2 760 X 6.25 4,750.00 3 1045 X 6.25 =s 6,531.25 d =1120 6 1130 X 6.25 = 7,062.50 4 7 1040 X 6.25 :s 6,500.00 10 1130 X 6.25 = 7,062.50 = 0.1338 3 OQ 1 — 1 11 1045 X 6.25 =s 6,531.25 R L 13 760 X 6.25 4,750.00 d = 880 L- 12 X 3. 125 — 37.50 1 ll 4 3, '262. 50 H = H = 0.10515 lbs R L - 31 - Case B: d = 327,902 B V V =25 lbs. R L d = 101,704.7 4 v^- = R 4 V = 0.3102 lbs. 0.6898 lbs. V, R 1 50,292.7 0.1534 lbs. 0.8466 lbs. Case C: d + ^ K = 97..47 J 0.078 X 3.125 = 0.24 2 5. 10 X 6.25 = 31.88 3 6. 60 X 6.25 = 41.25 6 7. 10 X 6.25 =* 44.37 7 6.03 X 6.25 = 37.69 10 6. 33 X 6.25 = 33.56 11 5.50 X 6.25 = 34.38 13 3.95 X 6.25 = 24.69 L 0.045 X 3. 125 = 0.14 254.20 M = M = 254. 20 X 120 = = 312.96 in. lbs. R 97,47 d = 6.90 = 5.90 4 8,495 in. lbs. 1 M = 7.264 in. lbs R R = 7.115 in. lbs. = 5.554 in. IbB L L Y = Y = 312.96 = 60.54 in. R L 5 . 1694 A. 1 Y = 63.48 in. Y = 63.08 in. R R =* 5 3. 16 in. Y^ = 52.82 in. L L -lB- ( * ' r'lrs ■ - '■ /• i ' ^ ■ '■ ' • A • *t, .• . /'. - , , ' X . > et’ 'V £ ■i fix- . - ,•■' . - c^ V , . . . ! . ^ X ■ - X ■| . ‘ . ♦ r • ' • - ^ ' » i V i' ' ' *• f ' ~ j •r •ar J I / , / T. ' i V \ -8S- Case Case B: d = 460,907 B V = V = 25 lbs. R L 143,269.8 0.3108 lbs. 0.6892 lbs. d V 1 1 ^ R v? = 71,075.2 0. 1542 lbs. 0.8458 lbs, C: d + ^ = 193.77 K 4 J- 0.079 X 3. 125 0.25 2 4.55 X 6.25 =s 28.44 3 5.80 X 6.25 36.25 6 6. 33 X 6.25 39.56 7 5.21 X 6.25 =: 32.56 10 5.40 X 6.25 s 33.75 11 4.70 X 6.25 s 29.38 13 3.40 X 6.25 =: 21.25 L 0.048 X 3. 125 0.15 221.59 ^ ^ ^ ^ 8 SI. 59 X 180 ^ 205.84 in. lbs. R L 193.77 d =6.12 4 "r = 5.685 in. lbs. M = 4.665 in. lbs. L di= 5-20 = -4.8 30 in. lbs. R = 3.600 in. lbs. L Y = Y = ,805.,.84 = 81.76 in. ^ ^ 2.5177 = 90.25 in. R = 74.06 in. L Y = 92. 30 in. R Yl = 68.79 in. I u '■± /: 1 .■s i —84— Case A: Span 50' - Col, Height 26* t = 38.5 ft. Class I. = 164,564 J 33 X 3. 125 =: 103, 13 H = H = 2 18 35 X 6.25 = 1], 468. 75 R L 3 2375 X 6.25 = 14,843.75 6 2660 X 6,25 =: 16,625.00 d - 7 2240 X 6. 25 = 14,000.00 4 10 2660 X 6.25 16, 625 . 00 "h = fL : 11 2375 X 6,25 — 14,843.75 13 18 35 X 6.25 - 11,468.75 L 33 X 3. 125 103. 13 100,081.26 ”r t = 0.6082 lbs. 2*475 0.01504 lbs. = 2,080 ^ 0.01264 lbs. Case B: V* = V = V R L 0.3125 lbs, 0.6875 lbs. = 25 lbs. V = R 1 0.15625 lbs. 0.84375 lbs. Class II Case A: d ~ 42,234 J: 19 X 3. 125 = 59.38 2: 1100 X 6.25 = 6,875.00 3: 1460 X 6.25 = 9,125.00 6: 1600 X 6.25 =10,000.00 7: 1400 X 6.25 = 8,750.00 10: 1600 X 6.25 =10,000.00 11: 1460 X 6.25 = 9,125.00 13: 1100 X 6.25 = 6,875.00 L: 19 X 3. 125 = 59.38 60,868.76 R L 1.4412 lbs. 1,556 0.03684 lbs. 1,276 0.03021 lbs. V ( I •' • ^ • r S r ’u- 1 . .11^ »r;. '..,i ^ O 'k ■ Case Case B: d =* 593,912 B Y =s V — 25 lb s . R L (i = 184,834S = 0.3112 lbs. R = 0.6888 lbs, L d-, = 91,857.8 1 = 0. 1547 lbs. R = 0.8453 lbs. L C: d + 4 = 325.81 K 4 J: 0.075‘x 3.125 = 0.23 2: 4.28 X 6.25 = 26.75 3: 5.45 X 6.25 = 34.06 6; 5.83 X 6.25 = 36.44 7: 4.75 X 6.25 = 29.69 5 10: 5 . 00 X 6.25 = 31.25 11: 4.40 X 6.25 = 27.50 13: 3.10 X 6.25 = 19.37 L: 0.045X 3.125 = 0.14 205 . 43 M =: M = 205.43 X 240 =: 151.33 in. lbs. R L 325.81 d = 5.68 d =4". 90 1 4 “r = 4.184 in. lbs. =s3. 609 R in. “l = 3.404 in. lbs. =2.679 Li in. =: Y T = 151.33 105.00 in. Li 1,4412 = 113.57 in. R y4 = 92.40 in. L 1 = 119.46 in. 1 Y = 88.68 in. L - 86 - Span 50' - Col. Height ol' t = 45.5 ft. Class I. Case A: = 124,285 J: bb X 3.125 = 10b. 13 = H = 0.9483 lbs. 2: 2175 X 6.25 = 13,593.75 R L 5 : 2800 X 6.25 = 17,500.00 d = 3,000 ol blOO X 6. 