Economical Design of Building Frames Citation Chan, Glenn Chew (1935) Economical Design of Building Frames. Master's thesis, California Institute of Technology. doi:10.7907/d779-h927. https://resolver.caltech.edu/CaltechTHESIS:10092024-171559234 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: 18 May 1935 Record Number: CaltechTHESIS:10092024-171559234 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10092024-171559234 DOI: 10.7907/d779-h927 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16784 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 11 Oct 2024 17:47 Last Modified: 11 Oct 2024 18:10 Thesis Files PDF - Final Version See Usage Policy. 21MB Repository Staff Only: item control page

E CON OMI Cb L D E S I GN OF 3 U I 1 D I N G F rl A M E S A Thesis Prepared in The Colle ge of Engineering California Institute of Technology Pasadena, California 1935 By . Glenn C. Chan Candidate for the degree cf Master of Science in Civil Eng ineering Pres ented to Adviser for ap pro val _.i.pprovea by 1935. - - - - T.AciviserT - - - - - Presented to Dean of the Graduate School for approva l .dpproved by 1935 . LETTER OF '.rdANSMITTAL California Institute of Technology Pasadena, California May 18, 1935 Dr. Richard c. Tolman Dean of the Graduate School California Institute of Technology Pasadena, California Dear Sir: In accordance with the regulations of the Department of Civil Engineering of California Institute of Technology, I hereby submit this thesis as a partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering . The title of this thesis is Economical Desi gn of Building Frames. tlespectfully submitted, T A B L E OF C ON T E N T S PAGE LETTEJ. OF Tl:l.NSIHTT.AL CH~PTE3 I ----------Introduction CK.\.PT}~H II ---------Slope Deflection Method--Symmetrical Bent with Bays of Unequal Width CHAPT:2:R 28-31 IV ---------Cantilever Method--Symmetrical Bent with Upper Stories Set Back CH.AP~ER 4-27 III --------Cantilever Method--Symmet:rical Bent with Bays of Unequal Width CH.:lPTB.~ 1-3 32-35 V ----------Portal Method--Symmetrical Bent with Upper Stories set Back 36-40 ''Modified" Portal 1Iethod 41-43 CH~lPTE ,1 VI ---------Continuous Haunch Beams CHAPTEi 1. Slope Deflection Method 44-46 2. Moment Distribution Method 47-49 VII --------Conclusion JI:SLIOGUPHY 50-52 53 ECONOMICAL DESIGN OF BUILDING FRAMES CH.APTER I INTRODUCTION As a result of accurate experimental and research works carried on to determine the actual conditions of stress distribution relative to its assumptions, engineers are now abled to know within a fairly reasonable margin of this actual stress which will exist in the finished structure. Since such varia- tions from the ideal conditions are so closely approximated, this exactness of results has paved the agitation in favor of increasing the allovvable v,orking unit stresses used in designs in order to develop as economical a structure as possible and with sufficient degree of safety. On the othGr hand, man:7 experiments give results of considerable difference between these two sti-·esses; theref'Ji"e, it must be borne in mind that conditions do vary and not necessarily according to any specific law. Even the most intricate is not :;eneral- ly the most precise solution. The complexity of the analysis in- volved the consideration of all possible factors that will exert any influence upon the finished structure, which will necessarily provide reasonable assurance to the ssL:ty of the design when the laws of nature act acc'Jrdingly and exclude factsrs beyond huDan comprehension. The main consideration in any structural analysis is to have the laws of statics being fulfilled and to be sure there are no forces being overlooked. Having all these considera- tions fulfilled to satisfaction, one may conclude that the most ( 1) economical method of analysis will no doubt be one for which the least amount of labor and material are expended . .At times one will be forced to sacrifice precision for time, not exactly at the expense of safety, but for structures of less importance which do not justify too enormous an amount of time and labor to be spent. In the past years, for buiidings of moderate height, the analysis of wind stresses is not of great importance in many cases. However, one cannot now deny that such stresses will de- serve a careful and thorough analysis, because of the growing tendency towa1"'d buildir:g of sky-scrapers, especially in populous cities where lanc1 values are enormously high, there exists a greater desire to tax a greater return from a unit area of land in orc1er to justify its utilization. The question naturally arises as to which method of solution to apply in order to obtain reliable results and still permits greatest economy of labor and materials. Among the various methods that may serve this purpose, the approximate solutions find greater application tlian those nexactn ones, mainly because of the increased labor of