x I | ‘ ! \lll } i l1 1 'l ‘! l } I" N l _x_: h—x 1 GUN TH? DESIGN QF CERQULAR REiNi:0RCED CGMCRETE FQOTWG ON ELASTIC FOUNfiATlONS T’hasis for flu Degree 3% M. S. MiCHEGAN STATE COLLEGE YiunaYuan, Haung 1,949 THESIS This is In ('f‘l‘tifl] that tlw thufis¢unnhd THE DESIGN OF CIRCULAR REINFORCED CONCRETE FOOTING ON ELASTIC FOUNDATIONS [rl‘c’St‘llh‘¢] lull Bun-Yuan Huang 11m lu-vn m‘rc‘plw] lmmnlx fulHHIm-nl HI 1110' l'c‘quil‘t‘lm‘nlx lnl‘ M.S. (it‘tJH‘d‘ in CoEo ‘liiit rl‘ lbl‘c th'SMu‘ lhm‘ 5/16/49 M-T: .7 TEE D3313: F CIZC LAB ELL FC‘CED FOOTIIG OE ELASTIC FOUIDATIOIS 3y _. VTTM' _Lv- YIUI lo \T‘f‘i I‘Tj .‘J Submitted to the School of Graduate Studiee State College of Agriculture and Applied in partial fulfillment of the require for the degree of “Psurvm'1 *1 7cm“ LnL‘lLR 0: LC; SCIE Department of 1940 \ fivil Engineering of Richigan Science ments Tl ’THESlS O flv’r- 'Il\- r . r: ’W "v—fisV—v‘ '1 A DAL». .TLLDJL i-..‘.ai-. TD The writer is greatly indebted to Professor C. L. Allen, Head of Civil Engineering Department of Kichigan State College for his guidance and direction in connection with the study and also to Dr. Richard H. J. Pian for his valuable advices. Professor Allen is going to retire at the end of June after his thirty years of untirin: service at Lichigan State College. Fortunately the writer is about to finish his studies toward Easter of Science degree at the same time and is proud of being Professor Allen's "The Student Behind the Closing Gate" -— a Chinese proverb which means the last student before the retirement of his mastering professor. I. II. III. IV . VII. TABLE OF COETEKTS IZ‘TTRO DUCTION FUR ALLITLL COISIDERATIONS 1. Basic Assumptions 2. Method of Superposition 3. Eotations 4. Basic Differential Es uations for Homents and "w Equilibrium BASI C C-"“’ FOOTIE; CARRV KG A VELTICAL LOAD RESTIIG OE ELASTIC SOIL (l) 1. Energy Hethod of Determining Deflection Curve 2. Shape of Pressure Distribution Curve FOOL ILG CARRIII G A VERIICfi LOAD RESTING n ELASTIC SOIL (2) 1. Load Moments 2. Actual Moments FOOTIEG CARQIIIG A VEIII CAL LOAD vvsmrvr or ELASTIC J COIECTYTRIC CIRCLES CF PIL E a 1. Distribution of Pressure on Piles 2. Moment Analysis FOOTIIG CARRYIEG A EOLEIT LOAD RESTIIG OE ETASTIC SOIL 1. Pressure Distribution ’3 2. PrOposition o: In troduci r.g a Symmetrical Distribution of Pressure in Practical Design to Lake Unsymmetrical Case Symmetrical 3. moment Analysis VIII. ’7 I410 FOOTILG CARRIILG A EOLEIT LOAD 3::TILG iI ELASTIC CC} CELI IC CIRCLES OF PILLS 11-..;le o XL ELLAL-I‘LES 1. Foot in3 Carryin3 a Vertical Load and a Ioment Load Restin3 on Elastic Soil 2. Footin3 Carrgrin3 a Vertical Load and a Moment oad Resting on Elastic P11 0) OJ C“ICLUSIU AP. “Fri 1 TV 7» fl ‘ :-n‘;f‘ U&.¢'1-L.LJ -U- L UHJL‘JL- q- -.-r~~—I . —r1‘n-\‘~1 Arr‘ DICE.) l. Derivation of D for Reinforced Concrete Section 2 . Bibl io 3rap1: y I. IKTRODUCTIOH In this paper is presented a practical method for desi'rir" a reinforced concrete footin3 of circular shape restin3 on elastic foundation based on t elasticity. Claarts as the time saver in desifn worn, for CL: solving internal moments under ifferent loaning and foun- dation conditions are prepared. The most reasonable srm pe of a single column looting is supposed to be circz ml r, as the foundation pressure under such a footing is symmetrical with respect to its center in every direction when its supportin3 load is central, which is the most usual ca e happ er aed to a single footing. E 3h structures like chimney, centrally supported elevated Later tank, silo, and etc., which are circular in stage and sensitive to unequal settlement in foundation are specially suitable to have a circular foot- ing underneath. Tie conventional des i3n of a sinfile reinforced concrete footing is of square, rectan3ularo Holy onal in shape and the foundation pressure is assumed to be uniform. The moments are in vestijated at several nair of parallel sections at which bendin3 is assumed to occur. (The method of analyzin3 moment in such a way is sunposed to be based on the Bulletin Lo. 67 of the University of Illinois Eng. Exp . St. 19l3 on tLe title of 'Reinforced Concrete Wall Footin3s and Column Footings' by Arthur F. Talbot.) The recent developments in theory of elasticity and soil nanics prove that those assumptions mentioned acove are not near from being true. Structural engineers are very much interested in the ae ign of a circular reinforced concrete footing on U] elastic foundation , but high mathematics which is essential for solvin; the problems concerning the theory of elasticity is too much involved and has kept many of them from being 0 F43 smiliar with those theoretical methods. In this paper, high mathematics is eliminated as much as possible in de- rivation of formulas and in the case of footin3 carrying a moment load a symmetrical desi3nin3 load is proposed to re- place the actual unsymmetrical one without sacrifice of the reality which makes the complicated problems simple and easy to solve without using high mathematics. Charts for solvin: moment in circular footings are prepared in this paper coverin3 the usual loading and foundation conditions. The cut and trial procedure is in- evitable in solving every statically indeterminate problem 0 so it is not the inconvenience particularly presented in his paper. 1. BASIC (l) (2) (3) (4) (7) K)! II. FUZTD.’