35 = 19,375.00 4 7: 2675 X 6.25 = 16,718.75 4 4 10: blOO X 6, 25 = 19,375.00 H = H = 0.02414 lbs 11 : 2800 X 6,25 = 17,500.00 R L 15 : 2175 X 6.25 = 13,593.75 d. = 2,450 L: bb X 3.125 = 103.13 117,862.51 1 = «£ = 0,01972 lbs Case 3: V = V =25 lbs. R L = 0.bl25 lbs. v' = 0. 15625 lbs. R R = 0.6675 lbs. J. V = 0. 84375 lbs. L Class II L Case A: d B = 32,569 J: 17 X b.125 = 53.13 H = H = 2.1094 lbs. 0 • 1236 X 6.25 = 7 ,725.00 R L b : 1640 X 6.25 =10,250.00 d = 1,755 r* - D : 1605 X 6 . 25 =11,281.25 4 7: 10: 1620 X 1805 X 6.25 6.25 =10,125.00 =11,281.25 h‘= = 0.0538 lbs. 11: 1640 X 6.25 =10,250.00 R L loi L: 1236 x~ 17 X 6,25 3.125 = 7,725,00 = 53.13 I H = d = 1,410 68,743.76 = 0.04327 lbs. R L J f., • t V- / -87- Case B: a B 271,377 = V = 25 lbs. R L d = 84,045.9 = 0.3097 lbs. R =: 0.6903 lbs. L Case C: d + ^ = 190.97 J 0.053 X 2 3.60 X 3 4.65 X 6 4.93 X 7 3.90 X 10 4.01 X 11 3.40 X 13 2.52 X L 0.02S X 3.125 = 0.17 6.25 = 22.50 6.25 = 29.06 6.25 = 30.81 6.25 = 24.38 6.25 = 25.06 6.25 = 21.25 6.25 = 15.75 3.125 ^ ,09 169.07 = 41,463.3 = 0. 1528 lbs. R 1 V, = 0.8472 lbs. Li M = M = 169.07 X 300 = 265.60 in. lbs. R L 190.97 d == 4.80 = 4.20 4 4 7.540 in. lbs. “r' = 6. 598 in. lbs. “r 5.860 in. lbs. L 4. 528 in. lbs. Y = 265.60 = 125.91 in. R L 2. 1094 II >-• 140.02 it^. II 15 2.48 in. R R y4 = 108.82 in. 1 104.65 in. L , ‘ - - • . *t--- I ’ - 88 - Span 60 ' - Col. Height 16 ' Total Vertical Load on Truss = 60 lbs. t = 31 ft. Class I. Case A: = 28,327 J 17 X 3.75 63.75 H = T 2.5383 lbs. 2 1090 X 7.5 s 8,175.00 K L 3 1420 X 7.5 3 10,650.00 A ^ 1, 485 6 1580 X 7.5 S 11,850.00 a 4 7 1390 X 7.5 = 10,425.00 10 1580 X 7.5 S 11,850.00 = 0.05242 lbs 11 1420 X 7.5 = 10,650.00 R L 13 1090 X 7.5 8, 175.00 L 17 X 3.75 63.75 1,244 71,902.50 1 H = = 0.04392 lbs R Case B: -R = V* = V = V = 30 lbs. R L 1 X 225 =r 0.3125 lbs. 720 1 X 495 = 0,6875 lbs. 720 Class II. V = 1 X 113.5 0.15625ltee. 0.843751bs. V R 1 = 1 720 X 607.5 720 e A? d 3 7,868 J 10 X B 3.75 37.5 2 745 X 7.5 S 5,587.5 3 990 X 7.5 = 7,425.0 6 1105 X 7.5 S 8,287.5 7 1010 X 7.5 7,575.0 10 1105 X 7.5 =; 8,287.5 11 990 X 7.5 — 7,245.0 13 745 X 7.5 = 5.587.5 L 10 X 3.75 37.5 50,070.0 fj =s H — 6. 3638 lbs. R L d = 1, 060 13472 lbs. = 844 H = H = R L ai0727 lbs Ak t I T ^ •• / J V ( '.-i - 89 ' Case C ase B: d = 439,954 B Y = V - ^0 lbs. R L d = 146,062.5 y* = 0.5108 lbs. R = 0.6892 lbs. L C: d + K 4 = 95. J: 0.072 X 5.75 0.27 2: 4.90 X 7.5 = 56.75 3: 6.55 X 7.5 49.13 6: 7.08 X 7.5 = 55.10 7: 6.10 X 7.5 45,75 10: 3.41 X 7.5 = 48.07 11: 5.45 X 7.5 =5 40.88 15: 5. 85 X 7.5 = 28.87 L: 0.046 x3L7 5 =: 0.17 302.99 Mn = M =-b02.99 X 130 = 380.72 R L 95 . 50 d =6,89 = 8.657 in. lbs. R = 7.455 in. lbs. L Y = Y = 580 . 72 ^ ^ 3.5658 = 64.26 in. R y"*” , = 55.17 in. L 1 1 Li L 58.85 in. = 72,425.1 = 0.1541 lbs. = 0.8459 lbs. in. lbs. = 5.69 = 7.150 in, lbs. = 5.602 in. lbs. = 66,65 in. = 52.22 in. - 90 - Case Case Case Span 60* - Col. Height 21* t s 36 ft. Class I. A: V 76, 182 J 22 X 3.75 = 82.5 H = H = 1.1919 lbs. 2 1360 X 7.5 = 10,200.0 R L 3 1815 X 7.5 = 13,612.5 1,885 6 2000 X 7.5 = 15,000.0 d = 4 7 1735 X 7.5 = 13,012.5 4 % = = 0.02474 lbs 10 2000 X 7.5 = 15,000.0 11 1315 X 7.5 = 13,612.5 13 1360 X 7.5 =: 10,200.0 di“ 1,570 L 22 X 3.75 = 82.5 1 i 0.02061 lbs 90,802.5 II B: V = V = 30 lbs. R L 1 0 o II > 5125 lbs. 0.15625 lbs. ‘ = 0.6875 lbs. L 1 0.84375 lbs. Class II, A: d 19,852 B J 13 X 3.75 = 48.75 H = H = 2,8610 lbs. 2 850 X 7.5 6,375.00 R L 3 1130 X 7.5 = 8,475.00 6 1250 X 7.5 9, 375.00 d “ 1,190 7 1100 X 7.5 = 8,250.00 4 10 1250 X 7.5 = 9, 375.00 0.05994 lbs 11 1130 X 7.5 = 8,475.00 H4 = 13 850 X 7.5 = 6,375.00 R L 980 L 13 X 3.75 = 48.75 56,797.50 1 = = 0.04937 lbs R L I I 't 1 i 'J'i Case B: d ~ 681,481 B V = V = bO lbs. R L d = 205,912.5 = O.ollb lbs. R = 0.6887 lbs. L Case C: d K + 4 rz 191.18 J: 0.038 X 3.75 = 0.26 2: 4.33 X 7.5 32.40 3: 5.76 X 7.5 = 4o . 