computation by these latter methods. Although these exact solutions sometimes yield re- sulting stresses representative of the ideal conditions, it should be remembered that such closeness of agreement between the assumed and actual stresses rest greatly upon the assumptions of the wind load rather than wholly on the differential of 11 exactnessn and 11 app1·oxirnation". Experiments have shown that wind velocities vary with the al- titude; furthermore, it shows that with a given wind velocity, the ( 2) average pressure is less on a large surface than on a small surface. This information, however, is not sufficient to enable the engineer to know just what approximate pressure to proviae for on buildings of different height, mass, and exposure. Consequently, the wora nexact'' solution will be exact to the extent that the above conditions are correctly provided for. With these various methods for wind stress analysis, it is the writer's intention to show by examples that the approximate methods will permit a more economical design of building frames than the slope deflection method which is the exact solution under consideration in all these discussions . .Additional examples will offer a comparison of the relative economy of tl1e cantilever and Porter methods. Included in this discussion of economy in design will be an example of the design of continuous haunch beams by the exact and the moment distribution method. ( 3) CHAPTEH II SLOPE DEFLECTION 11ETHOD SYMMETRIC.AL BENT WITH BAYS OF UlJEQUJ,.L WIDTH The exact method considered here is the slope deflection method which takes into account the relative stiffnesses of the columns and girders and that the point of contraflexure of each column and girder is not fixed at the midpoint which in this respect differs greatly from the assumption made by the approximate methods. The assumptions upon which the theory of slope deflection is based are as follows: 1. The connections between the columns and gi.rde1~s are perfectly rigid. 2. The change in length of a member, due to direct stress in it, is ne gligible. 3. The length of the members a re the distance between the intersections of their neutral axis. 4. The deflection of a member, due to interna l shear is negligible. 5. The wind load is resisted entirely by the steel frame. In order to show the importance of the distribution of moments at any joint in direct proportions to their stiffnesses, the writer has chosen the symmetrical three bay tenstory bent of unequa l vvidth as shown in figure 1. Since the cantileve r and portal met i1oa s a c•e approxima te solutions a nd in order> to exa 2;g erate the difference of results, no attempt is here ma de to readjust the column and girder moments by considering its relative stiffnesses. The cross-sectional areas of the columns are taken to be equal which in this particular case is very nearly so. The assumptions upon which the cantilever metnod is based are as follows: 1. A bent of a frame acts as a cantilever. 2. The point of contraflexure of each column is at its mid-length. 3. The point of contraflexure of each girder is at its mid-length. 4. The direct stress is directly proportional to the distance from the col1unn to the neutral axis of the bent. 5. The wind load is resisted entirely by the steel frame. The assumptions on which the portal metnod is based are as follows: 1. ..:i. bent of a frame acts as a series of independent :portals. 2. The point of contra-flexure of each column is at its mid-height of the story. 3. The point of contra-flexu.re of each girder is at its mid-length. 4. The horizontal shear on 9ny :plane is divided equally between the number of aisles. An outer column thus takes but one-half of the shears of an interior column. 5. The wind load i s resisted entirely by the steel frame. The bent of figure 1 was fi.::-st analysis fo::c m0ment s and shea ,:·s in columns and girdei~s by the slope deflection method ( 5) and as a comparison of the economy as resulted from the approximate solutions, the same bent was analyzed by the Cantilever method. The method of stress analysis by the slope deflection is outlined here in general as follows: Consider a strip of the bent one foot wide in a direction perpendicular to the plane of fig. l; the loads are due to a horizont':11 wind pressure of 30 lb. per sq_. ft. The columns are as- sumed to be fixea at the base. Arbitrary values of K for the columns and girders are shown. The chang e in slope of the members meeting at the joint is represented by~ and the deflection angles bye,(,. Prom the condition of symmetI,y the following relation holds: etc. The fundamental equations used to determine moments in g irders being M : 2EK( 20-n +6-.1 ) and for columns, M : 2EK( 20n +-ti; -3.£ ) Where L1 = moment in inch pounds E - modulus of elasticity K - stiffness factor = 1/1 B-n : twist angle near joint ~ = twist angle at opposite joint oG: deflection angle By application of the above formulas, the joint equations are as follows: ( 6) Joint No. AlO: Joint no. BlO: 2 ( 12. 