.IZZ'T-._- CCITF'T 31“.:IC'=“3 ASBUIETIOES: n this paper the followin3 assump- tions are made: The t1-ickne es of the footin: is assumed to be small as compared with its radius so that the gen neral theories of thin plates are applicable. The deflection of the footing is assu*;1e d to be small so that the ener3y method of determinin3 deflection curve is valid. The center part of the footing directly underneath the column or column capital is assumed to be ab- solutely rigid. F5 The intensity of reaction 0 the foundation, soil or pile, at each point of the bottom surface of the footin; is assumed to be proportional to the deflection of the footing at tlat point. The Possion's Ratio for concrete is assumed to oe constant and equal to 0.2. The wei3ht of the footin3 is small as compared with its supporting load, and it is a sumed to be U) uniformly distributed over the base area of the column or column capital in moment ar alysis. Those assumptions made in the general theories of reinforced concrete are also to be made in this paper. The particular case of 10? din3 of a circular footin3 is such that it carries a downward load from column to which it supports and up.Jard pressures from foun- dation on which t rests. Both of ts surfaces are loaded and its ed3e is free in all directions. But the basic ed3 e concition upon m 1 ch the analysis of moments is based in this paper, is simple supported. In each case, the moments, radial and tangential, due to foundation pressure under the oas ic ed3 e conditioz are deducted respectively from those due to column load under the same edge condition which is equal and opposite to and colinear with the re- sultant of foundation pressure, and the net moments are those under actual free edge condition. In ase the column carries a moment load besides a downward load (such as pro- duced by nind load on a chimney footing or eccentricity in a building column footin3), an equivalent symmetrical up- ward foundation pressure is assumed to apply at the bottom surface of the footing and an go ddit ional down: ward column load at tOp, so that it can maintain equilibrium. he mom- ments in the Moo i13 due to moment load are thus computed as in previous way and added to those from real downward column load In this paper moments due to downward column load under basic edge condition are called BASIC I KEITS and those due to foundation pressures are called LOAD hC"“"o Land-x . Thus Actual Mement = Basic Moment - Load Homent (In any case) or- Mr = 1.1} _ 1:; (Actual radial moment) and Mt lfi - Kg (actual tangential moment) \rl KOTATIOLS: " hr Tr- ‘- L,L1,L2 The following notations are used through- out in this paper. Constants in the equations for deflection curve of circular plate Steel area in reinforced concrete section Integration constants Flexural rigidity of section Lbdulus of elasticity of concrete, etc. Homent of inertia General moment constant in standard notation Radial moment constant in circular plate Tangential moment constant in circular plate} Total load in each concentric pile circle,or total upward pressure in elastic soil under Hement in general Radial moment in circular plate (actual) Tangential moment in circular plate (actual) .asic radial moment in circular plate (see definition in 2) Basic tangential moment in circular plate (see definition in 2) Load radial moment in circular plate (see definition in 2) Load tangential moment in circular plate (see definition in 2) Origin meow U) U) l’ 2 C1 1’ 2 <1C3 I l I"! N :4 .' r. L Total column load Unit Snearing stress in circular plate Radius of circular plate Areas Strain energy Volume or total shearing force in standard notation of R.C. Rectangular axis Radius of column or column capital Width of section in standard notation of R.C. Eccentricity Fibre stress in concrete Stress in steel Depth of section Constants in standard notations of R " Lbdulus of reaction of foundation Poisson's ratio Unit load or pressure -adius of circular section Ratio of pressures at rim and center of footing Unit shearing stress in standard notations of Deflection Rectangular coordinates Radius of concentric pile circle .?>>m?-Q.Q‘:} ? ratio a/R Ratio r/R Ratio z/R Angle in polar coordinates in horizontal plane Angle in polar coordinates in vertical plane Constants 9 4 - - up ’9' m‘__‘-__ ,‘T ‘4‘ V7 9,41 pgl—A‘ 7.1 ‘4“ — -\ , ‘w‘T‘ fl, ”4'137?‘ V' :3 I“ . L)" 433‘s IC D :M‘ .4- .44.. J..4..’LLJ QWu-JL- 4.»); D b‘C-J- ~'-L-‘:E--J-; *L‘D H‘JV'L'J"“ J". . -0 — ~‘ 5 Figure 2.1a shows a diamet- rical section of a circular /f - plate under bending and /- ’ j#¢;. F gune 2.lb an elcwentary , ‘ % segment of it with forces ,n' “ marks” "’ ' The differential eq ation Fig. ?.l (a) for moment per unit length arc: Mr$gg;.dr '+13.4r \ . . Mr - D(d¢4m 3: -D(d2w4m dw) gr‘\ ’ll" '7" ”(pl , \ >Q¢,J:,L dr r er r dr 2 ' 3') .I- I Eligizqtyiw = (2.1) _ 3‘ _ .__--..., A Mt - D(_‘£+ Ill %£_P) : r' .l1 0 ‘ - D 1(dw ¢g dgw) r‘"" " "' r dr r dr2 ---- (2.2) Cons de in 3 t1:e ele 3 nt aboa in Fir equilibrium, taking moments about the center 0 and neglect- in: the terms containing tize small quan tities of r‘v order, the following relation is obtained: hr. 4 5.1.“ r - lit 4 Qr = C ---— (2.3) ubstituting the values of Ir an d Mt in equa tion (2.1) and (2.2) the basic dif F43 erer tial for equilibrium becomes: D Ql. (2334 2’)* dr dr r Q l '3‘. +5 .- $3734 5‘6 .5 B are V I ) Th deflection w in the above equation is con- ive when it is downward. c"?- sidered to be posi For a plate of constant thickness the Flexural Rigidity D is constant, and equation (2.4) reduces to: ---— (2.5) or put into other forms: dl:l . algal] H I OLD d? r dr , ---- (2.5) a. i a_.(r g3 l: a ---- (2.7) dr r dr dr) D IfQ is represented by a function of r, these eouations ca be integrated without difficulty in eacb particular one III. BASIC I IEITS The problem is to solve the moments, radial and tangential, in a simple supported circular plate carrying a symmetrical central uniform load with its center portion, direct under the load, con- sidered to be absolutely ri3id. Referrin3 to Figure Substituting Q in equation (2.7) and integrating, we have: _3 : Pr 2 103 3 - l - Clr : C dr o'TTD( P ) "'2“ ‘2 ’) (1"?ng 2log_r_+l -C1 +C2 dr2 8WD ( ‘2 ) —2 From t: e bour dary r=a= 3.25120 dr 3 -- o . r = n , hr . C, or dew 9 3 33 = 0 dr2 r dr m a 0.2 for concrete we have c1 . r .c o.o7 [1 .32 (2 io~o<- 1 fl " line a. - m2 32 [2 lord- 1.57] “ SW D Thus l-L'r . _ D ((323; .. 33 id) 1"- I' Cir = P 3.: 0.0319 [1.5 + d2(2}_0j0‘ -l.j:3) - = x 0.0319[l.3 4&2 (310~o(4 1,5) .3 1.5 + 0&2 f0 :lOZd-‘2.5 4 l - 3103; J (1.5 +OL23 )[3 {3 (moment per unit length) --—- (3.2) ~, 7‘1 or “H‘s": " i : 3V'P‘~'d -‘ 1 7‘13”“ 17* AT T N :‘i 1’) . ' -‘:""I"""f1 I‘fo t‘OK/‘J. .Li.