35 6: S.15 X 7.5 46.12 7: 5.39 X 7.5 40.43 10: 5.70 X 7.5 = 42.75 11: 5.10 X 7.5 =: 38 . 35 13: 3.78 X 7.5 26.35 L: 0.043 X 3.75 = 0.16 272.07 - 91 - 102,b50.1 0.1547 lbs. 0.845O lbs. Y = M = 272.07 X 180 = 253.16 in. lbs. R L ' 151.18- d = S.OO d = 5.11 4 ^ “r = 5.649 in. lbs. = 4.811 in. lbs. L = 4.785 in. lbs. = 3.695 in. lbs. Y = Y =356.16 R L 2.861 = 89.54 in. "r = 94. 24 in. R = 97.45 in. T = 79.83 in. 1 Y = 74.84 in. L A ( ] I i: ) V » Span 60'. - Col. Keigiit 26’ t= 41 ft. Class I. Case A: = 131,137 B J: 28 X 0.75 = 105 II = 0.6809 lbs. 2: 1640 X 7.5 = 12,500 R L 5: 2160 X 7.5 = 13,200 d = 2,520 6: 2450 X 7.5 18,225 4 7: 2140 X 7.5 =: 16,050 10: 2450 X 7.5 18,225 K = 0.01440 lbs. 11: 2160 X 7.5 = 13 , 200 R L 15: 1640 X 7.5 12,500 = 1,920 L: 28 X 5.75 = 105 1 109,710 Hd = H = 0.01192 lbs, R T. Case B: V = V = 50 lbs. 4 R L V =0.5125 lbs. R 1 V = 0.15625 lbs. R V"" = 0.3875 lbs, L = 0.64575 lbs. L Class II. Case A: d B = 41,105 J: 16 X 5.75 60 H = H = 1.64o5 lbs. 2: 1000 X 7.5 = 7,500 R L ‘A • w • 6: 1540 1500 X X 7.5 7.5 = 10,050 11,250 d = 1,456 7: 1510 X 7.5 = 9,825 4 10: 11: 1500 1540 X X 7.5 7.5 = 11,250 1C,G6'0 = 0.05499 lbs R L -15: 1000 X 7.5 = 7,500 L: 16 X 5.75 = 60 a = 1,176 67,545 1 ' = 0.02861 lbs. R L * A ■( * ' _ iiola- V; “'J* * . - 94 - Span 60' - Col. Height 61' t = 46 ft. Cl as s I • Case A: d = 119,448 B J: 27 X 3,75 = 101.25 ^ • 1925 X 7.5 14,437.50 3 : 2550 X 7.5 = 19,125.00 6: 2850 X 7.5 = 21,375.00 7: 2525 X 7.5 =r 16,937.50 10: 2850 X 7.5 = 21,375.00 11: 2550 X 7.5 =: 18,125.00 13: 1925 X 7.5 = 14,437.50 L: 27 X 3.75 = 101.25 129,015. H = H = 1.0801 lbs. R L d = 2,750 = 0.02302 lbs R L d = 2,250 1 = 0.01884 lbs R L Case B: 4 V 4 L V R 0.3125 lbs. 0.6875 lbs. V = 30 lbs. Li y = 0. 15625 lbs. 'r = 0.84 375 lbs. L Class II. Case ffl: ^ B J: 15 X 2 : 1120 X 3: 1480 X 6: 1630 X 7: 1420 X 10: 1630 X 11: 1480 X 13: 1120 X L: 15 X = 31,150 3.75 = 56.25 7.5 = 8,400.00 7.5 =11,100.00 7.5 =12,225.00 7.5 =10,650.00 7.5 =12,225.00 7.5 =11,100.00 7.5 = 8,400.00 3.75 = 56.35 74,212.50 2.3824 lbs. 1,570 0. 05040 lbs 1 , c oO 0.04270 lbs I - 95 ' Case Case B: d : 388,565 B II > II ( > d = 120,631 .3 « I' 4 4 V = 0.3105 lbs . R 4 V = 0.6895 lbs. L C: d + II I-* OJ CD K 4 J: 0.046 X 3.75 = 0.17 O • • 3.35 X 7.5 =25.13 3: 4.34 X 7.5 = 32.55 6: 4.67 X 7.5 = 35.02 7: 3. 85 X 7.5 = 28.88 10: 4.09 X 7.5 = 30.67 11: 3.55 X 7.5 = 23.63 13: 2.65 X 7.5 = 19.87 L: 0.025 X 3.75 = .09 199.01 M = M = 199.01 X ;500 ^ ^ 188.27 lbs , d 1 R 1 V L = 317.11 d = 4.53 = 7.218 in. lbs. = 5.778 in. lbs. B M m: M = Y = 317.11 R L 2.3824 Y4 =: 143.21 in. R Y^ = 114.64 in. L> 133.11 in. 1 Y R 1 Y L 59,709.5 0.1537 lbs. 0.8463 lbs. in. lbs. ^ 3.94 6. 276 in. lbs. ^ 4.442 in. lbs. : 147.03 in. 104.03 in. / I 96 TABLE 1. DESIGN OE BENT SPAN LENGTH 40 PT, A L 1 Member * Section Area in sq.in. Length in in. A G J Same as column 1/4, 7.24 72.00 9.9448 J 1 2 Ls 2 1/4x2 1/4 x 2.12 75.00 35.3773 G 1 / •?2 is 2 3/4 X 2 3/4 X 1/4 2.12 103.97 49.0424 J 2 1/4x2 1/4 X 2^ 1/4 2.62 67.08 25.6031 1 1.06 33.64 31.6415 2 3 2.62 67.08 26.6031 1 3 2.12 75.00 35.3773 1 4 2.12 75.00 35.3773 3 4 2.12 67.08 31.6415 3 5 1.06 75.00 70.7547 4 5 1.06 75.00 70.7547 3 6 All members having the 2.62 6 7.08 25.6031 6 6 1.06 33.54 31.6415 6 7 same area have the same 2.62 67.08 25.6031 5 7 1.06 75.00 