An Investigation of the Effect of Bearing Area and Depth upon the Settlements of Footings Citation Housner, George William (1934) An Investigation of the Effect of Bearing Area and Depth upon the Settlements of Footings. Master's thesis, California Institute of Technology. doi:10.7907/2bkd-2s20. https://resolver.caltech.edu/CaltechTHESIS:09182024-211238173 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Civil Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Civil Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1934 Record Number: CaltechTHESIS:09182024-211238173 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09182024-211238173 DOI: 10.7907/2bkd-2s20 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16744 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 24 Sep 2024 16:43 Last Modified: 24 Sep 2024 16:43 Thesis Files PDF - Final Version See Usage Policy. 4MB Repository Staff Only: item control page

.A,N lNVESTIGATlO.N OF THE ,Ell'F'.60T OF BEARING A.RF;.\ lUID DEPTH UPON '111:& SETT.Ll!M.ENTS OF .l!OOTINGS Thesis by George Housner In Partial Fulfillment of the Requirements For the Degree of' Mast-er of Science California. Institute of Technology. