Y vv— _ r 'r—_r L‘ - _ 4“ 1 ‘ V ‘ o l . . :49“ 3.5, ’1 ULTIMATE DESIGN OF REINFORCED" .- CONCRETE” Thesis .fbrxfh‘e Degree-‘cjf'LMfs; ‘- ' MICHIGAN STATE “COLLEGE ' ‘ Lyte Clintonpavis __ 1 ; - . 195.3.- THESlS This is to certifg that the thesis entitled Ultimate Design of Reinforced Concrete presented In] Lyle Clinton Davis has been accepted towards fulfillment of the requirements [or M.S. C.E. degree in M ajnr prolesllnr [Date 12-3-1955 i Ultimate Design of Reinforced Concrete Py Lyle Clinton vais A Thesis Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil Engineering 1953 Introduction . Derivations of C'I'aphs o o o 0 Examples . . . Conclusions . Bibliography . “ACIE CF COFLTTT3 \.)J ACKNOWLEDGMENT The author wishes to express his sincere gratitude to Dr. Richard H. J. Pian for his patience and kind understanding of the student and his problems. This paper was prepared under his inspiration and supervision. INTRODUCflON Although present day specifications emphasize stresses at working loads (theory of elasticity) and present day calculations pretend to calculate these stresses, laboratory tests of reinforced concrete beams show that actual defor- mations and stresses at working loads only faintly resemble our calculated values. The theory of elasticity, as applied to design of sections, is too inflexible and inaccurate to be entirely satisfactory and give the most efficient use of materials. Actual stresses under working loads are affected by so many conditions that they are quite indeterminate. It has been provei that shrinkage and plastic flow entirely upset stress computations based on the elastic theory but they have little effect on ulinate strength because before failure the strains are great enough to cause redistribution of stress. There are several advantages that the ultimate design theory has over the elastic theory but onlv a few of the more irportant ones will be listei here. (1} The applica— tion of the ultimate design theory is much simpler, easier to use ani is bette supported by tests tean the elastic theorv. (2) Due to local bucblinr, load and stress are not always directly related, therefore it would be more rl‘ consistent to apply the safety factors to the load (ultimate design theory) rather than the stress (elastic design theory). The useful strensth of each member would have approximately the same relationship to the strength of the entire structure. The load factors can be varied to comply with the accuracy of predicting the applied loads. Dead loads are definite and constant, therefore a small safety factor may be used. T. Y. Lin and R. C. Reese suggested the value of 1.2. Live loads being variable and highly indeterminate, demand a higher factor of safety. R. C. Reese sucgested the value of 2.4 and T. Y. Lin, the value of 2.0. This paper deals only with the flexural computations of the various theories. DVRIVATTOV~ OF THE ULTIMATE “HEOQY The ultinate theory of reinforced concrete as ore- sentei by C. 3. Whitney is presentei here. The ieriva- tions will be limited to flat slabs and rectangular beans with tension steel. 40a? \3000 2 000 U/nf 5/7'6’56 , IE; Adi/DEW 57/0. \ b D nay 0p“! 