“ U-lJ.L.~4.-La.~\; ii urA-‘JLL... LU-XJ JJL'oi' Avid—1 J. ‘.~.T (w {'7 3. 1‘2". C ‘r‘. '7": l rn'f‘q‘ :37}: :1 7'5?“ 'flT 7.7" ‘5 TIT’T'TWT."‘,T.' r177 ' ‘57"? -1 ##41-2‘: Sky—Lg - -----J ~4Lo-I-CJ Ju'4n-m-‘J ‘J.a.~l_—~-:--V4I-.I.‘--. 4-.i--—J “- -"" "- ’i‘r ‘ 'i‘VnT' * *1 ~-\'?"~1."‘. ‘-' ‘T‘~’T'~' ‘“ ‘ ""‘1' " r321 Tm" CvT ‘.'t " m1. 'Q I4 .J._- . 1 I --C I I I | - | _ u ._ Q .4 ' n , f 0 . l . Adi-“-LM; -ahd&$_ U C: . ALL—‘J_.-"..L¢..~...4 .JJ-J.‘.'._l—‘J VJ. $1»- U--‘ all 0 .L:-e exact solution of the deflection curve of a diametric sec- tion of a circular plate carryinc a central load and est- n3 on elastic soil whose intensity of pressure is propor- tional to the deflection of the plate at the same poin under consideration, is: (refer to Fijure 4.1a) w - 4 Q Cl(l- x4 + k' 23.42 .8 _ ) 22. 42.52.52 . - i E c -,-2 _..2’:.__.__ X 1 3 Ian Ta'fiP-q a) " 4(‘* "' 2 ,2 " o . )0»? 9. H .L. I. J i '2 I , a (*b ) + 5 X0- 1034.10:_xlo+ .E] FLE- 4-1 3453 fiE2,338 '\ But for small deflections, the equation of tie curve may be expreszed on the form ‘u'f I :5. ‘f 3 T2 --" (402) % Timoshinko, Theory of Plates and Shells q; o 1" WulCu is of hiiher the plate to Figure 14 obtained from re'lectiej tne ter ms contain n3 x power than 2 in equation (4.1). In our particular case, the central portion of bein3 ri id equation (4.2) becomes: (re 4.1b) ferring W = A 4 B ( r '- (3)2 "-"' (403) _ g f) 3 i1 = ..__"_.I", d“W = 2..) dr dr2 From the above express one we can conclude that the deflection surface has a constant curvature equal to 2’? a.) the central and the plate is under pure bendinj. The strain energy of a plate und or pure bending U1 : 1/2 D (area of the plate) 317021: 32w 32).”! 4 621.1 4 2 "3x2 y2 «5—2. 2772 In our particular case,gy = _y = n and 93 by 'Dr -ortion of the plate bein: rigid, U1 . D (area of 432D (l 1 elastic portion of plate)52w (141" ) "—2 15 Referring to Fi3ure 4.2, the force exerted on the infinite- simal area of elastic founda- tion beyond the area direct ‘ 1 LR u. under the rigid portion of \\‘\‘ I ”Aha, J41, Shadea‘ ‘””‘b) ' ’J . the footing is pirtd [:’“'”B‘ ' k'.wr.dr.d9 r E and the strain energy stored is , 2 l/2.k.w .r.dr.d6 : and that within the area direct _. __ underneath the rigid portion Fig. 4‘2 of the footing is 1/2.k.A2.r.ar.ae Total ener3y stored in the elastic foundation 18 11K 11!“ U.D . 1/2)) 11' 7.72 r dr (16-: 1/2) 812112 r dr d6 1... 0 1, o a o 2 2 .- g’J‘RA 4 3(r-a)2 2r dr do. [a lit mm (4.6) 2 2 a. 0 Integrating and neglecting terms containing d.of hi3her power than 2 (in usual co se d is less than 0.3 and at} is small as compared with other quantities) we have, U2 -_- wk',[l/2A2 R2 4 3333401 4 323502] --.. (4.7) where Cl = [1/2 - (4/3)o( ¢cX21and 02 g[l/6-(4/5)J+ (3/2)d2J The total elastic energy in the whole system is, U - U1 4 U2 - PA --— (4.8) where P is the total load from column. As the total energy of the system in stable equilibrium must be a minimum, substituting (4.5) and (4.7) into (4.8) and putting jQLI = O, ‘aU = O and m = 0.2 for concrete __._ ’ “0;": ‘53 we have A : C1234 vk'Rz (1-9.5(1-ok2)2 4 3234) kl 9 B I -[ 01R“- ""'" (409) 3 3T [9.6 ( l -d2)2 + C2 k. where D is the flexural rigidty of reinforced concrete section (see Appendix I), k'the modulus of reaction and Cl, 02 as the same meninings as indicated previously. With the constants A and 2 known, the deflection curves is deter- minei. l7 2. ShAPE OF PRESSURE DISTRIBUTION CURVE: As we assumed that he intensity of founda- ¢l_.uf¢1il.l7_-li_l-ur_r ‘ tion reaction is proportional . i h A+B(A-&f) to the deflection of the foot- .J ing at that point, the founda- tion pressure curve has the same shape with that of the b*m_—*”Tm’ll ””m“““ “'* , deflection curve; Fijure (a) Deflection Curve 4.3a and b shows the diametri- cal deflection4pressure curves respectively. The ratio between the pressures at the center and the rim bears the same ratio between the deflec- ‘ i (b) Pressure Curve Fig. ‘03 tions at those points, that L is the ratio B/A. As we assumed before, the deflection is small, it is sufficiently accurate to assume straight line variation in deflection or pressure between outer edge of rigid portion of the plate and its rim provided that the area under the curve 2-3' - 4-O'S is making equal to the area 2-3—4-0' (81). - c.’ (ll—k) sl . J (A 4 Br2)dr (origin at 0') 0 Ar (1-4) 4 3 ( l-d)3R3 3 A(l+u) LR-a) 2 U1 [0 ll i...) C) u a [1 + 2/3 (1 —o{)2R2 g] -..-- (4.10) es at point C' and the *5 where u is the ratio between pressu rim under the new assumption. It is evident that the ratio u is a function of the physical prorerties oi plate, soil, and column which are known (or assumed) befo e design. *3 l. are internal due V. Foorii: ELAdTIC sOIL (2F LOAD l-EO LEI-T333 : to foundation pres E— CAHIYI.L A VET TICAL LOAD F 1-10. “EITTm “T ALYo'ISD l9 TING ON The load moments, as defined previously, moments, radial and tangential, of the footing sure when the edge is assumed to be Load exerted on a circular area with a radius r - 1TI‘2p [u 4 (1-1).) -1Tr'2P{(u-C) 4 (1-11) [2... Err R-a simple supported. Figure 5.la shows the shape of load exerted on the bottom surface of the footing and F gure 5.13 its diametrical section. Referring to Figure 5.1b, The volume of small pressure cone at tOp is ;_ E@¢3(l-u)p, 3 l-ok or 1 Earp, where'}’is equal to a3 l-u) 3(l-ok). Total load on bottom KM surface = up1TR24.lnR2(l-u)p 3 a IIrRarp 3 2 =flR.p 2n 4 1 - e- --- .1 [*3 I] .(5) 4 33:1rr 2p[l- u-(l-u) R13] - ""3213?" 1:1. -'%%I:37]} --- (5.2) 2O Q - p[(u-c) 4 (l—u) 3-CA] r - l-u r2p 2 3 — 3 l-d R =Wpr-epfi3 R where c}!- [(u-c) 4 (1:3) 3—d. 6 = (l-u 2 0 1-04 ’ 3 1-00. Since g_[1 a; ”who. dr r dr dr D Substituting Q into the above equation and integ- rating, we have, d3 = 23* r3 — n 6 rz‘L 4 Clr 4 g; --- 5.3) dr 8‘ 15? “E‘ r ’d2w : Eov'r2 - 4n€ r3 4 Cl - 02 dr2 SD 153 2‘ g: --- 5.4) r . a - R, d1 - 0 dr r - H, hr = 0, or dgw 4 m aw = o er P dr m . 