Secondary Wind Moments in a Five-Bay Ten Story Building Bent Citation Vanoni, Vito August (1932) Secondary Wind Moments in a Five-Bay Ten Story Building Bent. Master's thesis, California Institute of Technology. doi:10.7907/dssp-ea18. https://resolver.caltech.edu/CaltechTHESIS:09232024-200641053 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Civil Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Civil Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1932 Record Number: CaltechTHESIS:09232024-200641053 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09232024-200641053 DOI: 10.7907/dssp-ea18 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16753 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 24 Sep 2024 21:50 Last Modified: 24 Sep 2024 21:56 Thesis Files PDF (Thesis) - Final Version See Usage Policy. 33MB PDF (Plate 1. Solution for wind moments) - Final Version See Usage Policy. 2MB PDF (Plate 2. Computations of bent moments Hardy Cross method. Solution for wind moments) - Final Version See Usage Policy. 3MB PDF (Plate 3. Computations of bent moments Hardy Cross method. Solution for first secondary moments) - Final Version See Usage Policy. 4MB PDF (Plate 4. Computations of bent moments Hardy Cross method. Solution for first secondary moments) - Final Version See Usage Policy. 4MB Repository Staff Only: item control page

SECONDARY WIND MOMENTS IN A FIVE-BAY TEN STORY BUILDING BENT Thesis by Vito A. Vanoni In Partial .Fulfillment of the Requirements For the Degree of Master of Science California Institute of Technology Pasadena, California 1932 TABLE OF CONTEJ:iTTS Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Th.e Bent . . . . . . . . . . • • . . . . . . . . . • . • • . • • . . • . . . . • • . . • . . . . . . . . . • • 1 Assumptions . . . . . . . . . • . . . . • . . . • • . . . • . . • . . . . . . . . . . . . . • . • . . • . . 1 Loads . . . • . . . . . . • • • . • . . . . . . . . . . . . . . . . . . . . . . . . • . . . . • . . . . . . • . . 2 Procedure ........................... Calculation of Secondary Wind Moments ................. Figure l, showing column deformations ................. Table I., Typical Calculation of Fixed-End Moments .... Checking Solution .................................... . Calculation of Wind Moments - 3 3 4 ? 8 Discussion Foundations .................................. • . . . . . . . . 13 .......... 13 Secondary Wind Stresses in Wilson-Maney Bent Con cl us ion .................. , ................................. l;i. Diagram of Bent (Diagram l.)••••••••••••••••••••••••·••••••• 16 ... •......................... . l? Secondary Moments and Corrections (Diagram 3.) ............. 18-19 Ratios of Secondary Moments to Wind Moments (Diagram 4.) ... Wind Load Morr.ents (Diagram 2.) 20 Wind Moments, Secondary Moments, and Ratio (Diagram 5.) Discussion Submitted to "Proceedings" of .Am.Soc.C.E. ....... 21-22 23 . 1 .PROBLEM The purpose of this thesis is to determine the effect of secondary wind moments due to column shortening under wind load upon a five bay ten-story structural steel building bent. THE BENT The bent used was obtained by adding two twenty-two foot bays to the upper tens stories of the classical Wilson-Maney bent, analyzed in Bulletin 80 of the University of Illinois. The dimensions and sections are the same as for the Wilson-Maney bent except that the first story height has been increased to 16'-0". properties of the bent. Diagram. 1 shows the dimensions and This same structure was investigated for prim- ary wind stresses only by Rolland A. Philleo as a thesis problem for the degree of Master of Science in Civil Engineering at the Graduate School of Cornell University. ASSUMPTIONS It was assumed that the joints in the bent were perfectly rigid and that no relative rotation of the members occurred at their intersections. The foundations were assumed rigid against settlement but were allowed to rotate by connecting t hem with a beam. These two assumptions are not cons is tent , of course, but it was thought that allowing the footings to rotate would be a better assump- 2 tion than the ordinary one of assuming them as completely fixed. LO.ADS The load used was the srune one used in Bulletin 80 of the University of Illinois, i.e., a horizontal wind load of thirty pounds per square foot on a vertical surface one foot wide. 3 PROCEDURE Calculation of Wind Moments The wind moments were first obtained by the Hardy Cross method by the ordinary procedure. obtain the moments. Seven complete cycles were carried through to The wind moments are shown on diagram No. 2. Calculation of Secondary Wind Moments The change in length of the columns due to wind loads, the resulting differences in elevations of the ends of the beams and the fixedend beam moments were then determined. These fixed end moments were then distributed by the Hardy Cross method and the secondary wind moments found. Six complete cycles were carried through with the Hardy Cross method to determine the moments. In the upper stories of the bent the secondary moments were quite large and would also produce colu..ron loads and deformations and more secondary moments or corrections to the secondary moments which, in general, would be opposite to the secondary moments. These corrections to the secondaries would produce a second correction which, in general, would add to the secondary moments. What actually happens when the wind load is applied to the bent is that the columns deform and the stresses arrange themselves in the structure producing equilibrium. In the solution for wind moments the columns are not allowed to defonn and the column loads obtained from these moments are too large since it can be seen that allowing the columns to deform tends to ~educe these loads. 4 (-281) (t-227) -f(;.3 -f-37'1 Col.C. Co/8. Column Deformations and Moments Due to Wind Moments (a) (-112j {t-195) -12.5 r/9/ ¢ (-7-,) /fbtJ/. -125 --- (li) !I) -..L t- 'i.. ........ C.:7//l (b) \9 N ½ Co/.B. Co/C. Coltnnn Deformations and Moments Due to Secondary Moments (r39.T) /too/ .-t-....G::::7.::;;5':::::--. -;::: (t-1(,/) (r-14-8) - 187 f2S7 <j, I {t-6-,o) /-13{.) -f{,7.~ -137 Ill) ..!.. Col.B: (c) Column Defor:rmtions and Moments Due to .First Correction to Secondary Moments. Go!C. --Col/l (d) Column Deformations and Moments Due to Second Corrections to Secondary Moments. Figure 1. 5 The diagrams in Figure 1 show the column deformations, fixed end moments, and final distributed moments at the roof due to wind moments, secondary moments, and first, and second corrections to the secondary moments. The figures in brackets are the distributed moments and the other figures are the fi xed end moments. The moments are in foot-pounds and positive moment tends to rotate the joint in a clockwise direction. The wind load is applied at the left. The diagrams in Figure 1 are for the roof only but the floors below act in a similar way, the actual deflections of course being smaller. It can be seen from Figure 1 that one correction to the secondary moments will give secondary moments that are too small. When the second correction is applied the resulting corrected secondary moments (algebraic sum of secondary moments and two corrections) will be too large and t .he final wind moments (algebraic sum of wind moments and corrected secondary moments) will in general be too small. The magnitude of the error, however, will be small, as can be seen from the fixed end moments due to the second correction to the secondary moments. In the solution only two corrections to the secondary moments were solved for. The steps gone through in obtaining the secondary moments are a series of approximations identically analogous to those gone through in 6 applying the Hardy Cross method. Assuming that the columns do not deform, corresponds to fixing the joints against rotation and the calculation of each set of moments corresponds to distribution . .Allow- ing the columns to deform is analogous to balancing and the calculation of fixed-end moments gives the carry-over effect. Diagram No. 3 shows the first secondary wind moments,the first and second corrections to the secondary moments, and the corrected secondary moments, (algebraic sum of first secondary moments and corrections.) Diagram No. 4 gives the ratios of the first secondary moments to the wind moments and the r atios of the corrected secondary moment to the wind moments. The first ratio is given near the member and the latter is given over the first one. Positive ratios indicate that the sec ondary moments add to the wind moments and vice versa. Diagram No. 5 shows the wind moments, corrected secondary moments, and the ratio of the former to the latter. 