4%”3 4mxt “ll/'7‘ 57/8/27 I.” Covert/8,5, [fr/oer)”- Fig. l: Idealized 3tress~Strain Curve for concrete cylinder. ‘. "Dlastic theory of Reinforced Concrete Desicn" by Charles 3. Whitney, Trans. Am. Soc. of Civ. Vnprs., Vol. 1C7, Dec. 1942, pp. 951-396. ' The stress distribution in a concrete been at failure may be assused to have the same shape as the cylinder stressmstrain curve shown in Pig. 1. The total compression, C, is the area bounded by the curve and the line of action of C passes thrcurh the center of gravity of this area. For sinrlicity, the actual irregular stress bloc” is re- placed by a rectangular stress blocV of equal size. If the width of the rectangular stress block is C.DF f and, the depth ecual to a the location of the center of Travity of this rectangle will correspond closely with that of the actual area. The rectangle is an esuivelent replace— -ment°and does not mean that the theory is based on a rec— taneular stress distribu+ion. /: as:' C 3 nt» .1E%j=oavyfaé 3 C I 7. r734”; Fig. 2: Stress distribution on the crossm section of a rectangular beamo It is worthy to note in Fig. 2 that the denth a in less than the distance to the neutral axis kd, and bears no definite relation to it. The equilibrum of the internal forces gives: T - C therefore f' a.__.‘gs,;£_ . -------------------------------- 1 0.85bQ in which: A5 = area of tensile steel; Q, I yield point stress of steel; b = width of beam; fl; = standard con- crete cylinder strength. UNDER-REINFORCED DESIGN. mhe ultimate resisting moment of the beam as controlled by steel failure can be written: (d- Act; )--2 erc-AJr, (d-._g_)-A5f 3‘ lflOBfi also: M - Cc - 0.85 fish (a - mg.) ----------------- 3 0.95 rc’ b (a2 - 2ad) = - 2M .................. 4 ea - 28/3. “I” dz = - 2M + at ------------ C} 0.85171: a - d = 1' d 1 ~ , 2M, '7 ------------- 6 J ,0 Sfc’b Since a must be less than i, use the negative sign with the square root term. 47 a 2. 3RM ‘7‘ "[1 'v/1"*§267r* """"""""""""" 7 The foregoing exnressions are relatively simple and are inuecendent of n, (the modulus ratio). It should be noted that all of the equations aooly only to ultinate load conditions and do not nredict stresses under workins loads. The required steel area is A : '____7"T~_ ................................ Q 5 C f3 / TALAlCEU 0E511V: It has been shown by bean tests resorted by Slater and Lyse that for balanced desl*n, a equals 0.537d and c = 0.732d. Suhstitutina these values in equation 2, I M = _%L_ bdz -—-~---------—----------------- 10 and M = Tc = Q7 A5 (0.732dl ------------------- 11 or 0.7?2dfy «5 r .?a_ bd ---------- --~-~----- l2 tierefore I A5 = 0.4% if. bd ........................ 13 -J The following Graphs were drawn to sinnlifv the com- outations involved in checking or desisninq some types of reinforced concrete members. In design nroblems, fig, f, and the bendine noment includinq deed load with nrooer safety factors annlied are usually known. The followirc curves are based on the nroceedin: derivations. mhe curves in Pic. 3 were drawn usinJ equation 10. uein: all one way slabq are designed as rectangular beams 12 inches wide, the only variable which M is dependent upon are fg and dz. If we draw a family of curves with a value of’ig constant for each curve, M is denendent on d2. This nlots as a straivht line on losarithmic pacer. The curves in fig. 4-7 were clotted usine equation 13. A5 is desendent uoon hree variables (b - 12 in.). One variable £5 was eliminated by drawing four sraohs, one for each of four connon values of fly. If a value of f, is encountered other than one of the values used, the value of AJ-may be interoolated from the two adjacent sraohs. The curves of fie. 8 were drawn