0.2 for concrete 1 r: and neglecting the terms containingcx¥ and (moment per unit lenjth) . . (5.9) 6 I E tf’ (mor ent per unit length) . . (5.10) y 0 C)" H K's are constants. Curves for K values in equation (5.9) and (5.10) are plotted with different values och and u (see P.49-54) so that the designer can readily find the designin3 moments. (F In the p r icular case u . 1, it becomes the con- go ventional ca se of assumin: the pressure unre r the footin3 ein; uniform. . . - . _. ______,‘ .-~ Am~.-",‘ Q-fiju'r - -.: ‘7'“ fi-rfl ‘4- : '5‘ h um: V ‘II 0 EC, 3-1- -L-\-\—A‘ CLi‘kL... .5... .4 '-:1 V -44. —- _-. Us- J vC - L3 .v—4‘J .L A-.\A — — Wfi - 1 'q rw‘ q— _—1 flf‘ - - -‘ - s---1‘ « T f‘ m ..— d—r -"—1 ;—.I "l —-‘ "I v '1 C34. L.‘_l-\1~)i ..'-‘~J VK... v‘..s3.:il'.,i - . _‘y U...i LU._J41: C‘.‘ I .L 4...») l. DIM ‘IZUTlO PP usual CI FILES In investi, till; the distribution of pressure on piles the same method of minim m ener3y applied in Section V, is applied a3ain here. As su33ested in Section I the piles under a circular footing are to be arran ed alon3 the circumferences of concentric circle and in each circle the piles are suj3ested to be of ame nodulus of reaction (in practical case, they are of same diameter, length anl material.) But piles in 'ifferent circles may be of differ- ent modu"l us of reactions. Usually, piles in the circumfer- ence of inner circles have large modulus than that of outer ones as we can see the presences over there are larjer It is assumed tnat the intensitv of reaction is proportional to the deflection of the footinw at tiat point. As the load is grmnetrical with respects to es ter it has equal deflection alon3 the rc.n_er3 ce. Referring to Fi ure 5.1, 2's and k's represent radius and modulus of reaction of different pile circles respectively. In a pile circle the pressure is ass1aned to be uniformly distributed along ts Cil cumference and k values have the unit of Mo ce per unit length of circus- ference per unit deflection. The w's are p. eflections 0: he circumferences. cf- the lootinfi at .4 23 Total load on the circumference of a pile circle with a 1 raaius 2]. .- 2 ‘n' zl .111 .111 Energy stored in the circle = L.2.Zl .1‘:i."f12 : 1T 21°}:i‘wl2 A) Total elastic energy stored in pile foundation is I As assumed before, for small deflections W1 = A e 3 (Z1 - a)2 = A 4' (/u]_ ‘0‘)232 'tU where 21 =,u, R Hence U2 =TI’Z ki.fll.R[ A 4 3 (M-o{ )2J2 . . . (6.1) Strain energy stored by the fOOting is 9 U1 3 431) (1 + m)1rR2 (1 -oL2) AA 4r Ox 24 Total energy in the system is U-Ul'OU2‘P-A ( or ( Substituting (6.1) and d( .2) into (6.3) (m 3 0.2 for 3) .8) b0\ concrete) and puttin 521g . O and‘vU' = O, we have 7151 R2 MAM-M?- ‘B' 3 ' D“ r 4) A . . . 0. 4.8(1 a2) , sEZaM—owli . D/ki The same assumption in Section IV-2 is made here and the shape of diametrical section of pressure curve re- p esented in Fi;ure 6.2 is the same as represented in Figure 4.3b. I ' In the figure “[1 4 2 (1—002.2 3 3 A . . .(6 .5) r(4.l lO) Fig. 692 The unit pressure exerted by each pile circle of different radius can be proportionated from the ordinate in pressure distribution curve at corresponding radius. The total pressure from each pile circle thus can be calcu- lated from its unit pressure and its circumference. 2. LDLEIT AIALISIS: Considering one pile circle once only, the actual moments, radial and tangential, in a footing are represented by those equations for basic moments, (3.1) and (3.2), except replacinfilu, for/9 and P in those equations representing total pressure exerted on the footing by that pile circle instead of total column load. The total moments in a footing resting on several pile circles can be obtained by summing up the moments due to individual circles correspondinfily. 'quations (3.1) and (3.2) can be put in the Kr . Kr? (moment per unit length) . . . (6.6 Mt =-KtP (moment per unit length) . ( and.K values in (6.6) and (6.7) are plot with different values of Al ,cX and radius of pile circle so that the q moments can be Obtained readily at desirable point 0] (see 555—60) . 26 VII. FOOTIEG CARRYIKG A IOHEKT LOAD RESTILS ON ELASTIC SOIL l. PRESSURE DISTRIBUTION: On the column, or something otherwise, to which the footing supports, usually carries a moment load besides a vertical one. Like a chimney sub- jected to wind load and a column subjected to eccentric vertical load are common examples (Figure 7.1). The distri- '~ A bution of foundation pres- sure in such a case ‘is some- thing like that shown in Figure 7.2a, and its dia-' metrical section in the plane of moment is shown in Figure 7.2b. As the central portion of the footing is rigid, the pressure over F180 7.]. there is of linear variation. One side of the diametrical Moment in section of pressure distri- bution curve is magnified in Figure 7.3. If the footing is non-flexible, the pressure distribution would be a straitht line througiout and (v at the rim the pressure is ttraight p = Mc/I. Now due to the ins ..._ 1 A deflection of the footing the 27 pressure a the rim has been released somewhat be- ’(1F‘ coming less than p, and the I read::~_‘~_‘F:;~__~: distribution of pressure ‘I“ between the rim and the edge of the rigid portion of the (norm- —-- omn- —~ 7 «ma- footing is along a certain . u; 2 curve. is we assumed first, the deflection is small, \\ ,_ the curve is flat, and can i p =Ic/I l‘JL' be replaced by a straight i _ line, providedly, the areas g under those curves are set equal. The pressure at the rim under the assumption is 'up', where u has the same expression as (4.10) and J; in (4.10) is expressed by (4.9). A 2. PROPOSITIOH OF TKTRCDUCIEE A SYIHETRIGAL DISTRIEUTICN OF PEESSURE IN PRACTICAL DESIGI TO ZAEE UKSYEZETRICAL CASE SELL-LET RI CAL : The actual case in a circular footinr carrying a moment load is of unsymmetrical bending. The exact analy- sis of such a case is difficult, especially when it rests on elastic foundation. But the moment load, like those pro- duced by wind pressure on a chimney, does not act in a fixed plane and it will revolve in all directions. Thus we 28 have to design the footing in every section according to the maximum pressure distribution curve which is the dia- metrical curve in the plane of the applied moment. The tension part the distri- buted pressure under the % * footing is not interested in design and may be neg- lected, which is one the p safe side. The shape of the introduced symmetrical distribution of foundation pressure making the unsym- metrical case symmetrical is shown in Figure 7.4, where p = Mc/I as before. If the design is made according to the prOposed pressure it will sustain the moment load from any direction. In the case of a column carrying an eccentric load which produces a moment in a fixed plane, the footing is suggested to design with the same proposed symmetrical load and just neglect the reinforcements in that part of the footing (semi-circular) which is in the tension region of foundation pressure due to the applied moment. 3. HOMEKT AEALYSIS: Figure 7.5 shows the geometrical pro- perties of the diametrical section of prOposed symmetrical pressure distribution curve. 29 VO'LOf Pressure 04 amj Radius r :: ’t’r‘z'nEOJ'f 6 E] .. "(A) : , . L ;9? To+alv01 0+ Pr85>urg :41” 4'“: ’ =‘P)€U[0'+5J~_-_(B) htfijb”:§:}*r#m_®jfi wzdtnki 1 L- r W 2:333:23?! __ MT“ ' P 1 EM w 1 ==1 M] '2' 1516.15 The load moments, or the ra dial and tangential moments in the footing when it is simple supported are ob- tained as follows: Q . pg; 4 Egg? . . . . (7.1) where 0” amfl.e are constants expressed in Figure 7.5. 31; [11: it. <’" 31>] 9‘ E3233 Substituting (7.1) ir to (7. 2) integrating, we have d1_1=20"r3 +62 rZ‘L-pclr4c2 dr 16D 30DR. —§- 5— . . . (7.3) d2w Mir 1‘2 '. 46 9 P5 '9 Cl - 02 o o o (704) a}? = 16D BODR 2 ;2 Using the boundary conditions rzagR, éflgo dr r . R, Mr . 0 or d2w + m aw . 0 dr2 r dr 30 and putting m - 0.2 for concrete, integrating and neglecting U A terms containing ol and xk', we have Cl 2 - Elsi—22’7“ o o o (705) 4 A CO 3 R o o o (70U) _ EET-Fr whereN-[.015100’+d0.lié] 1.5 Substituting (7.5) and (7.6) into (7.3) and (7.4) we have x; _ - D d2w 4 a a! dr2 r dr -_- 13321770., 4 e) (Onooq 0.406 ) -o.2o// / .' "/ 5”"4 ll 1" Z ‘LV / / “3.: Z 1 ~ ")‘.{£g‘. #355“; ~ L._____-- _. ._ ___2.¢1.<25f_’u:a.': 38 V -¢ - 4..--..4 -A-—‘— 0.- 2. 300111;} 0113331120 .1 RSSTIEG N “LASTIC Data: Vertical Lo VERTICAL FILES ad 0 O O O O O Revolving Homent Load Size of Col Allowable Bearing Capacit Piles in Piles in Piles in Lbdulus of Pile in Pile in Pile in Modulus of is c = 1,350 f5 3 3,000, R.C. Code . (a) The general a The arrange 9.3 and the diameter 0 Zl = 3 ft., 22 I 6 ft., Z7 = 9 ft., umn Capita inner circle. middle circle outer circle Reaction of Piles inner circle . middle circle. outer circle . LCAD ETD K)! \O A 1102:3131 10110 Elasticity of Concrete p81, f3 = k _ 0,400, 3 = 0. rranfiement of ment of piles f the footing #1 /H2 #3 = 0.3 0.6 0.9 is shown is chosen to be 10 ' 0 20,000 psi, n = 10 3 . 1,200,000 lbs 1,500,000 ft-leo d_= 0.3 . 30 tons each . 40 tons each . 35 tons each . 26x1051bs/ft each . 54x1051bs/ft each . 50x1051bs/ft each Ec—3,000,000 psi 9 K : 235 + 1. in en figure .04. AU. 4O .31 ’ \J‘ V/\ - J . ,1 .--~-~~~za=9f-t. ----- - ff. Fig.9.} (b) Computation of Load on Piles A. Load on piles due to vertical load Referring to Figure 9.4a, Assuming u: 0.6 we have: Unit pressure at middle pile circle = p2 = [0.6 1 0.4 X 4/7113]- 0.83pi lb/lin.ft. Unit pressure at outer pile circle = p3 = [0.6 + 0.4/7 1131 = 0.657p1 lb/lin.ft. 1.111 VlOft._1.hM (b) .1 Fig.9.4 Total upward pressure _ column load + wt. of footing P . 2Wx3xnl 4 2TTX3Xpl 4 21TX6Xp2 4 2nx9xp3 (reaction of (reaction (reaction (reaction of center pile) of inner of middle cute pile pile cir.)pile cir.) cir.§ 41 . (pl 4 6pl 4 l2x0.83pl 4 18x0.557pl)7T = 23.8Wp1 a 90.5P1 Column Load . . . . . . . 1,200,000 113. Assumed wt. of footing . . . 160,000 lbs. Total P 1,330,000 lbs. _ 1,350,000/90.5 = 15,000 lbs/lin.ft. ’d H I ’d [U I — 0.83 x l5,000 g l2,500 lbs/lin.ft. 03 . 0.675 115,000 = 9,900 _bs/ lin.ft. Load on entire inner pile circle . Ll - 15,000 x 2ITX3 a 282,000 lbs. Load on entire middle pile circle . L2 - 12,500 x 2vx6 = 472,000 lbs. Load on entire outer pile circle = L3 = 9,900 x 2TTX9 = 592,000 lbs. 3. Load on piles due to moment load Referring to Figure 9.4b, ! I pl " 0.313 pi = (0.3 1 0.3:5/7) = 0.428p' p'3 -(O.3 4 0.516/7) = 0.558p' p' = Mc/I, I =TTr3/2 for each circle (assuming the area of the piles being uniformly distributed aloig the circumference of each circle) lo I = 5x1r(93 4 63 . 53) = a XTTX962 = 1,510 ft3 p. = 1,500,000 x 10/1,510 - 10,000 lbs./ft. 42 pi = 0.3x 10,000 2 5,000 lbs./ft. p; 0.428x10,000 g 4,200 lbs./ft. p'3 s 0.558x10,000 . 5,530 lbs,/ft, Load on entire inner pile circle : Li : 3,000x 2flx3 = 56,700 lbs. Load on entire middle pile circle : L m... l 4280 x 2vx6 = 152,000 lbs. Load on entire outer pile circle : L'3 5,580 x 2Hx9 : 315,000 lbs. Total load on inner circle = L1 4 Li 252,000 . 55,700 = 338,700 lbs. Load on each pile in inner circle = 337,800/6 Total load on middle circle : L2 4 L 472,000 . 162,000 - 554,000 lbs. Load on each pile in middle circle : 534,000/12 44,500 lbs. (<'40 tons 0.K.) Total load on outer circle : L5 1 L3 : 592,000 + 316,000 3 908,000 lbs Load on each pile in outer circle - 900,000/18 50,400 lbs. (‘1 35 tons 0.K.) Mr = Kr x Total Load on Circle, Ht: Kt X Total load on circle. 43 A Section E 0.51 : 0.42% 0.453 0.5n 10.6 a 1.0? T 5 5 E . ‘ r W2-.5 Kr(21666 9) 0.15 e 0 .065 --_ 0 0 ; o ,1 {PL E 69,500= 56 600 0 i0 1 0 ”F * 1 I f ; 15:.9 Kr(Flate 9) o. 206' 0 .15; _-_ 0.057 20.014 : 0 ' KWL 167, 000 116, 000 51,700 12,700 r Raul 61 Lonen ir E A J _ E _in lS/ln. 5_2565 mL1 4 60 51,700 12,700 i 0 -jM2_,5nt(riatelayfi0 .0204E0. 034 O 032: o 022 O 022; 0.022 ” K 6 15,900 16 100 17.100 11 600 ll,§00ill,800 “° ,=.9K (Platel2) 0 .045 0 .065 0 .071; 0 .062 0.046 0.056 W5 KtL 59 100L59,000 64 500L56,500E45,600 54,500 Taniential Homent in in- -lb. /in. .55 000E77 100 61 600E6610d 55, 400 46,500 (d) Calculation of Depth 0 =./nr/K J“ 56 ,000/"5 5=/41,100 = 55.2 and a ov rall t: :ick— ness of 40". " use 34" Wt. of footing :WTX100x402150/l2 157,000 lbs. as assumed Flexural Ritidity of the section 6.3 x 108 lb.