5+11.11-,.22. 21} 0o,. +12. 56-4,. .,.11. llBJ,. ,..22. 210-09 -3X22. 2loL,. 1O2.750.s,• .,.12.50-4,o -r22.2l·fo~9 -66.63oLlo =0 =0 The complete equations for the other joints are found in table 1. In addition to the joint equations, the additional independent equations for the determination of the a values are obtained by considering the equilibrium of a portion of the frame formed by cutting sections across the frame just belovv the top and just above the bottoms of all colUJ1ms in any story. The equations thus obtained are: For the tenth story: 22. 21 ( 0,9,, -,..e..,9- +-6t1/o -,..es9 ) -4X22. 2lo(: -33OXl4Xl2/12E /0 133.26(-9,.,,. -,.fj-,.,9 ,..e,a,., ,..&e~ }-533.04<40 : -55,44O/2E For the ninth story: 22. 21 ( -&11.9 ,,.e,,uJ +fu.9 +0ea -,..Q ) -4X22. 21 = -7 5OXl68/12E 133. 26 ( B-119 +e,9e +0e9 r&ea ) -533 .04 : -126, OOO/2E The complete eg_uation for the other stories are given in Table 1. The solution of the equation are shown in Table 2. The de- sired stresses are found directly after these unknown quantities are solved. The moments in the columns ana. girders are obtained by the two formulas given above. The shears in columns and girders are equal to the sum of the end moments divided by its length. resulting v8.l ues are shown in Table 4 and 5. ( 7) The Following a re the numerical values of the change in slopes and deflection angles found from the elimination of the 30 eQuations: 73-74) -1 .4281 -B-r1,o = 308 • 549 -Gf/tO: 73) -Be✓1, -7 .26 : 152.00 -8-.9,0 70) ~0 : 216 • 05 = 15 9. 26 U.2995(159.26)~0.2775(216.05)+308.340 : 47.693r59.954+308.340: 415.99 67) -9A9: 0.7233(415.99)-.0051(159.26)-0.2352(216.05)-'"233.867 : 300.893-0.812-50.827+233.867 = 483.12 65) s.89 - 0.0210(483.12)-0.1749(159.26)r0.5245(415.99)~189.112 : 10.145-27.855+218.192~189.112: 389.59 62) ,£.9: 0.3021(389.59)~0.2800(483.12)~551.699 - 117.695rl35.276+55l.699: 804.66 59) -eAB: 0.7079(804.66)-.0052(389.59)-0.234~(483.12)~297.254 : 569.700-2.026-113.485~297.254 = 751.44 57) --088 : 0.0218(751.44)T0.5247(804.66)-0.1749(389.59)r247.742 54) = 16.382+422.205-68.14lr247.742: 618.19 e£8 = 0.2989(618.19)+0.2775(751.44)r709.320 : 184.777T208.525r709.320: 1102.6 51) "tiAT: 0.7195(1102.6)-0.Q059(618.19)-Q.2343(751.44)r383.275 : 793.321-3.65-176.062r383.275: 996.88 49) -B-P1 • 0.0220(996.88}r0.5249(1102.6}-.1749(618.19}~319.785 : 21.932+578.351-108.198 46) cC7 = 811.87 = 0.296(811.87)f0.275(9'06.88)+900.00 = 240.3lt274.16r900.uo = 1414.5 ( 8} 43) OA6: 0.7190(1414.5)-0.0052(811.87)-0.2343(996.88)~45945 = 1017.025-4.222-233.58'+459.45: 1238.7 41) .Q,B6: .0224(1238.7)-0.1745(811.87)+0.5236(1414.5)~381.033 : 27.747-141.67lr740.632+381.033: 1007.7 38) o.'.:~: 0.2941(1007.7}+0.2735(1238.7}+1042.18 : 296.365+338.784+1042.18: 1677.3 35) ~~~• 0.6663(1677.3)-0.0042(1007.7}-0.2177(1238.7)r520.565 - 1117.585-4.232-269.665+520.585 33) = 1364.3 -0-85 : 0.0185(1364.3)-0.4769(1677.3)-0.1590(1007.7)+424.345 = 25.240-799.904-160.224+424.345: 1089.3 30) cL~: 0.2983(1089.b)r0.2764(1364.3)+1158.97 : 324.938T377.093+1158.97: 1861.Q 27) ~+ = 0.7104(1861.0)-0.0039(1089.3)-0.2328(1364.3)+575.517 -- 1322.125-4.248-317.609r575.517: 1575.8 25) 22) 19) ,0,M = 0.0170(1575.8)r0.6168(1861.0)-0.1722(1089.3)~481.322 = 26.789+961.765-187.577r481.322. 1282.3 ,£4 = 0.2V92(1282.3)+0,2760(1575.8)+1370.71= 2189.3 ,t),43 = 0.0009(1282.3)+0.7173(2189,3)-0.2395(1575.8)~704.945 = l.152+1570.385-377.404+704.945: 1899.07 17) ,Bg3 = -0,0030(1899.l)-0.1758(1282.3)r0.5274(2189.3)+600.043 = -5.697-225.428~1154.637+600.043 = 1523.6 14) .£3 = 0.293(1523.6)~0.2708(1899.l)T1954.13 - 446.414+514.268rl954.13 11) -9-AL: = 2914.8 0.6493(2914.8)~0.0021(1523.6)-0,2185(1899.1)~967.970 = 1892.580+3.200-414.947~967.970: 2448.8 9} --G82 = -0.0095(2448.8)f0.463(2914.8)-0.1543(1523.7)~812.965 = -23.264+1349.552-235.107r812.965 = 1904.1 ( 9) 6) c:L,z = 0.2683(2477.6)+0.2910(1904.1}+2384.92 - 664.740+554.093~2384.92 3) -011, = 3603.7 = 0.6228(3603.7)-0.212(2448.8}+0.0040(1903.9)~1060.89 = 2244.74-519.14+7.61~1060.89 = 2794.1 1) -9;,, = -0.0194(2794.l)+0.4369(3603.7)-0.1459(1904.1)~744.109 = -54.205+1574.456-277.80 8 +744.109: 1986.5 ..c, = +0.24g8(1986.5)+0.2498(2794.1)~1909.827 A} - 496.228-697.966-1909.827 ] 1 = 3104.0 ollowing are the calculations for the moments in column A by direct substitution into column equations: MlO-~ • 22.21 2X216.05+483.12-3X415.99 : 22.21(-1248.0+915.2) : -7390 M9-10: 22.21(2X483.12-ii216.05-3X415.99 - 22.21(-1248.0rll82.j) : -1460 M9-8: 22.21(2X483.12+751.44-3X804.66) : 22.21(-2414.0+1727.6) = -15,240 M8-9: 22.21(2X751.44-1483.12-3X804.66) = 22.21(-2414.0~lg86.0) : -9,530 M8-7 - 22.21(2X751.44+996.9-3Xll02.6} = 22.21(-3307.8~2499.8) : -17,500 Ll7-8 - 22. 