7 Span CC Span BC Span AB L = 22'-0 11 I/L = 7.7 L = 18'-0" I/L = 9.4 L = 22'-0" I/I = 7.7 Fixed-End Mr, + MR .£ E Fixed-End Fixed-End M_r. + MR ..9:. E ML+ MR d h E :~ :Moment Floor h Moment Moment h L Ft. x lbs. ltis. Ft.x lbs. its. lbs. Ft. xlbs. 171.5 Roof 27.1 360 22.3 77.6 163 32.6 121.0 379 77.2 162 79.1 377 61.2 358 53.8 120.4 10 170.7 168.1 355 88.2 76.0 160 130.5 118.6 371 9 101.l 154 179.8 114.8 141.5 162.6 341 123.2 73.5 359 8 322 159.5 69.1 145 234 .8 108.2 338 7 180.9 153.0 192.3 132 98.8 308 6 139.4 293 62.9 282.8 219~0 228.2 113 266 252 54.0 336.2 85.2 5 254.8 120.0 224 45.2 4 99 .7 210 265.3 95 390.l 71.6 293.7 71 446.8 168 154 297.1 33.8 53.8 3 328.5 73.5 21.1 44 527.9 33.6 105 94 359.5 2 45.9 302.6 Table - 1. (a) Calculation of Fixed End Moments Due to Wind Load Column Deformations Load Story lbs. 10-R - 27.1 9-10 - 88.3 8-9 -189.4 7-8 -330.9 6-7 -511.8 5-6 -730.8 4-5 -985.6 3-4 -1279.0 2-3 -1607.8 1-2 -2004.1 p Area Sq. in. A - 38.86 " " " 42.26 " 53.79 " 64.24 ti Column c. Column B. Column A. lbs. in.2 0.70 2.27 4.88 8.52 12.10 17.3 18.3 23.7 25.0 31.2 Load lbs. + 4.8 +12.2 +25.1 +43.4 +64.8 +91.5 +118.1 +146.5 +177.9 +214.? p Area Sq. in. A Load p lbs. Story - lbs. in.~ lbs. in.2 Height 41.00 fl fl " 44.39 ti 59.72 II 67.80 It A 0.1 0.3 0.6 1.1 1.5 2.1 2.0 2.5 2.6 3.2 - 10.3 - 35.6 - 77.9 -134.5 -209.8 -300.3 -408.3 -533.1 -682.8 -851.2 0.3 0.9 1.9 3.3 4.7 6.8 6.8 8.9 10.l 12.6 Table - 1. (b) Calculation of Fixed End Moments Due to Wind Load Column Deformations. 12'-0 ti II " " It "II rt 16'-0 8 Tables l.(a) and l.(b) show the calculations for the column defonnations and fixed-end beam moments due to wind load. The quantity, 1\. + MR which is the sum of the moments at each end of a beam, divided L by span gives the beam shears, the sum of which is the column load. quantity { E The is the difference in the elevations of the ends of the beams in inches , (d), divided by (h), the story height in inches times (E), the modulus of elasticity for steel, (30,000,000 lbs. per sq. in.). All the story heights except the first one are equal, so by reducing the first story stress to one pr.educing the same deforms.tion in a column having the length of the upper columns, (multiply by i~) it was possible to factor out the common story height as indicated. If the difference in elevations of the ends of the beams are given the fixed-end moment is given by the following express ion, M= EI d = E (I/L) 212 2 L d E h h = 1 E 2 Where h is the sto~J height (12'-0") and L is the span of the beam. Checking Solution Each solution for moments was checked by calculating the horizontal deflection of each column. Obviously the deflections of all the columns in a story should be equal since the ends are rigidly connected by the beams. 9 c(z-L&, 2 Ma I 0, Me. Figure 2. Let Figure 2 represent a column under load causing bending moments shown in diagram. 