-ft. (8) Checking the Shearin~ Stress The critical section for snear is a circular section at a distance 34" from the edge of the column capital. b a (0.3 x 10 x 12 4 54) x 2’W : 70'W - 440". Va = 0.03f5 = 0.03 x 3000 = 90 psi Shearing force due to vertical load . V (1,560, 000 _ 262,000 - 262,000/6) 1,200,000/1,560,000 6 (1,360,000 —329,000) x 0.665 . 912,000 166. (”7‘ 01. 44 earing force due to moment load = ‘0 - ( 1/2 x (162,000 4 360,000) : 231,000 lbs. (Only 1/2 of the total load is effective in ca - culatinj shearing force) Total shearinfi force at critical section . V a V1 4 V2 : 912,000 1 251,000 : 1,173,000 lbs. 310x.875334 3 ‘81 Co}:. *r Checkin: the value of u Modulus of Reaction of inner pile circle = k'l / . I. - ," O f. '7 1 V - 20x10 xo/enxj = 8.; x loJ lo./ft2 Iodulus of Reaction of middle pile circle = ké n ". "aw f) 0.8 x 103 lo./ft.¢ = 34:103xl2/2fl‘x5 Modulus of Reaction of outer circle a k' C‘ = 30x Ouxls/ 2 x9 ll 3 0 U1 ‘1 1 5.3x10°/€.3x100 = 76.1 ft.) U i} fiL ll , . o r , c.3xlOU/10.8::10Q = ol.0 U \ :‘7 r0 - II M ’ _. v' f —'" -l ‘ r—N- ‘ ‘~ 2‘ I: '— - " D/k7 _ 0.91100/9.3l10“ : 0J.3 rt.) J fl ,M rum-U2 ”(u-4f? ' mu—d )2 MZDEE‘I—{d fir O 0 4 0 4 0.0048 8.83:10‘ ; 7.85310- 0.0116 j 48.5310 ‘4 g 17.4x10- L P.- H II II 000 V) .(AUJ \N L———— O O 0 b1 0 R) U] 4:. 23% 57.43x10'4 l 25.251110"5 . _ RgT/4{A4“v<)9 D/k' 4.8(1 -ol‘) + PEZAQI-okf4 D “' .‘LL uni U1 = " Ci- :70 2' 'ZC-IO _, - ‘t 71.5123 . Cl + 1,000}: .233‘210-3- . -5.74 x 10-2 = - 1.23 x 10"2 u=1+g (1-0K)32§ ...(5.3) 3 A = (l - 0.537 x 0.49:100x. 3x110"2) = (1 — 0.4) = 0.6 (as es umed bef01e 0.K.) (3) The remaining coraputations are similar to those corres- ponding ones in Example 1 and they are omitted Mere. (l) (3) (4) The moments, both ralial and tan ential, in a circular footin¢ under any loai n: ans foundation conditions, decrease oreatly if Le size of tne COlULT n or COlUfiln ca ital increases. A most economical ratio of 0< at which the sum of the volumes of footing and column cap- ital is a minimum in the particular cas *nder cons ider- ation couli be obtained from several trial designs. The tanje1tial moment is always much more smaller than the ra lie 1 moment at the same .oir t under any loading tion coniitions. 91 and foun Tte maximum radial moment occurs at the edge of the column capital a-d the maximum tanje1ntial moment, in i" d‘ 3 C 7" . v1": 4‘ 1 ° *1 "- usual case ( < .J/ ccc11s at a section soaesnere between the edge of the cagital and the mid-point of the remaining part c: diametrical section. As those Q ' . maximum moments do not occur at tne so me roint, the re- inforcing steel in the footing will not be over-crowded in the section near column woital, which reniers the design practical. In any case, the r-uia l moment rlecr eases much more rapidly than the tan _entie l moment at the same point along the radius 0: the foot 1n: It shows only the central 1:.eart of the footing needs heavy reinforcement. TLe ratio u affects both tanjentia: and radial moments due to moment load more than those due to vertical load 217 on the same foundation, both of then decrease if u decreases. As we assumed the foundation pressure being proportional to the deflection of the footing at tha point, the pressure will be larger a the center than at the rim. *1 ~ v‘ . q —‘ ' . ~ -- r1 WV 3 3; 't. ‘ \ . ‘3‘ ‘ 0 LViuently tn1s kind 01 pressure rlstrioution, union is k. ~- ~ . O 1" “ h V _ F . ' . q: a, int uniform pressure miti tue ~aae Spdleled all nogpe soil or tile bearing canacitv. The fact st vs the foot- 'n' ' .‘ v I‘ .-v A V V ,A I" I. - ma 9 ing JGSlLL 1n conventional ta; nae -reater tendency o1 settlement as the central part of the fou-detion under- specified degree of settlement the allowable bearing capacity of foundation could be raised in design with the new method presented here, The new design *‘5 equires tni O 1ner s-ctior ‘ r quired by the conventional design of same Specifications as due to "oth tLe affect of u in (5) and the affect of considering bending in all directions. (The conven— tional design assumed the bending only occuring at certain pairs of parallel sections.) Plate Plate Plate Plate Plate Plate Plate Plate Plate Plate V'T 4L; . N? l “3"“ "‘ V,.1.L-1J.Q F03 D; fl“ - 1 T“). N) A—‘JL‘ 1. Radial Homent = 0.10 2. Radial Homent 0.20 3. Radial Homent 0.30 4 Tangential Homent 0.10 5. Tanuential moment 0.20 6. I“ngential Ioment 0.30 7. Radial Homent 0.10 8. Radial Homent 0.20 9. Radial Moment 0.30 Tanjential Homent 0.10 Tangential lomentCl20 12. r“'a1';er.=.tial homent0.30 13. Radial Ioment 0.10 14. Radial Homent 0.20 15. Radial Loaent 0.30 lo. gal ential Ioment 0.10 17. Tan entia 18. Homent 0.2 Tan; ntial I-Iomcnt 0 .30 Elastic Soil Foundation w 1 [*1 Elastic Soil Foundation ., 1 a Vertical Lo Vertical or Load astic Koaent Load, d ‘- 1-10 :11 e . J. flu Foundation Pile 11: .06 L7 4 a 1. 4 . _ . _ _ InHT—LI~1 - 19.1. T‘s 1 w a n . 1 . 1 1 m m . 1 . .-I.+-;1- -rgI.:It+-I.w;...II+-I1- .-- v I;.I.1II1.:IsII.v- 1-Iq:I1:I:¢IIIr-IFI:IflIILr1.4I1 _ . 1 . . _ H . 1 , _ . _ .11- .I...I_e -.bIJt-glIJII.fII--I1fiI 41II _.II+ Int-.41 . at -_fir.- — - fiIIL-I 1:4, L - {JI‘ _ . O r O 1 1 1 @I - -- -. . -- III-L L _1 1 a. . 1. . . «I II. . ‘ ifl' ma 1 1 .. 11 . . ..1 1h _ .-1s-+.1. I l I I 4~—-— -1! -_ ‘1 fitIIlltll. ...fl'u!v.0 1 L...__‘..——. ‘Y I “T I r f“ r I dog I I ~. I Sqil‘ I r. >- AL- 1L”..-§— _.—. - T. 2 9 l J I I r l at 1 --+ a I .. -1... 1.. I I I .. 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I I .i I ~ I I I 4 Load w-4--g-4-- Mom Found t; ‘ U i. \\ - .3 _ I IoL M . o. I . I "r ‘_# 1 v — . I I .-.~l————¢.————Ib I I I ‘ 1 X I I --§r-6----—~—-——-+‘~r—-—+——- I I I \K A I - r \ -I - I - . I \j 'l 7 I‘**vW'—fiP“L‘*m% I I I I I ’— - l I I t I I O I I v 1 N—«» -—-4~ ‘ I I II I I I ://-;;' I § .f. l ""I ..-. 4 [I ‘O I +— I I I .J";I I _I I——__—. I I L. ..-- - -. .4, ....— 5 ‘0I20 _ . I rIIIL IIIIII I .IIIArIIIII II“. I I i. I Avll IIII I 6 II IflI. A I IIv I + I II, yIIIIIIA II.+ I III~III IIIoIII. I Q , I ‘- l . ---‘———» or--———o-—-——-- . -+z-u t Ezi i_0 -- -... . . h . 