70.7547 4 8 section 2.12 180.00 84.9057 7 9 1.06 75.00 70.7547 7 10 2.62 67.08 26.6031 9 10 1.06 33.54 31.6415 10 11 2.62 67.08 25.6031 8 9 1.06 75.00 70.7547 9 11 1.06 76.00 70.7547 8 11 2.12 67.08 31.6415 8 12 2.12 75.00 35.3773 11 12 2.12 76.00 36.3773 11 13 2.62 67.08 26.6031 12 13 1.06 33.64 31.6415 13 L 2.62 67.08 25.6031 12 Z 2.12 103.97 49.0424 12 I 2.12 75.00 35.3773 K I 7.24 72.00 9.9448 For the 16,21, and 26-ft, coltiran heights the columns D J and B L are made of 1 PI. 8 x l/4 and ^ L3 3x2l/2x 1/4 with the longer leg outstanding. The moment of inertia of the column section about a vertical axis normal to the plane of the truss is 81.2 inches.^ The moment of inertia for the 31- foot column height is 222.0 in.^. ♦ See Pig. 1 M . I X . . .1 - * J- ... J. ‘.'.I. X.. TABLE 2 - COI^UTATION OP ^ A SPAU 40» - Col.Hgt. 21* Class I, Case A Class II, Case A Member * hL Ul Pi Ull Pg = s X P + Q .. . A. A s J -3.370 - 33.514 -0.8100 + 27.415 -0.9600 -0.9600 J 1 +4.230 +149.64 6 +0.6200 + 93.529 +0.9200 +1.9200 Or 1 +4.860 +238.346 +1.3862 +332.493 +1.3862 +1.3862 J 2 -7.496 -191.89 6 -1.8112 +349.248 -2.1466 -2.1466 2 3 -7.495 -191.895 -1.8112 +349.248 -2.1466 -2.1466 1 3 +4.220 +149.292 +1.2000 +180.793 +1.2000 +1.2000 1 4 +5.196 +183.786 +0.9000 +164.488 +1.2000 +2.2000 3 4 -1.875 -69.328 -0.5367 +31.859 -0.5367 -0.5367 4 6 +2.100 +148.586 +0.6000 +89.151 +0.6000 +0.6000 3 6 -3.745 -95.884 -0.7379 +70.187 -1.0733 -1.0733 6 7 -3.745 -95.884 -0.7379 +70. 187 -1.0733 -1.0733 5 7 +2.100 +148.686 +0.6000 +89.151 +0.6000 +0.6000 4 8 +3.100 +263.208 +0.3000 +76.857 +0.6000 +1.6000 7 9 +2.100 +148.686 +0.6000 +0.6000 7 10 -3.745 -96.884 -0.3364 +31.642 -1.0733 -1.0733 10 11 -5.475 —96.884 -0.3364 +31.642 -1.0733 -1.0733 8 9 +2.100 +148.585 +0.6000 +0.6000 8 11 -1.876 - 59.328 -0.5367 -0.5367 8 12 +6.195 +183.785 +0.3000 +53.665 +1.2000 +2.2000 11 12 +4. 220 +149.292 +1.2000 +1.2000 11 13 -7.49 5 -191.896 -0.3354 + 63.325 -2.1466 -2.1466 13 I -7.496 -191.895 -0.3354 +63.325 -2.1466 -2.1466 12 E +4.860 +238.346 +1.3862 +1.3862 12 L +4.230 +149.646 +0.3000 +43.697 +0.9200 +1.9200 K I -3.370 -33.514 -0.1500 +5.027 -0.9600 -0.9600 +2216.9^7 A,JJj 930.070 E inches _ 2216.927 ^ A E ^ E inches *See Pig. 1. 98 , TABLE 2 (continued) Class II, Case A Member* A A A I’e Pi t _ A SPU,i , A A Pg 1 A iiL. A A A A G J - 9.5470 - 9.5470 + 7.7331 + 7.7331 - 19.05 J 1 +32.5471 + 67.9244 +20.1792 +42.1131 +100.19 G 1 +67.9826 + 67.9826 +94.2375 +94.2375 +136.36 J 2 -64.9596 - 54.9596 +99.5428 +99.5428 -109.58 2 3 -54.9696 - 54.9596 +99.5428 +99.5428 -109.58 1 3 +42.4528 + 42.4528 +50.9434 +50.9434 +84.91 1 4 +42.4528 + 77.8301 +38.2075 +70.0471 +120.28 3 4 -16.9820 - 16.9820 + 9.1142 + 9.1142 -33.54 4 5 +42.4528 + 42.4528 +25.4717 + 25.4717 +84. 91 3 6 -27.4798 - 27.4798 +20.2773 +20.2773 -54.79 6 7 -27.4798 - 27.4798 +20.2773 +20.2773 -54.79 6 7 +42.4528 + 42.4528 +26.4717 +26.4717 +84.91 4 8 +50.9434 +135.8491 +15.2830 +40.7547 +186.79 7 9 +42.4528 + 42.4528 +84.91 7 10 -27.4798 - 27.4798 + 9.2167 + 9.2167 -54.79 10 11 -27.4798 - 27.4798 + 9.2167 + 9.2167 -54.79 8 9 +42.4528 +42. 4528 +84.91 8 11 -16.9820 - 16.9820 -33.54 .8 12 +42.4628 + 77.8301 +12.7358 +23.3490 +120.28 11 12 +42.4528 + 42.4528 +84.91 11 13 -64.969 6 -54 .9596 +18.4334 +18.4334 -109.68 13 L -54.9596 - 54.9596 +18.4334 +18.4334 -109.68 12 K 467.9826 +67. 9826 +135.36 12 L +32.5471 + 67 . 9 244 + 9.7641 +20.3773 +100.19 K L - 9.5470 - 9.5470 + 1.4321 + 1.4321 -19.05 +605.5137 +70^.9853 , „ 200.9432P + 427.3681 . , = g inches 605.5137? 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to • to lO CO cy> lo o c- c- rH in • lO o lO H- rH O CM C- C- CM O to CM CM n o CM CM IT- to o> CS CO CO t- O CO rH CO CO ooo r-I rH rH C- C- O in o at at o to to in to to £>0 IC • •••••• • • • . o o . • a sl<00 cc 00 in sdt o o CM sd< to 00 CO n 00 00 H rH rH a a 9 rH in CM CM CO CM to to to to • • sH CD CD lOH O rH rH O in rH sjt in rH rH in tO 'O rH CM CM l>-CM 00 CM rH rH rH + + + H" + o o 1 1 1 1 1 sdt + + + + + 1 1 1*> • rH lO • + I I + 00 O o o sdt O o sdt O o c- o at Sit o o rH rH O rH rH c~ o o c- o rH at o o to to o to to o o o c~ o <3> 1 — 1 o o CO CO o CO CO a a • in • a at • a CM CM o CM CM CM 00 to a s}t sit a sit CM a a a a • rH CO sit CM CM CM CM CM + 1 1 + 1 + + 1 1 H* + 1 + + oo o o o o • • w H • H- 00 ^ C3> 00 00 0» lO l>- <3^ CO 00 Oi Oi 00 00 (3^ lO Oi 00 00 CJ> lO 00 C- CO o O lO O O O O ^d^ ^ O O U3 O -<;ii O C- 00 t~O0^0>iOl>-v0l0C0C0U0Oii3t0C0L0t0l>-U3a>C3^^3l>00 rH • to lO lO rH O rH rH OJC>-(MCOW'^C^IO-^COttO'=d< tOCV2,HrHrH +rHH‘+rH H- I I H- H- 1 i I I H'id-H ^COcO'tl-'^Cv3WWtOHHCV2t0CO>5d<'4»lOtOI>I>00O'HrHC3^HrHHHHlWl-H»^ l>-0 rHrHCOCVjCMM Cb *-3 Cl> »T) 00 lO o to / I rH rH to PM UnI 03 ® rCj o P! •H O H lO c- • o 00 'st< t- rH t=> CO ^ Wil ♦H 0> CM R CM ra ft o o o M ft « • • +3 ft ft R ID VO lO 6u 60 + + + © •H © ^ •H S ft B • ft o o ra ^ ft • • rH 60 ra rH 60 • rQ O • R •H ft O •H jCi O ■p ft -H o 'Sjc o CM ra • • o o o CO • VO -H ft SC- R CM © *H ra R • • • © © © CO ID lO ft R ft R 1 1 I • • • R VO CO • R CQ O <3^ H CD ra O cr> CM vo ft O CO O VO ft O Hc to C- rH (M CO (M 1 — 1 lO VO R ft o o o ft ID vO rO ft • • • ft o o o © 60 O o o ra 60 • • • •H + + + p! •H O O o O ft o ft 'r + + •H •H -P ft O • O cd H VO 0> to cd • CM VO © O 0> rH CD © R sH CO ft O CO o vO ft O lO vO R • (M to W • o ID vO vO N • O o o (SI • O O O •H ft • • • •H ft « • • U o o o U R o o o o © 1 1 1 o © 1 1 1 ft R ft R • • 1 — 1 H o , — . o (M lO O o CM CO O • o VO CM O • CO O O © lO H O H ra ft ID R O ft ft rH CO CO M ft ft R CO ID rH ft • • • R 60 • • • ■*— ' 60 o o o ra ■ — • •H o o o •H + + + ra ft + + + ft g3 V» W R O o o •H •H ft ft • O O H cd • i od O © R © O CD «0 O ft O CD C- O ft CO l>- O O vo 0 > o • ft CD O • CD O ft R CD vO ID ft • CD vO ID p © • • • P ft • • • © R O o o © R o o o > •f + + > © + + + Ir 'd © © •H • •H iH rH O ( — 1 1 — ! • ft © ft R © o ft s:! 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r j I f r . ''I Q . r J5j:i s a r TABLE 7 104 . DESIGN OF BENT SPAN LENGTH 50 ft. Member^ Section A Area Sq. in 1 . ieneth in z. in. . A G J Same as columns. 7.24 72.000 9.9448 J 1 2 As 2 1/2x2 1/2x1 /4 2.38 93.750 39.3908 G 1 /»2 As 3 X 3 X 6/l6 2.38 118.208 49.6671 J 2V1 L 2 1/2 X 2 X 1/4 3.56 83.853 23.5541 1 2^ 1.06 41.926 39.5531 2 3 /f2 Ls 2 1/2 X 2 X 1/4 3.56 83.853 23.5541 1 3^ 2.12 93.760 44.2217 1 4 2.S8 93.750 39.3908 5 4 2.12 83.853 39.5531 3 5 All members having the 1.06 93.750 88.4434 4 5 1.06 93.750 88.4434 3 6 same area have the 3.56 83.853 23.5641 5 6 1.06 41.926 39.5531 6 7 same section 3.56 83.853 23.5541 6 7 1.06 93.760 88.4434 4 8 2.12 225.000 106.1321 7 9 7 10 9 10 10 11 8 9 9 11 Sjmimetrical with the other half of the Truss 8 11 8 12 11 12 11 13 12 13 13 L 12 K 12 L K L For the 16-ft., 21-ft,, and 26-ft. coliunn heights the colnrons D J and B L are made of 1 PI. 8 x l/4 and 4 /.s 3 x 2 l/2 x l/4 with the longer leg outstanding. The moment of inertia of the column section about a vertical axis normal to the plane of the truss is 81.2 inches^. Moment of inertia for 31-ft. column height is 222.0 inches^, *See Fig. 26. T-- ri 106 TABLE 8 DESIGN OP BENT SPAN LENGTH 60 PT. Member’*' Section A Area in su 1 .in. Length in. A G J Same as column 7,24 72.00 4.9448 J 1 2 is 2 3/4 X 2 3/4 X 1/4 2.62 112.50 42.9389 G 1 2.62 133.67 50.9809 J 2 2 is 3 1/2 x 3 x 6/16 3.86 100.62 26.0674 1 2 IL 2 1/4 x 2 1/4 X 1/4 1,06 50.31 47.4623 2 3 3.86 100.62 26.0674 1 3 2 ts 2 1/4 x 2 1/4 X 1/4 2,12 112,60 53.0660 1 4 2.62 112.60 42.9389 3 4 2.12 100.62 47.4623 3 5 1.06 112,50 106.1321 4 5 li. 2 1/2 x 2 1/2 X 1/4 1.19 112.50 94.5378 3 6 3,86 100.62 26.0674 5 6 1,06 50.31 47,4623 6 7 3.86 100.62 26.0674 5 7 1.19 112.50 94.5378 4 8 2.12 270.00 127.3585 7 9 All members having 1.19 112.60 94,5378 7 10 3.86 100.62 26.0674 9 10 the same area have 1.06 50.31 47.4623 10 11 3.86 100.62 26.0674 8 9 the same section 1.19 112.50 94.5378 9 11 1.06 112.50 106.1321 8 11 2.12 100.62 47.4623 8 12 2.62 112.50 42,9389 11 12 2.12 112.50 53.0660 11 13 3,86 100.62 26.0674 12 13 1.06 50.31 47.4623 13 L 3.86 100.62 26.0674 12 K 2.62 133.57 50.9809 12 L 2.62 112.50 42.9389 K L 7.24 72.00 9.9448 Por the 16-ft., 21-ft. and 26-ft. coltinm heights the columns D J and B L are made of 1 PI, 8 x l/4 and 4 is 3 x 2 l/2x l/4 with longer leg outstanding. The moment of inertia of the column section about a vertical axis normal to the plane of the truss is 81.2 inches^. Moment of inertia for 31- ft, column height is 222.0 inches^, * See Pig, 26, TABIiE 9 RBACTIOUS FOR UlTIFORM VERTICAL LO.U3 OF 1 LB* PER FOOT LENGTH OF TRUSS 106 M M 02 02 ra H O fH rH s p> h ^ & 02 \ 03 o m M CO 02 O I. cd cl) « w H r£3 ® ;i3 -P -H -P O ® «H Eh ifl O ,a p to , P4 ® * CQ »A CQ O W ^ CV2 00 CO lO ^ ci« 04 CO CM to CM 00 O ^ to 'M* l>- I> 04 00 CM rH ^ 5^ -- H •si* 04 04 to si* ■si* ■>;H -M* -st* CO 00 U5 ^ 00 0> H 00 C«- o CM O CO t>- CJ4 CM M2 CO O C- CO M2 04 O 00 o CM 00 H C- M2 00 C- H H M2 00 CO o 0> H 04 H iH M2 C- si* 00 rH si* 04 CM H rH 00 i> »o lO ca M2 CO CM to 04 M2 CO M2 O M2 CO M2 H 02 xii ^ iH C~ 00 <£> t- 00 C- H «x> lO ■sj< -sjl 00 M) 00 O to CM 04 M2 M2 to 04 M2 M2 00 M2 'M* lO 00 !> M2 M2 I>- M2 04 M2 l>- ^ t> 00 IC- M2 M2 00 W CM CM M2 CO CO M2 04 00 l>- M2 0> O H lO CD O O O to lO CO CM H O O .0943 .04267 .02344 .02788 .16299 .06842 .03629 .04354 .18138 .07516 .03743 .04849 .20528 .07947 .04008 .05179 C- 00 O £> U2 00 *> H tD lO 00 M2 00 O CO CM vjt 04 M2 M2 CO 04 M2 M2 00 M2 M2 00 C*- M2 M2 -sjl C“ M2 04 M2 C~ Nji ^ ^ 00 t>- M2 M2 00 to CM CM M2 CO CO M2 04 00 M2 O ■«;)< O CM O CO O O CM H H O O O .0462 .01959 .01063 .01382 .07296 .03074 .01589 .02088 .07591 .03096 .01680 .02180 .08168 .03311 .01661 .02348 lO iH ^ H H CM CM M2 M2 M2 *A • • • • CO 00 to 00 CM CM CO CO M) H M2 H CM CO to M2 M2 M2 M2 • • . . 00 to 00 to CM CO to rH M2 H M2 to CO si* -si* O o O O 0 0 0*0 U2 o 00 04 CO O O CO to o M2 CO M2 00 O O M2 O M2 O CM M2 C- M2 M2 !>• O O O O CM CM CM CM CO M2 M2 l> O CO c- M2 M2 CO rH CM CO lO H O CM O o> 1 — 1 rH rH fH M2 H M2 H H CM CM to M2 iH M2 H H CM CM to M2 H M2 H H CM CM CO M2 1 — 1 M2 1 — 1 H CM CM to O O o o CM CM CM CM O O o o CO to CO CO o o o o o o o o M2 M2 M2 M2 o o o o M2 M2 M2 M) 107 TABLE 10 REACTIONS FOR VERTICAL LOi-L OP 1 LB. AT POINT 1* Span Length L Column Height h Ratio h L Total Class I Height Case A Case B of Bent a/t =HR/t V^/t VR/t 30 16 .533 23.5 .001629 .03590 .00665 30 21 .700 28.6 .000687 .02961 .00648 30 26 .867 33.6 .000360 .02519 .00466 30 31 1.033 38.5 .000478 .02192 .00406 40 16 .400 26 .001875 .03245 .00601 40 21 .525 31 .000773 .02722 .00504 40 26 .650 36 .000415 .02344 .00434 40 31 .775 41 .000541 .02058 .00381 60 16 .320 28.6 .001590 .02961 .00548 60 21 .420 33.5 .000655 .02519 .00466 60 26 .520 38.5 .000328 .02192 .00406 60 31 .620 43.5 .000453 .01940 .00359 60 16 .267 31 .001417 .02722 .00504 60 21 .350 36 .000573 .02344 .00434 60 26 .433 41 .000291 .02058 .00381 60 31 .517 46 .000410 .01834 .00340 *See Pigs 1,23, £5, and 26. ) !> TABLE 10 (continued) 108. Span Length L Column Height h Class II Case A Case B Case C 2L/tf=KR/t Vl/t VR/t %/t %/t 30 16 .003260 .03613 .00643 .1704 .2508 .363 .534 30 21 .001484 .02974 .00535 .1052 .1538 .347 .508 30 26 .000799 .02628 .00457 .0660 .0978 .308 .464 30 31 .000969 .02209 .00389 .0912 .1534 .290 .488 40 16 .004235 .03262 .00684 .2041 .2844 .402 .560 40 21 .001790 .02732 .00494 .1214 .1686 .377 .623 40 26 .000932 .02351 .00427 .0801 .1128 .358 .504 40 31 .001128 .02071 .00368 .1070 .1686 .316 .498 50 16 .003690 .02971 .00538 .1949 .2549 .440 .576 50 21 .001562 .02525 .00460 .1075 .1442 .382 .513 50 26 .000785 .02196 .00402 .0696 .0937 .370 .498 50 3lL .000995 .01948 .00351 .1041 .1517 .349 .508 60 16 .003460 .02729 .00497 .1807 .2306 .435 .555 60 21 .001371 .02348 .00430 .1026 .1336 .416 .541 60 26 .000698 .02061 .00378 .0638 .0840 .381 .501 60 31 .000928 .01840 .00334 .0966 .1365 .347 .490 TABIS 11 109 RSACTIOlfS ITOR VERTICAL LOAD OP 1 LB* AT POIRT 4 * Span Length L Column Ratio Height h h L Total Class I Height of Bent t Case A %/t= %/t Case 8. 20 ^ r ? 10 .500 15 .008065 .04583 .02083 20 12 .600 17 .004833 .04044 .01838 20 16 .800 21 .002 055 .03274 .01488 20 19 .950 24 .000805 .02865 .01302 30 16 .533 23.5 .001886 .02926 .01330 30 21 .700 28.5 .000827 .02412 .01096 30 26 .867 33.5 .000443 .02052 .00933 30 31 1.033 38.5 .000584 .01786 .00812 40 16 .400 26 .002323 .02644 .01202 40 21 .525 31 .000974 .02218 .01008 40v 26 .650 36 .000600 .01910 .00868 40 31 .775 41 .000651 .01677 .00762 50 16 .320 28.5 .001942 .02412 .01096 50 21 .420 33.5 .000782 .02052 .00933 50 26 .520 38.5 .000391 .01786 .00812 50 31 .620 43.5 .000555 .01580 .00718 60 16 .267 31 ^001691 .02218 .01008 60 21 .350 36 .000687 .01910 .00868 60 26 .433 41 .000351 .01677 .00762 60 31 .517 46 .000500 .01495 .00679 *See Pigs.l, 29 , 25 , and 26. 4 corresponds to Point T of 20-ft. span. See Pig. 22. ' '/ 1 1 •’ \\" . V 0 V V.' ■i 11 110 . TABLE 11 (continued) Class II Span Column length Height L h Case A Case B Case C ®L/t=%/t ^ L/t ^R/t %/t %/t ^L/d ^R/d 20 10 .015720 .04647 .02019 .5055 .6591 .447 .582 20 12 .009734 .04091 .01791 .3948 .5077 .422 . 54«3 20 16 .004306 .03301 .01461 .2473 ,3124 .399 .504 20 19 .002644 .02884 .01283 .1754 .2224 .369 .467 30 16 . 003906 .02943 ,01312 .2237 .2866 .398 .510 30 21 .001756 .02423 .01086 .1382 .1761 .386 .492 30 26 .000941 .02059 .009257 .0874 .1132 .352 .456 30 31 .001165 .01799 .007984 .1226 .1703 .325 .451 40 16 .005131 ,02656 .01190 .2777 .3350 .451 .544 40 21 .002139 .02225 .01001 .1658 .1999 ,431 .519 40 26 .001131 .01914 .00863 .1048 .1275 .386 .470 40 31 .001374 .01686 .00753 .1513 .1946 .367 .472 50 16 .004696 .02420 .01088 .2496 .2981 .443 ,52y 60 21 .001880 .02057 .00928 .1393 .1697 .411 .501 50 26 .000957 .01789 .00808 .0884 .1087 .385 .473 50 31 .001238 .01587 .00712 .1347 .1733 .363 .467 60 16 .004346 .02223 .01003 ,2398 .2793 .460 . 