9 1 can be cal cul ated from the known conditions at the base, i.e., the foundations do not settle. Take statical moment of the moment diagram f or Col. 1-2 about 2. MB L 2L EI 2 ;, MT L L EI 2 - = 3 2 MB - MT L 6 (1.)E = d2 - L 81 L Multiplying through by E/L, 2 MB - MT = E d2 6 ,r) r L - S1 E (1) 10 adding the known value of 91 we get the value of E d2 which should be L the same for each column in a story. The value of 92 may be f ound by adding t he change in rotation between the bottom and the top of the column 1-2 to 91. This is eQual to the area of the~ diagram of the column 1-2. = MB - MT 2 E I L Multiplying through by E we get: = MB - MT 2 cf) (2) The sign convention used in these expressions is the same as that used before, namely, that moments tending to rotate a joint in a clockwise dire ction are positive. Angular rotation in a clockwise direction is also considered positive. Table II. gives the computations for the deflections of the columns in the first four stories of t he bent due to secondary wind moments. No ot her calculations besi de those appearing in the table are necessary and afte r a little practice the application of the check become s very rapid. 11 Story A 1st.Story B 4th.Story C A B C A B C .A B 5th.Story A B 2nd .Story 3rd.Story C C I (d - L9lE :a.E L L ELI. e L + .087 + .364 + .162 + .142 +l.044 + .690 + .770 +l.735 +l.192 +1.272 +2.364 +l.703 +l.675 +2.965 +2.026 +2.025 + .293 + .343 + . 04.-4 + .360 + .352 .128 + .439 + .341 .283 + .412 + .385 .514 + .423 + .525 + .451 .455 + .485 +l.088 +l.050 +1.122 +l.607 +l. 631 +l.613 +2.081 +2.115 +2.060 +2.451 +2.449 +2.550 + .957 + .528 + .628 + .691 + .502 + .502 + .629 + .511 + .403 + .601 + .323 + .350 + .560 + .171 + .445 18.8 19.9 19.9 18.8 19.9 19.9 18.3 18.6 18.6 18.3 18.6 18.6 14.3 14.6 14.6 Column E 9 + - Table II. Calculations of Horizontal Deflections of Columns The values of E 9 for the first story columns were calculated from the first story beams connect i.ng the bottoms of the columns. Since the fotmda tions i,ere assumed to be fixed in elevation the value of d in equation (1) is zero and the expression gives the desired Ee. It will be noticed that the moments are given in foot-pounds so that if the deflections are wanted in inches, where E is in pounds per square inch, the result must be multiplied by 12n. The values of d E are merely L proportional to the inclination of the column from the vertical and serve only as a check. The above method of checking results was found to be very valuable. The expressions for the rot ation and deflection are very simple and can be applied even with rough mental calculations and are often useful in 12 checking and predicting results at least in a qualitative way. The same concept may be applied to any indeterminate structure and should prove to be of appreciable assistance at least in interpreting results. The calculations for wind moments and first secondary moments by the Cross method were made on large sized diagrams as used by the Engineering Department of the Institute. Subsequent calculations, however, were tabulated as sugge sted by E. A. 1.fucLean, Esq. in his discussion of the Cross method on page 421, Proceedings of the A.merican Society of Civil Engineers for March, 1932. The latter method was found much faster and easier than using the large diagrams and is recommended very highly. The flexibility of the Cross method in general was demonstrated on several occasions. In some cases the moments at a joint would not balance due to errors wade in computations. The error was easily corrected applying it as an unbalanced moment and carrying over perhaps once or t wice,depending on the relative size of the error. In calculating the fixed-end moments due to the secondary moments an error was made which affected all the fixed-end moments in the center span. 11he error was not discovered until the entire solu- tion for the first correction to the secondary moments had been made. The error was corrected by running an independent solution using the error in the morne nts in the center span as the fixed-end moments. correct moments were then the algebraic sum of the two solutions. The 13 DISCUSSION Foundations As was stated above, the foundations were assumed fixed against vertical movement. This assumption of course does not seem reasonable since t he foundations a ct as an elastic body and deform under load. The movement of the foundations would tend to increase the fixed-end moments and hence the secondary moments. Assuming that the foundations deform 1/8" under a unit load of 6000 lbs. per square foot, and proportioning the footings