21(2X996.93+751.95-3Xll02.83) : 22.21(-3308.49+2745.81) : -12,500 M7-6 - 22.21(2X996.93+1238.7-3Xl414.5) = 22.21(-4243.5r3232.56) • -1010.9X22.21 = -22,500 M6-5 = 22.21(2Xl238.7+1364.3-3Xl677.3) = 22.21(-503.L9+3841.7) = -26,420 ( 10) M6-7 = 22.21(2Xl238.7+9 96.93-3Xl414.5) -- 22.21(-4243.5+3474.3) = -17,100 M5-6 = 22.21(2Xl364.3+1238.7-3Xl677.3) = 22.21(-5031.9+3967.3) -23,600 :: M5-4 = 25.0(2Xl364.3+1575.8-3Xl859.3) = 25.0(-5577.9+4304.4) = - 31,800 LI4-5 -- 25.0(2Xl576.8+1364. 3 -3Xl859.3} I.14-3 25.0(-5 5 77.9+4515.9} = -2 6 ,500 = 25.0(2Xl575.8+1893.0-3X2183.7) ,. 25.0(- 6551.1+5044. 6 } = -37,700 IvI3-4 = 25.0(2Xl899.1+1575.8-3X2189.3} = 25.0(-65 67. 9+5374.0) = -29,800 113-2 = 21.42(2Xl899.1 ➔ 2444.8-3 X 2914.8) = 21.42(-8744.4+6243.0} = -53,500 Li2-3 = 21 .42(2X2444.8+18 99.l-3X2914.8) = 21.42(-8744.4+678.87) = -42,000 1'12 -1 = 20 .82(2X2444.8+2794.l-3X3603.7) = 20.82(-10811.1+7683 .7) - -65,200 Ml-2 = 20.8 2 (2X2794.1+2444.8-3X3603.7) = 20 .82(-10811.1+8033 . 0 } = -58,000 1a-o = 18 .52(2X2794.l-3X3104.0) = 18 .52(- 0312.0+5588.2) = -6 9 ,000 I✓I0 -1 = 18 .52(2794.l-3X3104.0) = 18 .52(-8 bl2. Gt2794.l) = -121, 000 Moments in the B-Columns Ml0-9 • 22.21(2Xl59.26+389.59-3X415.99) = 22.21(-1248.0t708.l) : -12,000 M9-10 • 22.21(2X389.59+159.26-3X415.99) = 22.21(-1248.0~1397.4) : 6,870 M9-8 - 22.21(2X389.59~618.19-3X804.66) : 22.21(-2414.0+1397.4) M8-9 = -22,600 : 22.21(2X618.19t389.59-3X804.66) : 22.21(-2414.0+1627.0) : -17,500 M8-7 : 22.21(2X618.19~811.87-3Xll02.6) : 22.21(-3307.8+2048.2) : -28,000 M7-8 : 22.21(2X811.87t618.63-3Xll02.83) : 22.21(-3308.49+2242.37) : -23,700 M7-6 - 22.21(2X811.87tl007.4-3Xl414.5) : 22.21(-1243.50+2631.14) : -35,800 Ll6-7 : 22.21(2Xl007.4t811.87-3Xl414.5) - 22.21(-4243.5+2826.67) : -31,500 M6-5 : 22.21(2Xl007.4i-l089.3-1677.3X3) : 22.21(-5031.9+3104.l) : -42,800 hl5-6 : 22.21(2Xl089.3~1007.4-3Xl677.3) = 22.21(-5031.9+3186.0) : -41,000 M5-4 : 25.0(2Xl089.3tl280.4-3Xl859.3) - 25.0(-55779t3459.0) M4-5 = -52,800 : 25.0(2Xl282.3+1089.3-3Xl861.0) : 25.0(-5583.0+3653.9) : -48,230 M4-3 : 25.0(2Xl282.3+1523.7-3X2189.3) : 25.0(-6567.9+4088.3) : -61,990 ( 12) M3-2 = 21.42(2Xl523.7tl904.l-3X2914.8) = 21.42(-8744.4 ➔ 4951.5) -- -81,250 IvI3-4 = 25.0(2Xl523.7+1282.3-3X2189.3) = 25.0(-6567.9 ➔ 4329.7) = -55,960 M2-3 = 21.42(2Xl904.1+1523.7-3X2914.8) = 21.42(-8744.4+5331.9) -- -73,200 M:2-1 :: 20.82(2Xl904.1 ➔ 1986.5-3X3603.7) = 20.82(-10811.1+5794.7) = -104,300 1.11-2 -- 20.82(2Xl986.5+1904.l-3X3603.7) = 20.82(-10811.1 ➔ 5877.l) = -103,000 Ml-0 = 18.52(2Xl986.5-3X3104.0) - = 18.52(-9312.0+3973.0) - -99,000 M0-1 = 18.52(1986.5-3X3104.0) = 18.52(-9312.0+1986.5) = -136,000 Moments in the Girders ( Bay AB) MlO M9 lfl8 117 M6 M5 M4 M3 = 12.5X591.36: 7,390 : 12.5(2X483.12t389.6} : 12.5Xl355.8 = 16,950 -- 12.5{2X751.44 ➔ 618.19) = 12.5X2121.l -- 26,550 = 12.5{2X996.93t811.87) -- 12.5X2805.73 -- 35,100 • 12.5(2X216.05+159.26) = 12.5{2Xl238.7+1007.4} = 12.5X3484.8 = 43,600 = 14.58(2Xl364.3tl089.3) = 14.58X3817,9 = 55,600 = 14.58(2Xl575.8+1280.4) = 14.58X4432.0 -- 64,700 -- 15.62(2Xl899.ltl523.7) -- 15.62X5321.9 = 83,400 M2 : 15.62(2X2444.82~1~04.l} : 15.62X6793.7: 106,200 Ml - 16.67{2X2794.1+1986.5) : l6.67X7574.7: 126,000 (13) MiO = 12.5(2Xl59.26+216.05) : 12.5X534.6 = 6,580 M7 = 12.5(2X389.59+483.12) : 12.5Xl262.3 = 15,800 • 12.5(2X618.19+751.44) : 12.5Xl987.84 = 24,800 = 12.5{2X811.87t996.93) • 12.5X2620.67: 32,700 M6 : 12.5(2Xl007.4+1238.7) : 12.5X3253.5: 40,600 M5 : 14.58(2Xl089.3~1364.3) M4 : 14.58(2Xl282.3+1575.8) : 14.58X4140.4: 60,370 M3 : 15.62(2Xl523.7+1899.l) : 15.62X4946.5: 77,500 M2 : 15.62(2Xl904.1~2444.8) Ml : 16.67(2Xl986.5~2794.l) : 16.67X6767.l: 112,600 hl9 M8 = 14.58X3442.9 = 50,200 = 15.62X6253. 98,000 ~oments in Girders BB' MlO : ll.11(3Xl59.26) • ll.11X477.8 • 5,320 M9 : ll.11(3X389.59) : ll.11Xll68.8: 13,000 MB : M7 - ll.11(3X811.87) : ll.11X2435.61: 27,100 M6 - ll.11(3Xl007.4) : ll.11X3022.2: 33,600 M5 : 12.96(3Xl089.3) : 1296X3267.9 - 42,300 M4 : 12.96(3Xl2824) M3 : 12.96(3Xl523.7) : 12.96X4571.l: 59,240 M2 = 13.89(3Xl904.l) : 13.89X5712.3: 79,400 Ml - 14.80(3Xl986.5) - 14.80X5959.5 = 88,400 ll.11(3X618.19) : ll.11Xl854.6: 20,600 = 12.96X3846.9 - 49,860 (14) ~ ~ "i /('=-//.// \ " ~ . -6).,e, J/1/0 JLJ/() J ..7,49 .;- Z,Z, 21 r .2.Z.ZI +/2.S-O ,&,49 ..789 -66.6.3 -r/47. /7 oCe Y/.33,.26 + 22,Z/ --61,,7 -t ZZ.ZI /llt:1 . 6'. r Z2.Z/ -1-/..3.3..Z 6 :I-ZZ.ZI ~?' -,.//3, 84 -I' l,,C, 5'0 t91'96 .,./.3~26 -;/3.3.26 -f /,C. ~() ., /4 7./7 -0e, .s- ro,,.e y -66. 63 -66.6'.3 o<:.6 S TO/e Y /VtJ. 6 .L oC:, -0,;/.s- .;.z 2 . 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'l'f) ~ "t ' Q) " It) ~ ~ l ~~ ~ ~ ~ l6 ~ ~ ~ ~ i ~ ~ - ~ ~~ ~ ~ ~~ n ~ " ~ ~ ~ ~ ~ \) ~ ~ ~ ~~ ~Iv \) ~ ~ ~ - ~ ~ ~ ~ ~ ~ ~ ~ ' -... I() ~ S#£,,,,,,,e /N 1'1rTo.P OF G/,e.O.c:,e .S/-0/ZY CHAPTE ,~ I II CANT I LEVEa LlETHOD SY1,II,m:r :n cx::. BEIJT r/ITH B.AYS OP mrnQU,\L WI DTH I n a na l y zing the ten-st ory be n t s hovn:1 in fi g . 3 , t he wri ter ha s found it very fe as ibl e to d etermine t h e c onst ants* fo r h or iz on ta l s h ea r, direc t st re ss, a n a moments i n t h e colunm s by the use of fig. 2. un i t loud a t A. The met h od of doing this is t o apply a j oint .A a n d t ake momen t a t a dist a nce 1 foot below Thus t h is is equilva l ent to as s u ming t h e poi n t of c ont r a- flexu re to b e 1 f t. below A and t h e t o t a l s he a r i n t ha t E:to r y i s t hen 1 poun d. The s olut i on of t he \}e nt. is ou tli n ed in g en e ra l as follows: As s umi n g a re a o f colunms e qu a l, t he moment of i n e r ti a of t he bent :[l(J)~+ 1(25))2 = 1412 ft:~ ' '\I /:>=. / # , ;f'JV:: / /6_ A--1= / ✓I# -A 't '- L t,,. ~ '-5 ~ ..8 ~ \} .4' .8' tJ./444 0,2/6.36 t::J./4/44 (:-a. 6'58 56) (- o. 5cJ,:;J76 J Gcu4z96J ~~ ~~ ~~ ~ '-:, 1-)'"' ~~ '<)'<} ~ ~ lei \) ") ~ 'v)~ c;;s ';$ ~ 0 ~ t:1./4/44 I;;) '"\ .a35T80 i-- i----- /6 ::t:J~.,.. ~ - /8 '-o ~ c;:i \!.,, "' ~~ ~" ~~ '-~ ~ 43.576'0 - - - /6 -'t:7 ~ (fi g . 2) (*s ee Sc hne i der ' s Pra c t ic al Willd Br a ci ng ) ( 28 ) .J.; t:J./4/44 Direct stress in columns .n,\ and A' = 25Xl = 0.01768 lb. 1412 Direct stress in columns B and B' 9Xl - 0.00636 = 1412 Vertical shear in AB = 0.01768 1 b1• tt " in BB' rt It in B'A' lb. -- (O.Ol7 6&Ju.oo636) = 0.02404 . = (01768-.00000) = 0.01768 lb. Moment in AB and A'B' = 0.01768X8: 0.14144 ft. lb. " BB' : 0.02404X9 = 0.21630 ft. lb. It II columns A and ;,, 1 II II 11 = 0.14144 ft. lb. Band B' : 0.35780 ft. lb. The direct stress in the roof beams ~re: AB= (l-U.14144) = -0.85856 lb. AB' : (0.85856-0.35780) : +0.50076 lb. B'A' : (U.50076-0.35780) : f0.14296 lb. With these constants found, the stresses in the columns and girders a r e re a dily solved. Thus the shear in column li. and A' of = 47 lbs .. for Band B' is 0.35780X The moment in column .A and i'i' is then= 7X47 = the tenth story is 0.14144X330 330 = 118 lbs. 329 ft. lbs., and for columns Band B' : 7Xll8 = 826 ft. lbs. direct stre ss in columns 1.i. and .A ' that for Band B' The = O.Ol768X2310 = 41 lbs., and = 0.00636X2310 = 15 lbs. The vertical she a r in girders .AB and .A'B' = the direct stress of column ii and A', while that for BB' = vertical shea r of AB plus the direct stre ss of column I3 which is 41-15 = 56 lbs. Simila ::'ly the other stresses a.re obt e.ined. Inspection of the resulting dire ct column stress ,~ s will imrned iatel y re veal the economy of this method . It will be noted that the column moments a.re quite less than that obtained by the ( 29) slope deflection method, thereby resulting a saving of material in regard to bracing and connections. This difference in the column moment and direct stresses are due wholly to the difference in assumption of the point of contraflexure at the columns and gi rd ers by these two methods. The resulting stres- ses will have closer agreement had the stiffnesses of the columns und g irde.rs as used in the slope deflection been t aken equal. (30) CHiiPTErt IV C.AlJTIL?:VEJ. METROD SYMME'.L~ICAL 3ENT wrrrH UPPE,;_{ STOlUES SET B.ACK The bent is shown in fig. 6, with the upper five stories of bent set back. The analysis for wind stresses for these stories were made simili a r to that mentioned in the preceding chapter. Since the same wind pressure as well as the same di- mension were used, the c onstant of fig. 2 thus applies here. The stresses for these upper five stories are taken directly from fig. 2. Now by the use of fig. 4 below, the direct stress at the sixth floor line will be calculated and carry down the columns immediately below as reactions. ~ '"'\ ~ V ~ ~ ~ ¾ /~= 6~12CJ /-/. # ~ ,l,, ~ ('. t~ ~ ' I '"' ~ ~ ~ ~ '\ ~ t\J "\ ~ "f -!) u✓d- .s;le,cY ~ ti/" (5-ij,C/. _!.., ._,,I ~ I\J ~(\i ~ ~ ' --!.I 6/q_ ,C/4pr L/,?e ,a~o'" ~/6~0" ,4 ,/3 /6~~-- A/ (fi g . 4) The vertical shear in the Gth. floo r beams are for AB, the direct stres s in column;_ at the 6th. floor line, minus the direct stress in the sc:1me colli.mn at the 6th. BtOl"'Y, or ( 1223-1025) : 198. For beam BB' it is equal to the a l geb r ti ic sum of the di- (32) rect stresses in the same columns at the 6th. story, or (1223+ 4391)-(1025-378} = 259, and that for A'B' is the same as AB The moment in the 6th. floor beams are 198X8 = 1584 ft. lbs. for beams AB and A'B'; and equal to 259X~ = 2331 ft. lbs. for beam BB'. These are the moments necessary to bal ance the moments in the 6th. story column. The constant factors are next determine for the bent below the 6th. floor as shown in fig. 5 below. 0\ ~. C L ~~ ~ ~ ~~ v ~ _g-' fi •' t7.//CJ48 Cl./~ '.t/,2/ C),//048 (-'7. 