usins equation 4, - 2 , letting l = 2 ad -d . The curves of fie. 9 were drawn \ usins the earlier substitution of N - 2 ad - d . The curves of fis. 10-13 were drawn usinz oouation l. The curves nertaininq to beams are very similar to those of one way slabs excent the width of the beam b is a variable. The curves of tie. 14 were drawn usine equa- tion 1C. The curves of fie. lF—ld were drawn usinfi equa— tion 11. n The curves of fie. 19-26 were drawn using equation n, and the curves of fis. 27-30 were drawn usins equation Q. To desisn one-way slabs using balanced design orocedurez (enter fix. 3 with the bending moment and n; to obtain, d, ‘the effective death of the slab. With d and f; enter either fig. 4, E, 6 or 7 which ever corresnonds to the siven £3 and set, A5, the area of the reinforciny steel. To design one-way slabs usine under-reinforced desien procedure; enter fig. 3 with the bendinv moment and f to set a value N. Fow enter fis. 9 to set values of d and a. In some cases there may be a choice of values for these quantities. Enter fie. C, ll, 12 or 1? cor- .0 J reseoniirs to the siven value o“ 39, with -é and a, g 6420):, C"‘g-")1(§Q to E‘PT’l'JQ 2+, vqlzfr f". If.” ’55.. Tleo 4'“/\r:1 "nef- eeononical set of values For the final desiyr. To desien rectansular beans usirs balanced desiqn procedure: enter fly. 14 with the herdins moment and Q; + l ~12 ‘ -~ n ~ so obtain 3 . Compute values of d wito assumed Values of b. Now enter fig. 15, 15, 17 or 13 with the bendirs moment and values of d to set values of A5. Use the most economical set. To desiun rectangular beams using under—reinforced .. I design nrocedure; enter fie. 10-25 serendins on fit, with 2 9 the bendins moment and hd that you have choser, to set n . - . -;- cl will be a difficult value to arrive at until experience is gained in this rroceiure. The cross section lfiill be very nearly the same as that cotten by elastic fiesicn so evrerience in other design oroceiures will be ! ‘halcful. Keen in mind when nickins bd2 that the under— I¥9inforced desien will dictate failure of the reirterce- " ‘— “'“' ‘-—K-mum ll. ment. Therefore a theoretical limit would be a steel beam incased in concrete. The steel varies inversely as the concrete area, so the desisner should make an attemnt at an ecomonical balance between the two materials brovidine there are no other restrictions. Compute c from the values of _%. and 5. Now enter fig. 27, 23, 29 or 30 with the beniins moment and c to obtain the reinforcinv steel area A3. 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I' a t' L L » .d‘.\h -YL .. .- .. 1 L: U .. . . - ,u H -L . L — - . . L . I .-. . IL .. '1 . . . . Fly. 2/ 7 Rec _‘- ‘ lllv-< . ——.y<- g- ... ..Q a ... I. 02‘ Jill...‘ I! ‘ 7mm 3/ £79.22 .' Ham 0 70 OM 0.60 0.5:! 0,510 1:. a: '- ...." I — WW”—— ..fl. Co‘- 750,075 L t +—' -—. -....- O l .1-.--“ . . . 1|. _ L L L L. . *’ ‘L” I < L L L «-4. _- ‘0 9:0 1- L — L L L l!' ”-LL m L 1‘ I .-.L1-. \ 0-6.. I ....... .| .0 .. ...‘. m. u _ . D 1A4! f u. JLIAY. . o nLl -. F L i u: 50,000 V. .44 -..L..+_ ,_ —+ /J‘ M m . _’44 .le .u: t .L. _L u ' ... EXAVVLE PROQLVVS Two illustrative Droblems are includei, one way slabs and rectangular beams, comparitive qualities of the elastic theory, involvirq to summarize the and the ultimate theory with balancei’desi;n and under-reinforced design. PROBL?M l - Design a one—way slab with the given conditions: f; : 3,000 psi, f5_ = 20,000 091., r, 40,000 091.. clear snan = 19' O", L. L. = 200 lbs./ft.,+-T =.