to carry the allowable load of the first story columns at the above unit soil d(,le .lo W/nc,/ /e,qq_ pressure, the deflections of the footingsAwere found to be from 25 to 50% of the column deform tions at the roof. From this rough calculation it would seem that the settlement of the foundations would add appreciably to the secondary moments. Secondary Wind Stresses in Wilson-Maney Bent The secomary wind moments in the twenty-story Wilson-1fu.ney bent were calculated by Mr. M. P. White as a thesis problem and were found to be ne gligible. This result can be justified if it is noticed that t he unit wind stresses and hence the column deformations are very nearly proportional to the distance from the center line of the bent. All the fixed-end beam moments act in one direction and the resulting sway will tend to decrease them so that when equilibrium is reached 14 the beams will have very little moment in them and will be practically straight. The column defonnations in the bent analyzed in this problem were alternate lengthening and shortening (See Fig. 1.) and t he fixed-end moments were in different directions. Side-sway in this case increases some moments and decreases others with the result that the secondary moments are q.iite appreciable. 15 CONCLUSION The conclusions to be drawn from this investi gation are not very definite or far reaching. Even the justification of the results obtained was at times quite difficult and certainly any very general prediction regarding the magnitude and effect of secondary moments would be hazardous. It is believed that the following general statements may be safely ma.de regarding secondary wind moments in building bents. 1. Secondary wind moments of appreciable magnitude occur in the upper stories only. 2. If the wind stresses in the columns vary approximately as their distance from the center of the bent, the secondary moments will be small and if the columns are alternately in tension and compression under wind load the secondary moments may be large. 0 z 3:: )> JJ )> G) 0 ~ ~ :-... ~ "()) ~ $-0 -i:,. r,+'I ,, A=ro7.80a" {:/3.9 -~ ~ "~ .. A =(.780a" f "I9.9 'I 'J\.i ~ .~ NJ I:, I\) ~ ~ I y=/3./ 12~0" f =/$.2 "'"~ ~ ~ l-0 A=4/0oa· '-l 'N "''-i II 1---fi., s.o" A=43.3.9d' t=/4.'1 ~ ',I "11-.. ,, }=13-2 A=4!0oa· '-I "'"" • 'N -;:. ~ "I~ ,, A•4!00a" f.=/3.2 '-I II '>-I "-I"-. A:4/.00a" -f=/32 ·•'N " A=38.8(,J:J'' -I =-/.3./ "i'-i ' 1z!.o" ' - II i=> l-o "i"-1 ,, A=4/.0tJa" f=/3.2 " ~ "I~ A=4t.CJOa" f =/.3.2 'I ~ "l"-1 ·11 A=38.8(, n'' I y=/31 12!.0" f=/3.Z " ~ ':Cl "\t-i fl= 4100 a " t=/.3.2 " 'N 'I "II-; A=4/.00a" '-l ~ ,~ ,, ..... ' A~388t:.o• f =/3./ 1z.:.o" " II -~ II 1'-\1-i 'N '1 :"-1"-i " ~ f'..fi-, ~ ~ ~ 6°" ~ ~ '-! I~ ~ ~ ~ "°~ ~ I~ ~ ti;) ::i::. ~ (,, ~ :"'I tb ~ ~ ~ C'\ ~ ); ~ ~ ~~ - - - - - - - - ----- - - - - - - - - - - ~ - - - - -·I...I~ · - - '1' L...--- ~ "h II :-0 A=43.39a" { =/4.t,, ?::t "'"~ A=4839a" f:-=14.G A=4J.39 a" '-l 'I -f=/4-.~ N N~ II f "/4.3 12'--o" A=3BB,a" A=3l3-8C, a" '),,I I~ - A=42-2c.a· IJ ~ - ~ " ~ "1"1 A=SJ.72a" f=/8.(.. '-I 12~0" f=/4.3 "I._. 1=!! - - " 'l'-1 "h A=59-7Zd f=/8.C, '-I '),,I "-1"-i ... A=531.9o" I 7:r:/8 ..3 /Zfo" VJ • ~ II "'~ A =59. 72 a· f:JB.t;, 'N 'I . 'N ~ "I "l"l " ~ A-59.72 a" f=/8.C, "I~ II A=,l80a" "I " £=/!J.9 'l-, II ,..,..., /1=53.7!JO N "I~ • II /l=-64.24d' f=/8-3 I z,,18-8 l'-11--t " fl='l80 o" {=!.9.9 " " .'),.,J "lf..t ,, A:h4.24t:t 12.:0" 1z.:o" ·- - - - - - - - · ' - - · - ~ ~ ~- , ~ ~,_ ~ ~ I, ~ I'\) ~ ,, I z=1s8 1(/0" '13\ 17 360 LBS. - 245 317 t-- 219 - 0, 0) 700 (J) 0 ti) 647 co <O 1080 <\I ..., 1155 O> (\J 1440 v 1618 '° co t-- 0 0 co 2063 '° It') t-- '1069 '° 2500 0, - 0) v - It') 0) 0 2520 (\J 2892 - CW) co C\I CW) 3321 - co - (J) (J) 3240 -- co - 3740 (J) (\I an (\I co -- 4431 -co- Ill 0 an ,.,, v ti) 588 ~ --- 1496 - 975 v ;! 