7.5224} (-t:J..:j 00..:S-S) to. 246'6'6} ~~ ii)-,; ~~ 'c;;sl-1 ~ ~~ 0. "\ :'\ ~'vi 0.d66'64 ~ ~ ~ ,4 0.1766'64 (-~9J/36J J" i~ ~ ~ ~ "l 1-1 ~~ . ~ ~ a/79/2 ~~ c--. I a2sl ~ ,. \\ '-> V ~ ~ 4' c/ 0.66844 f-Cl.06974) i~ t\ ~ ' ~ I\) I\) ''i:l . ~ ~ I;) ~'<) "()~ lsi ~ CJ.25/69 . !;;ii t:7./79/2 t:7.t:7 -9~ 25", (fig. 5) c. Assume a horizontal loa a of one pound applied at the 6th. floor line of figure above a nd assume point of inflection of the columns to be one foot below the 6th. f'loor line. The total wind load is then one poU11d and a unit overturnin1; moment in foot-pound. The center of g~avity coincides with the cent e r line as the bent is symmetrical. to be all equal. J ssume the cross-sectionul are a s of columns Then I :[1(9{°·+ 1(25)~f- 1(41)'}2 = 4774 ft: ( 33 ) Direct stress in columns A and .n ' = 1X25 = .00523 lb. 4774 r,· tt rt rt B and B' = 1X9 - 0.00188 lb. 4774 - ' n ff n. C and C' = 1X41 = .00858 lb. " 4774 Vertical shear in CA = 0.00858 lb. ,\ - fl tt tt AB tt It n BB' I = (.00858+.00523-.00188) = 0.01569 tt It rl rt ( .00858+.00523) = 0.01381 - ( .0085r.00523+.00188-.00188):: 0.01381 " .A' C' -- 0.00858 n B'A' Moment in CA and C'A' ff ll rt rt = SX0.00858 : 0.06864 ft. lb. AB and B'A' - BX.01374 = 0.11048 - ~X.01569 : 0.14121 BB' I.Toment s in columns C and C' = 0.06864 ' -- ( .06864+.11048) = 0.17912 ti " ti A and ', .ti IT " If B and B' = ( .11048+.14121) = 0.25169 Horizontal shear in columns C and C' = 0.06864 lb. fl a nd~• = 0.17912 tf II If A ti " J.f Band B' = 0.25169 Direct stresses in beam: AC= (1.0000-.06864) = 0.93136 AB - (.93136-.17912) : 0.75224 BB' = (.75224-.25169) = 0.50055 B'A' = (0.50055-0.25169} = 0.24886 .A'C' : {0.24886-0.17912) : 0.06974 The resulting stresses in the colur.:ms and girders of the bent are shown in fig. 6. The direct column stress in the first floor are for C and C' = ± 1625 lbs., for A and .A' = i 2214 lbs., and for B and B' = i. 796 lbs. (34) CHAPTEH V PORT.A L 1,1ETHOD SYMMET;.-UC).1 BENT WITH UPPE lt. STO~=t.IES SET B.ACK .B1 or symmetrical bent with equal bays, the Portal metbod possesses a unique feature in thnt the interi or columns of the bent will not be subjected to direct stress. Since the bent under consideration are of unequal width, the moment in the joints will not balance. Therefore, in order to overcome this complic a tion, use is made of the solution as proposed by Professor Smith. The scheme utilized is outline he r e in gene ra l as follows: The four columns with their girders which form the bent of the up:per five stories are assumed as three separate portals with a lo ud of 1/3 pound applied at each joint. The horizontal shear in each column of ea ch portal is e qual to one-sixth of a pound. The moment in each column a unit distance below the joint is one foot pound and is shovm in fig. 7 below. ,8 (fi g . 7) '.i:he direct st ress in colurnns .A and B of the first portal and the columns A' and .B' of the third p0rtal = lXO.3333 er 16 (36) 0.0208 lbs. The direct st1.. ess in columns B and B' of the se- cond portal - lX0.3333 or 0.0185 lbs. - 18 The vertical shear in the girders of each portal equals the di r ect stress in its respective columns which are 0.0208, 0.0185 and 0.0208 lbs. for AB, BB' and B'A' respectively. The moment for girders are BX.0208 = 0.1666 ft. lb. for AB; 9X.0185 = 0.1666, for girder BB'; and 8X0.0208 = 0.1666 for B' A'. These stresses are then combined al gebrically to be used as the constants for the upper five stories of the bent. The resulting stre sses thus combined are shown in fig. 8 below: l;S ~~ ~ /# ~ ,4 l;i g ~/667 (-C'.6'.33.5) ~ ',, ' \) ~~ ' l ~~ "\:i I;) 'I-.::, ~ CJ./6"6 7 (-<7./oo 7J ,,,,/ ~~ ~ ;; '<l \:I ~~ ~~\:I ~ ~ '-:i -..., 1-: Cl.3.53.3 /6 ~(7 ,,, ~/0'.'.'..t:7 ,, <;S t:J./66 T '1.J cJ./067 8/ fa5p,::uJ) ~~ ~~ . ~. '<l \::i ~l ~ , -!I i?.3.d'.3.S P./ 667 /6t.::,'~ (fig. 8) The direct stress for c clumns il and .ii.' remained the same; but for colrunn .B' = -0.0208+0.0185 = -0.0023 lb. and for colunm B: r0.0208-0.0185: r0.0023 lb. The direct stress for ;.B = 1.00-0.1667 = 0.8333 lb.; for BB' = 1.00-0.1667-0.3333: 5.000 lb.; and for B'A' : 1.00-0.1667 -0.3333-0.3333 = 0.1666 lb. The moments and vertic a l she u.cs for the beams remained the same after combining. The moment for columns Band B' • 0.1667,i, 0.1606 - 0.3333 and for columns .d and A' it is the same as bef0re. ( 37) Hav i ng these constants , the bent i s mechanic al l y sol ved and the resulting stresses are shown in f i g . 11 . Start ing \"Ji th the fifth story, a new set of constants are obtained by means of fig . 9 . The proceedure i n obtaining these constants is precise l y that outlined above . Jeferr i ng to f i g . 9 , the six colwnns with their girders are considered as five portals; therefore, the wind l oad at each portal is 0 . 2 lb . applied at the joint . The shear in the columns of all portal s are 0 . 10 lb . ~ "i ~ 0.,2# \~ I '" 1, I ~~ ' l ~ 0.2<¥ A ,4 C P . /t70 (-0./Cll:J) ~ ~' ~ 'li ,;i· ~ '-~ \\ ~ -!.I ,?./,:7 ,:;7. /0 - -/6~" '~~ .8 t::7-2.., 8 0./00 ( -O. /dtJ} ~ ~~ ~~ ~ t;:i ~ ~ <:i ~ ~, I:)~ ~~ '-li:I ~-~ ~ 1:1· ~ Cl./() /6~'' ~ 1!7./o ~ ~ ~ g ' az' LI' ~ ~ A' o. z"41' ?J,/cJO 0./C::,(J C'- / dO (-CJ. /CJO ) (::-~/OOJ (-t:J./eJCJ_) """\ ~ ~~ <:I'- ~'-:i ~ t;:i ~~ ~~ 0 '~ ..!.., ,:;7./0 - /8.:t:J~ 0./t:J \)~ ~~ ~ ~. ~- :!:; C}JC:> ---✓6~,, ~ ~1'< \) ~ ' -.!..,· ~',S ~ ~~ <:)t\j ~~ . 1:1 ~" '<I~ --- '7./0 ~~ ~ 0 a/o t?./o /o~t:7" (fig . 9) The direct stress in the columns of all porta l s except i ng the center one is 0 . 