0. c .81.- 0, 0 0 \x .0 . . 00. . - 0 . . \ 0 R m0 0 00 0 .000 mm 0 . - . 0 0 0. H. E - ..n.4.2 Tfi ab.04:: 00k .1-YI 0-.. fi---d!-a 0;P0-4: m5 T; . M “n m0 " V0 0 . .0 \ -5. e.;.i r 03-.7 -O- .-0 M, 0 0.0.04 .00 0.. @- - --00 . f0 C. ‘ 0.0. ... fl ..0 0 . _ . . 0 .. - --.-I039}? - . $41.10. 0 .P . x4, ...-0 0 0, 00+ . A . II-.. 1401!..- -..: M Au0n a 1w“ . 00-x ./ .fi//0 Tr , /M- 0 .4 0.00. 0.0:: .00... 0.... . - 0 - / .é: .- 0 .0, 0 .. .. . - f . . - . . Y .. -. v . I - . a-W/ . “01/10 -- . 1y 0 o h -T . . 0 D . 40 u. 0 _ 0 0 J 0 0 0 0 0 0 m . H -0 . ,.- ................ I0...- - -:0- .- - - . m 0 00 00 00 00 - 000.. 1 0 0% 00-01....- i- ..- .500 --.-0.0.4:...403 0 - 0 - 0 0 0 J 0 a , 0 .T- . 0 -0 0 0 0 .. -. 0 --0- 0 . . . 0 _ 0 , * .- ...-l... . . .0. u ,0 0- .0 .0 000 0‘ 00 0. 0 0 . 0 0 . 0 h- ,. .00. 0-0 .---0. - ,1 P 0... e 0. $0- . 0 . ,_ 0 w . a 0 M H0. #0 0-0 .0 0 -_ ...-..-rr.--0f--, -0. 1. 0 P M 0 n” A - 40 w 0 Eu 0 0 .0 0 0.;-. : -.- 000 .Q ;0f. . 09;.-.OM - 0 .0 0 ---:0- fi ...0 .- 0,, ..-0 0 0 0 0 0.0 0 0 0 0 0 .q t o - . - ! 0 50. I}? .07 |u A x 0 T v +-.fi n. 1 m w d. 0 ILL; _ r P M} . . h ail .I» W «I , L w tutu-.1... pitic 8L 67 XII. APPEKDICES 1. Derivation of Plexural Rididitg of Reinforced Concrete Section . Referring to Figure 11.1, The fiber stress at a distance 2 from H.A. Figure 11.1 in compression region T ! Theory of Plate and Shells P.2) 1g; n.(l-k) h.hp 10 of the section. S 2 ;_ d2w z dz . 0x2 FIG. 11.1 i h3 d2w 00 .me) dxz fc(l - k)2n.h.hp . - Eckh .(1—k)2nph202w k (1 - m2) k a"? : - 3c (1 - k)2 (pn)h3 dew dxz (1 - m2) pl: - Eh} [£3 9 pn (l " k)2] o 0 0(1) 3 -2 d w (1 - m2) dX2 kh.g% z pn.h. (1 - k) h (taking moment about N.A.) pn k2/2 (1 - k) . . . (2) Sub. (2) into (1) we have: M = - Eckej h3 dew - -D d2w _— '2(l-;§) EEE 0x2 Flexural Rigidity = 0.0 Eckgj h} 2(1 -m2 ) o 7! . . . . r 111 11 1 111-11111111111111.1111 11 1 111 411-1-111-11111-0 , 1 111-11.111-11-- 111 1 1 1 . . 0 0 1 1 1111 1.‘ 0 0 _ 0 fi . 0 0 0 . _ v- 1 1 .. - . 1 - 1. .1 1. 0T _ . . 1.1-.:.. :0 - . ,. - -1 _- 0 . 0 01- 1 1 1 1|..- .r11 -0.. 11; 11. .. .1 .- 11101- 14 1 9 y _ .. _ 0 0 fl . u _ 0 .. . . 0 0 0 _. 0 0 0 . 41 11- - ..11- 1.- 1 ..... .o 1 L1 11 411114 1.0. 1 - .... 1p .1- +1.10?- 1-».1 .1. - 0. 4.. 1 .. 1 T 11 --.- 1.10-1001-- ..01111... 1411- 310 . . 1 1 T 10111 . .- 11 1 0.1 . 0 4 0 0 _ . 0 0 - .. . _ 0 _ U. U .0 . 0 w h ..11. 1111.? 1111+.I1Il1o01- + 111-r1 o. .4.- I. 1.F 0o. 11 0. . . 0 _ 0 0 ... . 1..-1w {.0. -.. p11. 1- - 1 1..-.11 1 0. .- 1-11 .1 100 1 .. 0 O _ .. ...-11.4.1141 :14 11 -. 1.1 .1 .1 ._-- ,11#1.- 1 0 H. . _ 0 _ h 0 . _ H . .. 0, h _ 0 0 0 0 1 1 4. 1111. .11.. 1o 1P1 4111' 1 1- 1 .1 1 1 1- 11 1 1 .- - .1 - 0 10 1 1. .- 0 S 0.. _ 0 i _ - . . 0 - . , . 0 0 _ . . ._ -. 0. -.--1.1110.*-.. .1 1 .11- - - m . u ,1 - 1. o 1 11. 1 r-1 111-11 5- .+ 5. 0 11¢- 1\\. I V a J . . . .---1--.... - 101:0 _- 10- .- .-- 00 _ 1 .0 . _ _ 0 00. _ . 0- .0..- . 0 -21.... .. L -1. - 00 -111--.. 0-, - 0 - - . , 00.. \ . 0.1.- -..1-.11 1 .. _ . - , 1'1. -1---..w11 _fi 0 . . 0. 0 0 0 0 _ \ 0H 0 0f .1? 1. .9. 111.1% +1» 111. 6 .0 O 91 .1 1 - 0 G , m . . 5 1 1. 11L11 1.1? 11v - $ 11 1 1 9- Y - .111 6+1!.. . - 8 . -1. N0. 7; 10.3 .3--.- 1 11. . -n. 1 , H 0 . 0. .Do .I 0 e ,e 1. . ,0 -..1 111-11 -0 11. - 1 01 11. 1 - . 11 T 1 0 11 1.1-1m -1 mrll. 1 111. 1 1+ 111 111 11110111111111.1111m\ H .1111? m 41 *1 0 0 _ 0 . ,_ r _. 10.1 1. C .1 . .0 . . 0 .- _ . . .. . 0 0 1. . .. -01- - . .0 0+barP t V “1 . . - 110%.- 1.--1.-. ...\. k 1. \ 0- .000 41-1-1414..- 4 0 . 1---F0011 n.1,. . 1- 1.01: .10 .0 1014-1. 1- 0 _ . x _ 0 _ _ 0 0 0 . 0 . 0 . w & em C c ._ _ 3 HA1 1 u 0 . \ _ \1 ._T 1. F r 111? 11L1 1.11%! -+1-v. I 5111 I 11w 1| 1 11.1111 m 0 m0 1 1 ...a. 0.0.1-.. 00.... 1-- -. :0 1.,- - . - 10 - x 0 0 0 0 _ 0 - I-flMw-1f. 0.. at: . .. 0 0 . 0\\ 0 0 _ .0 V Y .11”.- .- L. t“.c 80 h m r b . 100-1 .111-1.-.\\ 111.01 +1 1.0.11-1 111w1__.1 4.1.1.100. . ...-U0?- r 0nal .- 10...C. 0.1:. -1-.. 1. -.1 . \0 00 0 0 0 0 0.. 6. c. axon. ar e 0 . \1. 0...... - 11.- - . .1 H Vo_0.0 .0.-rd-.. 0 -1 . 1 1 -11.-. .- 011 . . 0 R o. 11.-0.1- . - 1.2. 1. i . .. . _ _ 0 .. _ _ .10 0 td o r _ 0 \ 0, 0 0: 1 1-0. 0 .- - .-.. -. {1-1. 0 _ C _ 0 . _ e . 0 -. . \ 1. .. 0 . , _ 0011 .1 :0911 u 1|. b 1- h .1 1 a. e 1 1 . - q. 1» ..1 . 1 \ .11 .. m .0 _ _ n . r . N T 0 L: n c 0 _ 0 . \.- 0 0-. 0-- 1. .1 .- . 1-.. 0- .. 1.3-0 0 01. . g . 0 3.11 - 1 1141-19 1 . 01-. .40 . . - : T0 Y1-" . #9. 1 . $11 ”-11}. .0 .o m m m M H . \1 1%.. _ f .L - _ fi . u 0 1» M" _ n . u . _ H . , x. -1 1. L. .r .1-1 1..., ...1-. -11 1.0 - . 1 T.--H-100.- -.-.t0.a.b.o.1 . - -.\\0, _ .0. -\0 _ . 0... 0 .. q 0 0 . 4. 8 P: 1m. _ 0.1 _ R K V o T P \- 0 0 4.1 0 0 0 _ 0 _. 0 1-1111111..-m-111.-11110 M ...... 0 1 1 . 111 1 0 . 1a. .111 1. - 1... . _ _ _ . 11111100 .11 T 4111“ 11.1, . - q a 1 0 0 - \O 0 . .10 0 2 0 u "d . .. . — . n \1 T $— 1 11. 1‘ \k .- .... I 19 I 1.171 1.9 .11 111 11 . .7 19.. f 10 1 1 11 1A 0 1. ,0. =1. ..-... ..- --0- - 1.--11 - 1.\ =- 0 - 0 . “O - I. 19L ..1 1“ - V \- \w \\. . H \\ [Ml . .9 1. 6.. - .I .L - 0F .. 1Q - -M.r K - L. x 0 xx. w 1\ + m 0 1.1. 0 , 0 T- _ _ 0 _ \. ,021-1 - -1 -... 0.8..-. _- 0. .. - .900. -. m - .0 . 0- 0,: 011,-. 02-; _w 0 0 .. G _ u \ ~ PM 1.1 h...” _ >1 _ 111 A LAT 1. (”'1 111-Ar 0 1 1 _ .1111 i ...-111 . 1K 1 \1-- ”1 1 1 4. 1V1 . . - 1M .y ..0 / T_ H - . I . ...19... .. . ff! “- -. +1 1 ,1 .* . . .9. L o I 1. 0 1.1 11 1.7 07 ..1 ’10! 1 a. 1 . 11 III f l 3.]!1/1'11 1. . . 0 . 0 _ 11./I11 “- .10 . .1...- 11.1111.11 0 .1-1 ..1 .111..- . . - 1 ... 0. - - 0. .- 0:- .. .. 0 - .0 .- 0 .0. 0 00 _ . 0 0 0 0 . 0 0*: - -0.- ...--0111 - 0.0.. 1-0 .0 fl . 11.+ ...1. .... 11 r -1-1 111111 A 41 . 0 _m 0. 1 00 1 0 0 . 0 . 00 m 00 0 0 1D? LSP ‘11 _ 1 D- 1 - 5 P1 - 1.1..- 1 .1, 1.1 0 1 -- 1-1-.. 00-, - - .... . ... o. .. .0. .... _ .... . .. .0 _ . . . . .0 -- ..-- 100 - 7,-0-0 7.0.. - f . 141 11 I § 1- 1 O 1 A II 41 114-111 3* . 11 f l 1..-114111 t 1 10 l ..1 1.11 |1¢ fl 1. 1, G. . m “ 0 0 m _P 1 11‘“ 1 _+. 1 1 .W w s u 0 1 1- . I -00 -1 . -1 . 0.1 ...0 -- _ 0 0 0 0 . , _ . 