1366 0 1766 ...,- 1345 co <\I · 1744 0 C) (\J t-- (\J 0 2521 (\J t-- co (J) ~ C\I <O (\J 2545 <\I co - <\I <O CW) 2499 3026 ~ ;;; .... '° C\I co 2940 3141 C\I co V C\"I C\I -2'897 3261 .. ,.,, 3978 t-- v co co v 0) 2786 3152 WI ND-LOAD MOMENTS FT. LBS. DIAGRAM NO. 2 v co '° ..., (\J 4021 3216 0, 0 8 v 0 co 3511 ..... ('I) CW) 3488 ~ '° -~ '° 2113 <\I co <\I v (\J 4225 0 (\J P) 3505 t-- I 0, 2 103 ") - -- 1618 - 2128 2317 V an C\I 1175 Q) 0) 0 I ti) (\J <\I ~ 0 <O I 0, t-- C\I 712 I 965 co 0 1917 2714 v an v '° co an 0 (J) an C\I 3660 595 (\J ~ CW) 2880 0 ~ (\J 2160 '° t-- t-- 1800 246 v 0 co (\J v (\J P) 120 294 an 11) cit) 3930 (\J 2771 I I I 4751 -- P) Ill ! co I 3371 j I I I I 18 FIRST SECONDARY MOMENTS, FIRST AND SECOND CORRECTIONS AND CORRECTED SECONDARY MOMENTS +179.4 +197.7 -245.3 +227.0 +112.2 CORR.SEC.MO~. -216.2 -136.0 +39.7 2-ND. CORR. -13.5 I-ST.CORR. +194.8 -275.0 +146.0 I-ST. SEC.MOM. ~"'il'~(W) ~ ~1~= I+ I l +144 .7 +45.2 -84.5 +184.0 0 co (\J. l"--1 . It). .,.._inco co •~T• o rt) O>lco g~~ r.: •+ •135.7 +38.6 -72.9 +170.0 q ~ (J\I• 0 • "ii' co <Ort)~~ I + 1 I +130.3 +33.8 -64.5 +161.0 0-"'il' (Y) C\i r-: d coI (Y) + - I ..... I ao +121.5 +28.5 -56 ~0 -149.0 0 I() 011() ~~~ ~ I+ I .. -281.0 0~lO m +219.9 ~ui'° "t ~ ~ r:> (\J +161.0 + -222.I +172. 6 0 ~(W)q +207.6 <O (Oi'O) -249.0 (W) (\J - "ii' + 1++ +214.0 -+204.0 o...... Cl! ~1~ -co -99.3 (W) "ii' (W) (W) -tl+-t +55.5 +160.2 -229.3 -134.I ,q-_ "1 Q C\J mco- +189.8 ~~11 -285.0 -296.0 q~(Q ~ +216.5 (W) lO (W) ~ -155.7 ~+~ ~ -235 .2 +176.7 +193.9 '-.;tl"'il'•CX) Q r-,: • Ill "1 lO (W) -235.5 ~<}I++ +218.0 +202.0 0-"'il' (W) io en tO -96.2 C\i "ii' "ii' (W) +54.8 +•+ ~ +160.6 -227.1 ~~rt)Q -123.2 co co lli (W) +174.1 !i?~t1 -218.0 -287.0 0"" -11") "It <d +203.0 i C1l+~ ~ -146.5 + -230.5 +174.3 +178.1 ~1~ q -215.8 o <\J I"-: "'> m rt) (\J - (Y) + ,++ .f.212.0 +196.0 Q-:IO ~ -88.7 O>0~ rt) ~ (W) ~ +49.8 + I + + +157.I -214.0 -107.2 ~ rt) '°"'il'0 • . • rt) 0 0 +154 .2 rt) ,_,"'!t~ + + I + -261.0 -280.0 ~~~ r,; +186.1 (\J - (\J ++1 ~ -134.4 -228 .3 +163.2 <O(\JOO +156.3 comr,; • (\J - - rt) -192.1 + I+~ -I99.0 ~~~1~ -263.0 q LO 10 0 +164.6 ~idio + - (\J (\J -116.6 + I + -215.0 + 151.7 +131.2 q l ' t ' t q -162.5 CD r-,; 0 (Y) (\J - - t!) +-183.0 + I 't i-236.0 00.~~1'1: - CO lO +140.3 ~+T+ -97.4 -193.1 - I~ +135.7 - - 0 0 +42.0 i o<»cw; -79.3 (0 (\I t!)+ <O I I I +173.0 0 IO lO • ·O 0,• "ii' co • It) -C\I~ I I -t I +200.0 0 . (0. <O. C\I. I() (\J co -112.2 ..... <O co c9 -67.1 +1+, + +154.9 -238.5 "'il'CO(\J0 -144.9 co C\i • «> +205.4 "ii' It) ;2; It) ++I+ -299.0 1 +186.0 -79.1 ~~~ (W) -t43.I . + +150.0 -195.7 -89.9 ~1,_, oq ,._ 131.2 ~d'°~ ~ ~rp ... -237.0 +170.0 ~ (Q-t <O -66.6 C\I <O rtj (\J + l}l (\J + + +36.2 +139.6 DIAGRAM NO. 3 I. I . I . I. I 10 I. I I . !9 "° . I . I . I8 I It) I I I I7 .I / I 1 . 6 I- I I 19 + 111,5 N -mo +22.3 en· o...: o -45 -8 ,q- -(\1<0 I I+ I +135.0 om (\JIP') ari M (\j rt'i ...... (\J - '° I+ I I +95.8 +17.9 -35.1 +113.0 or--m (\J d .....,: «> ...: I+ I co-•'° +71.5 +122 -24 .7 -+84.0 - ~+•~ ~~:,~ 't-1"¢ I t -170.7 -70 .7 ql<QCQQ ...... (\J ...... (\J +104.0 (\I (\J (\I cY) ++I+ -204.0 +150.0 00-¢ "¢ -52.2 ~ ,q: 0 co +26.5 ~~~ + +125.3 -141.3 com-o -54.9 oi oi ,q: ~ +81.6 (\J - (\J cY) ++1+ -168.0 +123.0 q~~1~ -41.4 = !:2 co co +21.5 + + + + +103 .0 -105.1 -37.8 C\J CO co 0 r-,:tr)\00 + 56.7 -124.0 I +91.0 0 .It) . ('1)1~ ·o + -28.6 cY)=(J) +•+ +14.7 +77.1 I' +130.2 +104.6 OIOQ en ('I) «> cY) -130.4 ~ ,+~ +156.0 It) II 5 -206.0 q~1111~ +110.6 !::: = ~ (\I -76 . I ++I -171.5 +110.6 +81.1 IO COcYl 0 lri _;...,: o -102.5 (\J ('I) + I~+ -f-132.0 + -170.0 +86.1 -58.4 -142 .....3 000>1--: ex, ai o co +• + +85.S +57.6 oi co ,q: rt) - 71.1 -1+<\l + + +99.0 (\I 'It <O 0 -124.0 q~ ~1~ ...... <0 0 +59.2 (\I + + I + -40.6 -105.4 +44.8 + 7.0 -14 .2 +52.0 -63.5 .-21.6 0 0) (\J ('W) +33.1 C\J+-<\I + I + -75.0 0 '° (\I ...... ...: '° ('I) r-,: '(+, +59.0 q "": (\J I'° C\'l M .....,: -16.8 r--. I I + I +8.5 +50.7 +56.6 +32.4 ....:io10<\i -39.8 (\I I T (\I + + +64.0 -77.0 q ,q- en I ,qiri (W') r-,: +29.0 m I+ 1 I -23.2 -71.2 -7.4 mmoo -0.9 (\i+"('t)~ +1.5 + I+ -8.0 -7.7 (\I 0.0 'tC\JO tri d_;co +0.3 + +1+ -8.0 ('I) ~~Q (\J I -4.e -0.5 +0.7 -5.0 -6.0 +1.5 -1.0 -5.5 t----¢-0 -8.0 +0.9 -0.4 -7.5 I I ' I I4 I I I I3 I I I2 I ' I I ' II I N-OTES: ALL MOMENTS ARE IN FT. LBS. +MOMENTS TEND TO ROTATE JOINT IN CLOCWISE DIRECT. FIRST SECONDARY MOMENTS, FIRST AND SECOND CORRECTIONS AND CORRECTED SECONDARY MOMENTS DIAGRAM NO. 3 20 RATIOS OF FIRST SECONDARY MOMENTS AND CORRECTED SECONDARY MOMENTS TO WIND MOMENTS -.35 _,46 .:-0 • I +.88 -4-1.12 - .61 -.77 -.72 "ii"~ -:56 + + +1.14 ~ co 90 ":i q SYM.ABT. ~ R + +. (\J m!::: _.21 "il"CO +- .40 _ ~ -:24 '1--- •-•~--·2_s_ _ _ _ _ _-__ +_+~·i-4-_._s_o________ :;_ · :.: ....