%%§:_, - M ’ LY%EH, ani safety factors 0f 1.2 ari 2.4. A. I"lastic Design 7.5 in. slab. 0. 1, = 7.5 x 1? x Lth- x 150 = 93.7 lbs./ft. 12 L. 1. 200 " m. L. 203.7 lhs./ft. h ,2 +3t- L{8*‘LLLI;EL.LL¥§LL- 70,:00 iJL-lbs. (1:) 2 -fl = LQEQELZLLLlEl_Lngll = 115,5CC in.-lbs. (11) 32:— -LLX‘I“ a --glj..:. <00) 400' SO 0 1‘ no Kb 7,3) (12 A = 4,10 ir., use a . h.‘ ir. A = v = LLLL1115.60r> .- L = 1.03 fisii (20,000) (0.363 737F5 55- (7OLFOOV 0.706 sq. in. ’K?0,0007_Y0.865} (6.57 B. Ultinate theory — Ralanced flesicn F.“ in. slab 2 0. L. 5.5 x 12 x 1/12 x 1s0 x 1.2 = 82.? lbs./ft. I L. L. = 200 x 2.4 . £39__ " m. L. = 662.5 lbs./ft. r- \ 2 4-M = £“49o‘l (101 = 12,700 ft.-lbs. (16) (:59 a (long a _.\1‘[ g k... --°.- __'-+ ./¢_ 2 13,500 ft.-lbs. <11) for negative moment: from fiv. 3: v . 1R,EOO ft.—lbs., fig - 3,000 psi. d s 4.5 in. use d = h.5 in. from fig. F; d = 4.5 in., f; - 3,000 psi. 33: 1.85 SO. in. for positive moment: +2 eliminate the necessity of keeninq 1.85 so. in. of steel or varying the denth d to maintain a balanced desicn in view of the smaller moment, it woulfl be more nractical to revert to unfier- reinforcei design. from f-.. 8; M = 12,700 ft.-lbs., f = 3,CQO psi. V = 9.95 from fig. 9; N a 9.Q6, fl - 4.5 in. from fig. ll; a = 1.20, fb = 0,000 psi. C. Ultimate theory - unéer-reintorcei fiesirn 7.5 in. slab 0. L. = 7.5 x 12 x 1/122 x 150 x 1.2 = 112.: 1bs./rt. 1. L. : 200 x 2.4 = £99-- " T. L. = $92.? lbs./ft. 2 -+M - F92.5)_(191LL a 13,400 ft.-lbs. (16) 2 (:0? Cl (10\ Q a d ...": ‘aé—- : 4C f. ,. - b . ‘7113. ..A— ]_,, O Lt l s .. NY A for negative moment: from fi:. 8; v = 19,:00 ft.-lbs., fig = 3,000 nsi. N = 1F.30 from $13. 9; r - 15.20 d : 0 q 7, g E 6 in. /’ "’ a = 0.89, 1.0 , 1.1“, 1.10, l.bfi 21. from fi7. 11;13= 0.63, 0.78, 0.01, l.CO, 1.1] K2. in. rrohahly use d = 6.: in. and A5 = 1.00 so. in. For nositive moment: from far. a- v = 13,400 Ft.-lbs., :5 = 3,000 nsi. 9 N - 10.91 A brief summary of the results wouli be as follows: Elastic theory Ultimate theory-balancei design N " under-reinforcei design 0.6? 1.00 4?. +A5 “£53 4 0.71 1.02 6.5 Co 9 1001!: 40: ("A U1 PROBLEM ll - Uesiqn a simply supnorted rectangular beam with the given conditions: fg = 3,000 psi., f3 = 20,000 psi., 19 2 40,000 081., clear span = 18' A. Elastic design D. L. = 1121 ’ 251 LIFO) 144 - 0", L. L. = 2,940 lbs./ft., 8.—, and safety factors of 1.2 and 2.4. = 310 lbs./ft. =3250 lbs./ft. 1,580,000 in.-1bs. biz 2' -.1”. - laiqpfiooo = 67C”) K 240 use b = 12", i = 24" A : _ELL 1 ‘90 0C0 =3.80 so. ir. 3 £311 =720, 0007 (0. a6 7 (217 ' 2. Ultimate theory - balancefi desian 0. L. = 310 x 1.2 = 270 lbs./ft. L. L. 3 2940 x 2.4 =Zcfig " T. L. =7430 lbs./ft. 2 =LLZEEQlL11L1LLL = 101,000 ft.-lbs. (a \/ - 4 :‘QV ., 5‘ 2‘ ' h . T._. . L . ‘ “-3-! — —- - “-4 -~ -L _ _-'__-a_'_.-~.5L_t L- L... L __-_ L 44, from fig. 14; M - 301,000 tt.~lhs., g; = 3,000 nsi. bi - 3610 in. use b = 10 in., d = 20 in. from fie. 16; M - 301,000 ft.-1bs., i = 20 in. A3 = 6.17 so. in. C. Ultimate theory - under-reinforced desimn 0. L. = 610 x 1.2 = 1'70 lbs./ft. T. L. = 2940 x 2.4 = 70:9" H T: L° = 7430 1bs./ft, 2 M = LiliiQiLililLLL 201,000 ft.-lbs. 2. try b = 12 10-. d = 24 in.; b1 = 6910 1n.3 from fir. 21; M = 501,000 ft.-1bs., bi? = 2 = .9; LIL 0 14 c = (0.804) (24) = 21.2 in. from fie. 25; M = 301,000 ft.