0 _ 0 K 1 . . - 10 - . . .. _ 0 . . .- .. 0, ., _ . --. ...... - -0 . I1 «1 l v 1.0.11 .. 1. - .0 1 . 11.4 .7 o 4 o. . n ‘ b . H 0 _ . _ » 0 _ 0 . , 0 - 0 1 - _ - - 11.-1- 59 0 . . 0 . _ 1.11111 1‘11 .11“ .. 11.. 1~1 111-..:.1114 1... 1o 101 _ . . . 11.11.11‘111 I it! Lt...L-..|1.."11*ul 1.314.111. 111%- .1111 1. 0 . . 0 v 0 0 _. 0 1 i 0 1 J 1 k “1..-.- ”Y V TI} ‘ l .-. 1 l 4 --+ ...? 0 . _ 1 4 11% r L1 1111 L1 4 . . 0 _ . 0 0 0 4 . . 0 . . . . . 0 . . 1 1 1+1 1.1%: 111.11 .1.-1.9111114111114111 :- l1114111 1 r1 . +1 11 1.1%. ..1111 . 1*.11I1 . .111 1.91.111 .0111 1:1 ..... «.1.. “.11. 1 +1... 11' 11 1119.1. 1191- . . 0 n . _ W . . 0 . 0 0 . , 0 _ 0 0 .1 \I . . M’11 .19- . 11 Aw! I ...!1 I1 #11.11 Tit-0 Lrill .1 11.11%! - .111“ 1.! A 1. .111-1:1r . 0 _. . . n 0 II.11*'1|II1'|JAI1"161111II.101 .. 0 . . . . . . 11. 111i... .001 00.1111. .1.. 10111101110 11 41 411-‘11‘.11il" 1 ent10_ iles 1 Hi b -0 CULAR F0 I I .- 1 .0. q‘rfi if. ‘ . 1+-1_—.—.Y_ “6“ -—.-....7-1._. ~1”~W~—.— +0__.. ..4 1 ‘ . 1 1 I Come. Girdles of Vértical‘ from;eabh- om curve Elastic P LB 1“ 1- .- O | 1 1 0 . I 1 1 I 1 V I . .. . .H _ 111.. 1.- r 1111131.!111141111 -- .w/o . .4 . . Z . .. / 0 Y 1 1 1 1 j _ 3 ‘ 1 i 1 1 l 1 .10 1* ad ,0-..— I | 9 1 Q 6 l T ENT INC Value at each aéctiofi - Dbtained f Tofia111 F— bM. 1 ~ .- _ 0+. . 1 1 L ‘1‘ 1 I \ V 1 s ".MQ/R,=_0.aof 1 J \ 1‘12 =1, 0. 405’ 1 11....-\\...-...1.:1 11101-. 1 -- n: ,//0 - . nwxx11111- -111-1:- -1\ \.1 p116*pirc1 1 1 11 1 ? A L a 9 ' 1 Foundation.—--- '9 0;... 1 1 . - ;_1000000000.9 _ i 1.11.11 1..—4-..-.-— 0 14 \11 .\ \ - 9 ‘/ \\_ j z / .. . . 4.1 1-. 4 / 1 0 ’, --¢ 1 — 9 1 1. 1 \\ . : ‘. ' ' V : __._.+.—————4FM_¥——-Q---— . M- 1 9 1 '— _ . 1 .711}. iv 0 .1 1 1 l 13-..— 1. -- I 3 I ,_ 1N \ \ , b 1 \ 6 1 I 0' | ,. 1,»- 1 v . ~ ..1+11 1171.1...1T. / l I 1 o— I ,1 1 4 -+-+‘———.+1— 1’ /‘I -TANGENTIAi M 1 _ .4 ...—.0 00.00—_-_1y_.__.+-— -“ -—--§— ~-. 4’ m.C(1J.I.1.. .1.. rah... 1.. 33.001; ”0:. .. 1L. .40.. 7.4. v Urn 1! 1 l 1 1. P1111110 -1.; < -- M‘— .10 j. 0,. -1 .. 16/11" .1 . . 9 1 .49_ -91 . 20 .1-0 1:1 1 1 1 I 1 ' .000.JL.0_._A- 0...... .- 3 10.000.-- 60 I‘. I .. II I I .AIII I... .III. . 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I V \ . . _ ‘K . \\\ V 4. _ V . I O I Lfl. 9 II + I OI Afi I I I I I I L7I I *IVI I. O. m.I : #! OtI I'II Igws. III fl.9 N I . . x _ z \x p. \x % . . .. . . I. O \ . . .-_ -‘ Y— I fl. I l I | I .... -+-...._ 5.... _qr __.....4L.._....~,... __4... --_.,_.—._1,...__‘-.__+._—_.+.~. —4 I , I . . ' . I I l I I ' l I I I , I V o————1L~-—--+—-«—L- I ‘ Y I - I . I . i . I I Q - I - ~ 1 1 A . ' . 1 I f I I I I \ ( I I I o I I I I I I . =:a/R ~-4-- 0.30‘ + . I Foundation ---- Conc. Cir¢les .--k Value at each- Load §C§ Kt J 0 3-. I A, I .3 xb-- ..-IYII 1 . \ /.\\ u . _ _ IIIII I NIP I1 I 0 I I I .I-l I \m I1 > .I I4? I. ..u \ V x _ \.\ d 0 c I W O a. ...I I O I 4+ .\. A I I5 II ¢ k IT IV I II LYIII I I. .r IIfil \IF. I .4 I“, .4 VII _ x. . \ _ H _ .. . 8 _ . _ _ \ _ R O 1 .-- . .- . V- , \ .{ .V. I4,.,II,VVV,-I:.IV .. . w _ z .. . a 1 _ Ih . _ \ _ L \H 1 I Q I t . I I 0 o. w I. I o. 9.2II I t I I w. . II. I I Q. .I I. . n.“ . x V . r . .V g “H _ . IHI I /.I I M / . I.I K I. [W I. 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T V H V V _ . H .. . V V _ V V . L + 14 w on w r ; V. #1 o , . . V V . . VV _ __ _ V _ H V M n V V V . -V - . V V . . - ,o d _ O 8 _ V. . V b IL- is- , r t k , ”.1. 4 0” m 4 “w L m 4 V. O o _ V V .V VA JCV HMO a 0 O a O V .t 0 w .I ”I . I ‘a lw 0 v 0 . ¢ 9 . I . O o. . u . .V _ fl . _ _ u a V V _ w .t 4 w . : V . w _ V V _ run V V M .. k V. V v : v V. m V V V M w V V 1 I- l , - 1 L. V I--- 3 ?L: - \ w 1 l t. - I. I #11. _r : I34 xii: V -- 1 t - 71 _Pmm :18. H U Vr Q“ ‘7'?"' —'/"( ‘1 4‘”: I . o‘LrP-‘L-.. ‘ lvuo l. Derivztion of Flexural Riviiitg of Reinforced Concrete 5e Figure 11.1 in compression region 1? 0 SK : - 440 ' Z dgw. (1 -m—) dx‘2 . .. (Timoshenko: Theory of Flat and Shells P.2 LI: ssxzdz 4 f (l-k) n.(l-k) h.hp o o k where p is tte Stfifi} ~atio of the section. 9 h V 8,2 dz 3 — VV “y: 2 oz k (l_m2) f2 o 0 _ - EC k3 h3 d2v 3 (1- mg) (11' ’3 fc(l — k)2n.h.hp - - Eckh .(l-k)“nph2'2w Sub. (2) into (1) we have: x = - Eckgj h3 d2w — -D d2w '2(l-m2) K2 dxe Flexural Rigidity = D.: Eckhj h3 O\ CO Bibliorrapfiy: Cummin3s, A. F. 'Distribution of Stresses under a Founiation' (ASCE Transactions Vol. 101, 1935) Eoppl Tech. mechanic Vol. 5, (Cha§ters on Theory of Elastic Plates Holmery, E. O. & 'Analysis in Circular Plates and Karl Axelson Rinjs’ (Vw ZE Transaction Vol. 54, 1932) Newmark, Iathan I. 'Simplifiei Computation of Vertical Pressure in Elastic Foundations' Bulletin No. 24, Univ. of Ill. 1935) Prescott Applied Elasticity (Che pters on Elastic Plates, Dover Pu olications 1946) Richart F. E. & 'Tests of R. C. Slab Subjected to Klugle, R. U. Concer trated Loads' (Bulletin H0. 314, Univ. Cf 111., 1939) Roark, S. J. 'Stres 9ses Prod.uced in a Circular Plate by Eccentric Loa iin3 anl by a OTr€ Vns1rerse Couple' (Bulletin .74, Univ. 0: Wisconsin, 1932) Russel, G.M. 'Elastic De1lection 0: Thick Plate Under Unif orn Load' (En9ineerir9 (Eng land) Vol.123, 1927) Stoker, J. J. enain9 and 3ucklin3 01 Elastic Pla test (Few Vork Univ. Summer Session notes, 19 'l Talbolt, Arthur H. 'Reinforced Concrete wall Footings -and Column Footinffi ' (Bulletin Ho. 67, Ur niv. of 111. ,1913) Timosherl :o, S. 'T? cry of Plates and Shells' (a zapters on Fla es) 'rleor y 0: Elastic Stability. (Ch9ntegs on Buckling of Elastic . Plate— Vetter, C. P. 'Design of File Foundation' (& Transactions Vol. 104,1939) Wahl, A. s. a 'The 11211; T933099 d Disc Sprinf' Brecht, W. A. (ADC: T1r9nsacction fol. 52, 1930) Westerjard, H.E. ’Stresses Concentr9tion in Plates over Small Areas' (ASSET rans— action Vol.108, 1943) 'LMo ents and Stresses in Slabe' -(A.C.I. Procedin33 Vol. 17, 1922) Wise, Josep1 A. 'Circular Flat Slab, with Central Colunn' (A.C.I. Journal Vol. 34, 1933) Joint Committee 'Iecommended Practice and Standard -Specifications for Concrete and Reinforced Concrete, 1940' M'TITI'ITIEHTIL 311M:flfflitflflflflilflflwflfimfi“