·+--_ .3_0_ _ _ _ _--1 10 twlO -32 <\J <\J • .50 rt> I a \ ·o .. -.. C\J ~~ - .12 • I <; ~ ... .40 0.. .. ~~ 4-.23 i2--.1s ;25 I 1--- '-•·+---_ . IS _______ +_+~·1-+_.2_9_ _ _ _ _ _ _+_·.... ++-,...._1_8_ _ _ _ _--11 9 -; ~ ,. a --J9 ~ -.15 + + a~ +.3o (\J (\J -4-.24 ~ ~ dd I I twl\O -08 +.,1 -.11 ,.,.__-;'·----'-._-._11_ _ _ _ _ _ _+_+_+_.2_0_ _ _ _ _ _ _+_·_+,..-_.1_3_ _ _ _ _ _, 8 I ~~ co <O -:13 ,... 0 u, -:10 <\J o 0 -:08 + + 9- 9 ++ +.2, ~~n~ l ++ ,...~ -.06 ~~ • .12 ~ ~ --.oa I , ~ ,· -.08 -t + +.15 + + - .09 7 ..--,- - - - ' - - 1 - - - - - - - - - - - + - - - - - - - - - - - + - - -, :0 I' ,J C?, C\J -.11 C\J (\J + .1 7 • .15 o o +.12 + + (\J <OCO -.05 00 +.09 00 -.06 0 '1-'- . . J ~L....-.:., +--._0_6_______ +_+_·+-+_.1_1__________ ·,..-_.0_1_ _ _ _ _---i 6 +_+__ C\J I -.04 +.07 -.04 I +---·_o_s________ +__ 1-+_.0_8_ _ _ _ _ _ _+_.......• +--_.0_5______ 5 ~~ ' r r :0 •I ~ll) 0 -:07 0 0 - .06 -ti ~ +.II +.09 00 00 1-'-.....: ~ _ ..,_ .. j\ tw) ~ 0 O O ++ +"'"I· ~ 'q' oo ,· .- - .06 -.06 -.03 0 0 Q O ... + +.08 o 0 +01 +·q I +.05 0 0 -.03 +__+~·1-+_.0_1_________+_+_·...-_.04 _ _ _ _ _~ 4 '1-'_....:. ~ - .:...,·1---._0_3 _________ ~~ •· r - - -.01 - .01 0,. 0,. ~04 -.0.3 ~~ o. o. +.02 1 02 • + + .... -.01 ~q o o -.01 -4-.07 +.07 ~~ 00 -.01 + + -. 01 ._ 02 oo o 0 .... +.02 ' 2 1---'--'--------------------------0r,. o Oo n ~q .00 .00 00 oq ..00 .bo Oo ~q .00 .00 .00 .00 DIAGRAM N0.4 .00 .00 21 WIND MOMENTS,CORRECTED SECONARY MOMENTS AND RATIO OF CORRECTED SECONDARY MOMENTS TO WIND MOMENTS -.35 RATIO +.88 +I 12 COR.SEC.MOM. -216 -317 WIND tvtOtvt.FTLBS. -245 -279 ~ - C\I I'- C\I C\I +155 C\I CW) 4 I cY)I• Ill + -:-56 + = -.21 _CW)_ -+145 en com 1· I + -700 0) C\I 0 oco .. <0 + I - .12 -+ 136 -1155 0> C\I Ol C\J I'- 0 CX) I I. + f-: - .24 co -..r .f.173 = CW) <") +· + -+ -712 -647 -..r C\I C\I C\I CW) 0 •J60 en+ • + -:25 + -588 0 Ill I") C\I 0 -235 <O Ol + +· -f:40 + ....23 Ill C\I ~ -229 qcY)r-,. + +-+ -975 -1069 OCOI") 'q' cY) 0 -4-161 CW)+ +· -:15 -.IS "'"177 OC\I~ +· + + -1175 -965 co co C\I Ol C\I O -231 !!}+ +· •.24 + +.17 -277 QC\')= ... + + -1366 -1496 0 cY) C\I +157 LO CW) 0 !:::+ +· - . II + C\I -.II CO -1-174 OcY)= f\') I'- C\I co co co I'- co 0 0 I 1· -+ -246 0 CW) CX) -222 't ~ 0 tO + • +.90 + ... ....40 Ol -239 'It <O -: 'It IY) + +-+ -595 + -.08 +136 -1618 -.61 +179 -294 0 'q' I'- l I ' I I i10 ' ' I I' -..r CW) 0 +· + + -1618 -1345 0 CW) C\I CX) C\I 0 -228 I'-+ +· 4-.17 + - 9 I I I I i8 I I I ' -.06 ococo +130 1· I t -2063 coo co r-,. r-,. 0 O tO I'- 0 C\I I I' + -1917 ... ,so -.08 -.OS <O Ol &O +122 Oto en 1· I t -2500 O)cY)~ ~ coo -IO+ +.12 -214 +·+ + -1766 C\J CW)~ 0 CW)~ I ,· I'-- - co I") O C\J + +· + 09 - ....-196 C\I -..r -t "t -t -2317 -1-140 -.06 -1744 -215 "':12 OtoCOC\10 C\I + +' C\J + I'-C\I -2128 -.06 -coo -1-152 q C\I C\I ... + + -2545 'It C\1-C\I 0 io-2103 C\1-0 ~ q cY) 2? -.08 +163 + + .. -2113 I'- C\l s--- Ol q C\J !! ~+ +· + DIAGRAM NO. 5 -193 <O + +· C\I +.09 + I I7 I I I I I6 I I I I 22 CX) -.04 +112 -2892 ~ +.07 - ..... (\J -171 0 (\J (\J .... + + -2521 rt) (Y) ~ -2714 l'coC0 - 0 +125 0, + +· -.05 (\J + co coo C.0 I I. -+ -.03 +96 -3321 CX) - rt) o, Ill 0 0) I 1· + -.02 +72 -3740 o,-(\J (\J ~ 0 - I 1· <.}:I -.01 +45 -4431 -ex>- CX)(\Jq (\J I I + +.OS -141 -oco q (Y) (\J +++ -2940 -3141 (\J CX) 0 +103 co+O ~ -.03 (Y) + . 