53t> 60 21 .001665 .01913 .00865 .1329 .1569 .444 .624 60 26 .000852 .01679 .00760 .0805 .0963 ,394 .471 60 31 .001096 .01499 .00675 .1256 .1569 .382 .477 / 1 112 IT5 CD CO o o CD o H CO c- o o> CO <3» cb CO CO O lO CO lO CO o o 1ft O) J5- Sii cr> CO lO o> CO CO ft; O l>- CO CO '4' 'Slj CO CO LO a CO 00 a Lft lft to 00 C3> o> c- a CO co CO CO CO CO CO Ift lft lft Ift -i- + + + + + + + + + + + + + + + + + + + M/ FJ m rd pj •H © o • 1 +» •H ft a o c- CO 00 o CO o CO o o> ■cil CO CO t- CO o ^ o a CO (7> CO lO c- o CO Oi CD 1 — 1 Gt lft 00 o »d S H Oi CO a O lO CD I>- to t- CO 'Cjl lft t> CO CO a 00 ' — • o3 • o © H 1 — 1 to to uo co CO 'Clc Ift CO CO •cil lft CO CO ^ CO CO CO O CT* o « a + + + + + + + + -»■ + + + -t- + + + + + + + H H 02 © a > © 10 C5> o CO IJ~ IC- O CO m •H ft © »o CO a CO CO CO 1 — 1 CJ» o CO Ift a •Cii to c- CO OO E- to CO a CO lO CO lO o CO c- o to a £> CO lft CO o a E- Eh • •H H o» to CO a 00 Ift a o to 03 o to o CO o CO Ift a ->fl to c~ CO 00 c- to CO © O »d 'd •H CO CO a CO CO CO a CJ^ Ift a t>> Oi to Ift CO o a E- n cd 05 ft a CO Gi lO CO lO o CO c- a 00 Ift c- a CJ> lft 00 CO a CO Oi © o o w a 0> lO to '=^ 1 1 1 1 1 1 1 t 1 1 1 1 Ra © 1 1 1 1 1 1 1 1 • lO H CO 1 — 1 lO c- o> CO Ift o -cji to 00 a o lft ra CO a O a? 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IP lO IP IP IP IP IP IP ip IP IP IP IP IP IP IP rH CVJH C- C- O' O 00 tO CM to IP O' iH to to CM -til s 0H HCVJ l> iO IP !>• lO O It- to to IP • Ip • • • • H 1> Oi e- o to c- to O' CM IP to CM IP ^ii IP o rH H H iH iH O' H H tO O' CM ^ O rH rH 5;i ^ O' O' 00 CM IP IP +> M O»0 0> lO C- l>- CO to 'si* to to O fr- O' o H CM o o i 0> GO CM to tO H to O' IP CM O' to O' O' til to IP IP W ^to CVI C\J to to 'ti* to tit IP t- IP o^ii IP O' to Ip to + + + + + + + + + + + + + + + + H + + -t- c-to o w O' ^ o c» CM IP C- IP H IP to IP O' H CM CM lOQO CV2 00 IP It- C- to I> to to til to CM 00 to c- O' H vO to to 00 oo o IP CM IP to O' ^ to IP t> til IP IP 00 OO CM to IC) lO O H CM O' to IP O' O IP to O' a ooo> to I>- CM O' O' O to 00 H H to CM H O H 00 o HH CM CM to to tj* >tlt IP 00 IP iro to O' to tO to O + + + + + + HI- + -♦- + + + + £ H- + + + ■^H o i-t + rH CM 00 00 lO o o H l> t> til to IP to CM H OHO' to H to to tl< IP o O' c- c- o iH !>• C>» sji GO O to H lO ,jl ^ IP CM to O' til til to ^ O ^il 'til • • • • • • • • • ♦ • • • • • • • • • f-t rH * o • • • Oo O 00 O CM CM to O 00 to <;il to CM 00 to O to ''JI 1-3 CM • sf 00 • to IP C- • O CM CM «>- O' O O' til tO • Pi • • Pt • • • Pt • • • • • • • • O IP O IP CO rH 00 • o» 00 O' *>- IP o i>- to to 00 'tji H • • 00 • • to o lO c- O CO H ■s}« O O' H to O to H H to H O lO ^ H H iZi H H iH rH H H 'til 00 H CQ rH rH p tOH *> lO O' O' CM 00 00 O O' tl< CMh C- C- til CM O' o rH 1-3 00^ CM «0 O' O' -Mt CM (S' O' O' O H CM O' 00 to to ^ IP o CMO 0> 0> H CM 00 00 to IP to tP H CM to tO H o IH CM to CM ti' to to IP to IP to H to IP to + + -» + + + + + + + + • + + + • + + + lOO U3 ^ 0> 00 H O' O CM O' H to H O' til to H O' H O OOO to «o tp to to l> to IP tP O' 00 O' til O' to CM CM O CM CM £> o> CM CM 00 LP O O' H -tit H H i> to O' *>■ S> O' CT>tO O' -d* H to O' H H ^ tO tiT^iii IP* to CM*^ 'til H IP CMCM CM to O CO H • O' t- C- H CM H CM IP C- CM td< to CM til to H ''ii IP 1 + + + + + + + + + H- 1+ 1 + + + 1 + + + OO o o to O l>- to o IP o IP o o o o £> O to l>- o oo O lO to o to to o CM IP CM CM CM CM to Ip to H •H iQtO 00 O' IP 00 o IP to I> to ^ Ip to CM to -til IP -P" • ♦ • • • • • • • • • • # • • • • ♦ • # c3 jd OO o o O O o H O o o o o o o o o o o o OCM M3 O' to H to H to H tO H to H to H to H to H Hi— 1 H H H CM CM to H CM CM to H CM CM to H CM CM to OO O O O O O O OOO O O O O O O O O CMCM CM CM to to to to O tjt -tji til IP IP IP IP to to to to 'No point of oontrafle(Xure TAfiTiK 17 ( contlimed) VSRTICilL LOAD 76 LBS PER SQ. 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