-1bs., c - 21.2 in. As-z 4.26 so. in. A brief summary of the results are as follows: As Vlastic fiesiyn 3.90 U1timate theory - balancel flesian 6.17 " " under-reinforcei design 4.26 21% 200 233 C OT‘C LUSIQ‘CS The ultimate theory, under-reinforced procedure tends to give approximately the same cross-sectional deminsions as the elastic theory but the balanced design nrocedure gives a smaller beam section with more steel. The econo- mical values of the materials involved indicate that the balanced desiqn procedure is often more costly than the others when limited to an investigation of the member only. If we take into consideration the entire structure, other factors are involved. The reduction in dead load thrcuqh- out the structure, due to a smaller d in slabs and beams, will allow smaller supporting beams, columns and footineL; thus a possible reduction in the total exnenditure of th (D D; ( structure. If a tall building is involved, a small-r will shorten the floor to floor heiaht and consequently the height of the buildirg. Also there is an advantafe in using less steel than called for by balanced desimr, in that incipient failure will be noticed, by increased de- flection and cracking of the concrete, in time to reduce the load or strengthen the beam. One way slabs are desisned as rectangular beams to carry the enclied bending moment. Unusually heavy loads for short scans require a check for shear stresses. Slabs .l|l.l.|I. \ 1 iii: ’lillllll {1115: with very long spans should be checked for deflection. Ultimate design theory has been used in Europe more extensively than in the United States. Although our city buildina codes are quite comprehensive, they vary a great deal and are often based on imperical formulas. The design procedure must keep pace with the improvements found in the methods of evaluating service loads. The graphs are presented to improve the usefulness of the ultimate design theory. With the proper size of scales and arrangement of linework, Jhey are as accurate and much easier to use than tables and slide rule compu- tations. The ultimate design theory of reinforced concrete as used here is not a perfected standard but is a good basis for research leading toward modernization of standard practice. An attempt was made to ieve10p a practical design procedure that was consistent with all of the per- tinent facts that have been discovered regarding the nature of the materials, and not with a few incomplete character- istics selected for convenience. A7_ EI?LIOG?APHY H. J. Allcock and J. Reginald Jones, "The Nomosram". H. J. Cowan, "The Design for Ultimate Strength of Rein- forced Concrete T - Beams", Civil anineering and Public Works Review, August and September, 1951. q, J- COVE“ and R- C. Reese, "Discussion of a paper by R. C. Reese; Practical Design at Ultimate Loads", Journal of the American Concrete Institute, V01. 24, December 1952, Part 2, pp. 831-893. T. Y. Lin, "Load Factors in Ultimate Design of Reinforced Concrete", Journal of the American Concrete Institute, Proc. Vol. 48, June 1952, pp. 881-900. V. T. Mavis, "The Construction of Nomosraphic Charts". Dean Peabody Jr., "The Desimn of Reinforced Concrete Struc- tures", John Wiley and Sons, New York, 1950. Portland Cement Association, Concrete Information, Struc- tural and Railways Epreau, "Ultimate Design of Reinforced Van Den Brook,"Theory of Limit Design". lastic Theory of Reinforced Concrete 3001 p. 25 (3 V 1 ty of Civil Pnsineers, -126. Transactions,