0 +.03 -105 +· + + -3261 (Y) -1'1' 0 (\J (\J -3488 +77 -.02 o,-o -+o 0 . ~ + co I.I) l+.02 -Ot'l -64 q (\J (Y) + -t + -3978 -4225 +51 -:01 +130 ~ (\J (\J + ·-+ + -3026 -2499 ~('1)0 -172 ~ + q +.07 rt) + - 0) l'CX>O ~•q co ~ + - -.04 co 0) rt) ~ (\J I I I.5 I -.03 - co CX) +111 0 (\J (\J +· + + -3511 -2897 ~coo CX) + ~ -142 Ill +.05 (Y) 4 + -.02 " +86 ff) -m C\J ~ ;- ~ -4021 -327ij -105 +.03 o,-o IO-t 0 0 ~ + 0 Ill -.01 +57 +- + + -4751 "o -3930 -ff) I 0 . -71 Ill +.02 + - (\J IC') 0 C\J IY) I3 I I. I I.2 I I I I I I I I Ill .00 0 ~ -5 0 IC') CY) ·+ + -3505 co .00 0 -7 0-IO ·+ + -2786 (Y) (Y) 0) -3152 -6 .00 (\J I -2771 -8 .00 I NOTES: MOMENTS ARE IN FT:-LBS. +MOMENTS TEND TO ROTATE JOINT IN CLOCKWISE DIRECTION. WIND MOMENTS,CORRECTED SECONDARY MOMENTS AND RATIO OF CORRECTED SECONDARY MOMENTS TO WIND MOMENTS DIAGRAM NO. 5 I .00 ~ -8 0 I l l o _co -3371 '+ + I I I 23 WIND BRACING IN STEEL BUILDII\1GS Discussion This article was submitted to the "Proceedings" of the .American Society of Civil Engineers as a discussion to the report of the Committee on Wind Bracing. A rather sketchy investigation disclosed only one reference to the subject of Secondary Stresses in building bents due to column shortening under wind-load. Bulletin 80 of the University of Illinois covers the subject in a paragraph. The shortening of the first story columns of the twenty-story bent is computed and the fixed-end moments in the first story girders determined. These moments are small com- pared with the original wind moments. However, it must be remembered that the effect of column shortening is cumulative from bottom to top, while the girder moments due to wind become smaller so that the effect of column shortening should be much greater at the top than near the ground. In its discussion the Committee on Wind-Bracing states that in the case of a high, narrow building secondary moments require investigation. In the relatively high and narrow Wilson-Maney bent secondary moments are negligible. Apparently there is another criterion. The importance of secondary moments depends upon the relative size of bays and upon the relative stiffness of columns and girders. In the Wilson-Maney bent both columns on one side of the center line have the same kind of stress under wind load and this stress and, therefore, the shortening 24 is roughly proportional to the distance from the center line. Since all the fixed-end moments in the girders due to column shortening act in the same dir-ection, they will all be reduced by the resulting sidesway and the girders in their final position (after equilibrium is reached) will be nearly straight. When we have alternate tension and compression in the columns, the g irders are constrained and may therefore have large bending moments after equilibrium is reached. In such a case sidesway will increase certain girder monents and decrease others. The Cormnittee does not consider the possibility that the secondary moments as first obtained may require correction. For example, in general (not always) the secondary moments in the girders will be opposite to the primary moments. If the secondaries are large, say 50% of the primaries, the resultant moment will be one-half of the original. But the secondaries will also produce column shortening which will cause more secondary moments, ordinarily of the same sign as the primary moments. If the first secondaries are 50% of the primary moments then the second secondaries will be about 25% of the primary moments which will make the resulting moments 75% of the original moments instead of 50%. In general the first secondary mo!ll3nts will give results which, for girders, are on the unsafe side. Calculati ons were made on two bents; one, the twenty story WilsonManey bent, gave negligible secondary moments. The other bent which was obtained by adding twenty-two foot bays to t he upper ten stories 25 of the Wilson-Maney bent gave very interesting results. Under wind loads t he columns of this bent were alternately in tension and compression. The accompanying diagram*gives the ratio of the first secondary to the original wind moments (lower figure), and the ratio of the secondary moment after two corrections (algebraic sum of secondary plus first correction plus second correction) to the original moment. V. A. 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