20 = 0 . 0125 lb . '.1.:he ai re ct stress for 16 co l umn B and ::3 ' = 0 . 20 = 0 . 01 11 lb . 18 The vertic al shear i n each port a l equals its respective col1Jmn direct stress which is 0 . 0111 lb . for BB' and 0 . 0125 lb . for tne other girders . These portals stresses are then combined by add i ng alge braically the results shown in the above figure from which the constants are obtained for fig . 10 below : (38) c/ ~ ~ ~';) ~ ~"-l 0 . /00 0./00 (-tJ. 7cJ() } I::: &>. S-cJ,:J ) ~ ~~ ~~ v l;) ""'I ~~ "-l '-I 'I:) ..., l\j 'I:) ~.-~ .., l,i~ 1· ~ - - /o .:o ~ /6 .:c7 ,,, C ;7 0./00 0./00 ~o. 3otJJ t;:-o . / cJt) J ~, 667,c/ L//?e f~ I;) ~ "'\ ~ ~ S:)~ .. ~\) \) ',) i-...~ ~ ~ J.i <;:l~ ~ .8" /o .:.tJ ~ a/oo aZoc, o..zoo - - /8:.o :._.>--J.--.8 ~ ',) c7. ZcJo o . ZOt) 0 . /00 ~ .~ ~ ~ 0./00 (-d. 900.} ~ ~ ,i - /o ~eJ ,_, - 4' c' (fig. 10) This algebriac summation is ca rried out exa ctly as be fore mentioned . Then before proceeding to apply these con- st ants to t h e bent, it is 11ecessa r ;r to calcul a te t he ove r turning moment a t t he floor of the sixth story from which the direct stre ss a t t h e floo r· line may be carrie d clown t be columns of t he floor i mmedia tely bel ow a s re a ction . r:i: h is is expl a ine d i n det a il in t he cantileve r met hod ~na will n ot be r epe a ted he re . The fina l stre ss e s a r e s h own in fi g . 11; the d i 1~ect st re ss at t h e fi ..:-s t floo r col1i.mns are fo r C ancl C' = :t 2366 1 bs . , for J. and A ' is± 1438 lbs., and fo.c B and B' it is ±424 lbs. ( 39) MODD·IED po ;_~T),L METHOD In order to retain one of the features of the Portal method, the writer has slightly modify the assumption to arrive at this desired result. The assumption as modified here being to assu.~e the shear as distributed in proportion to the unit volume of each bay in the bent and not equally among the bays irrespectively of its width. The basis for this assumption may be erratic in character, nevertheless, the st~esses obtained will not differ greatly from that of the Portal method. In the desi gn, the wind load is assumed to act on the tot al surface of the building in the windward side, but it must be r emembered that the building is not a surface, and that the wind could blow in any direction; thus under these circumstances, the above asr:mmption is justified. The procedure of calculation of the stresses is exa ctly as outlined for the Cantilever and Portal methods. The only step which need to be mentioned here will be the distribution of the columns shears. Thus for columns .A and A' the shear is 1/2 X 16/50 X 330 = 53 lbs. and for columns Band B' the shear is 1/2(18/50 + 16/50)330 = 112 lbs. from which the respective moments are 7X53 or 371 ft. lbs. and 7Xll2 or 784 ft. lbs. Proceeding from here, the shearing and compressive stress for the girders are obtained as mentioned in the previous chapters. The resulting stresses are shown in fig. 11-A. For this particular bent, there is ha r dly any variation in the results (41) obtained, but for bents with outer bayers whose width are proportionately greater than the center bay, the moments in the outer columns will be greater than that got by the original portal methoa and vice versa. (42) CHAPTEJ VI CONTINUOUS HAUNCH BE.AMS Haunched beams have recently come into extensive use because it offers two essential features not characterized by ordinary straight beams; namely, the artistic appearance it presents to the structure and the relative economy of material. This reduction of material at t~e center reduces the moment due to dead load of the girder, ~nd in effect to decrease the moment at the centec while increasing the negative moments at the supports. Since such beams occur in building frames v ery frequently, the writer in this connection have chosen such a continuous beam ovei-· fou1· supports to illustrate the possible economy in material as required by the calculations of the slope deflection method and that of the moment distribution method. The columns which suppose to form the frame work are purpo s ely omitted and substitute for it with simple supports. The general slope deflection equations used for the calcuhi t ion of' tlle moments are: ( 1) for unif OI'm load ings: M, = AE I, (:00, rCB-z. -3..c) +Cl!' = AE L (BG-2. +CB, -3,.c)-C]' T M2 T ana (2) fo2 concentrated loads, the term Fin the above equation is replaced by 2 D-P.€ and z D.zP-'?, where D, and Dz.are coefficients depending upon the posit i cm of the concentrated loa d P. A. B, and Care coefficients depending upon the proportions of the haunch and 1!1 is the fix end moment due to uniform loaa and = 1 W 12 See fig. 14, for the di me nsion and lo8din~ of t h is beam. (44) The moments are calculated by the aid of constants from Turneaure and Maurer's Concrete Construction Book. The complete solution by the moment distribution method is shown in table 6. Fig. (6) shows the 1/I curve and fig. (C) shows the moment diagram. The notations usea are as follows: L: length of the 1 curve for each span I - depth of beam at centroid of the 1/I curve ti= L/tfI X = distance from centroid of the 1 curve of I the span to: 1 d L z. 12 I = . bo rtJ x2 m • moment from moment diagram of fig. (C) - kern point of the haunched beam X: s = stiffness factor of the members = carry-over factor M'= fixed end moment The resulting moments obtained by the two methods show little variations from each other. Nevertheless, the moment distribution method will seem more feasible though not a marked saving of material will be affected. As the continuity of the structure becomes more complicated, this method will offer a considerable economy with corresponding reduction in labor of computation. (47) CH.APTE~1. VI I CONCLUSION As the maximum moment is the criterion in determining the colu.11n cross-sections, the maximum economy is obtained when the bending moment at each end of the column is equal; i.e. when the point of inflection falls at the center of the columns. Since this is one of the assumptions upon which these approximate methods are based, it will be seen that a greater economy is obtained by using these methods for wind analysis than by using the slope deflection method. From actual ex- p e riments conducted in tall buildings for the location of the contraflexure point of columns, it was found that this assumption holds t :cue f,:;r all columns in the mid-storios. This point of zero moment being less than the mid-hei ght of the column as measu1~ea from the bottom for those columns of the stories close to the top of the building, and is sufficiently greater than the mid-length of the columns for those at the base of the building. J!•o1" the 5 irders, the approxima te methods likewise offer greater economy a s these inflection points are also taken to be at the cent e r. For the middle bay due to symmetry, this condition a lso holds for the deflection method, but as these moment must hold the colwnn moments of any joint in equilibriwn, it will be greater t j1an those obtained by either approximate met r1od. Since the direct stre sses in the c olwnns are a function (50) of its moments existing, it is also gx•eater than those obtained by the approximate methods. Solving the same bent by assuming all stiffnesses of the columns and girders to be the same and cross-sectional areas of the columns as equal, it was found that the resulting column moments and direct stress are still greater than those obtained by the approximate method, thus showing that it is less economical for the simple reason mentioned above. ls to the comparative economy of the two approximate methods, the portal method will be the more economical since it distributes the greater part of the burden of resisting the external moments to the outside columns, and all direct stresses are then taKen by the outside columns for· symmetrical bent with bays of equal width. For this particular bent analyzed, the direct stresses obtained by the portal method shows quite a distinct difference from those obtained by the cantilev~r method. If the outer bays haa been chosen very wide relative to the center bay, which condition generally occurs, there is practically no economical advantage over the cantilever method. Precisely, such theoretical economy is a mar~ed feature as the height of the building increases. It seems to the writer that the Po~tal method will be the most favor~ble for wind stress analysis in practically all cases for symmetrical bents, because of the economy in labor and its underlying assumption affords a greater possibility of economy of material as well. This method will meet with complications when the bent becomes more perplex especially if it is treated for architectural value. Therefore, it must be concluded, that in order to arrive at the maximum economy, arrangement of the building space as well a s the method of' solution should be carefully considered. :B,urthermore, with such an impossibility of knowing just exactly wha t wind load should be a ssumed, it remains doubtful as to what degree of precision such us supposely exa ct solution will yield. (52) BIBLIOGR..i'l PHY 1. Cross and Morgan: Continuous Frames of d.einforcea Concrete; 2. Fleming, 3.obins: John Wiley and Sons Wind stresses in Tall Buildings John Wiley ana Sons 3. Johnson, Bryan, and Turneaure: Structure; 4. Schneider, H. E. Modern Framed John Wiley and Sons Practic a l Wind Bracing; Juffalo, New York 5. Spurr, Henry: Wind Bracing; McGraw-Hill Co. 6. Turneaure and Maurer: Principles of a.einforced Concrete Construction; John Wiley and Sons 7. Wilson and Maney: Wind Stresses in Steel Frames of Office Building; Bulletin 80 of the University of Illinois Engineering Experiment Station. 8. Wilson, aichart, and Weiss: Analysis of Statically Indeterminate structures by Slope Deflection Method; Bulletin 108 of the Eng. Exp. Sta., University of Illinois. (53)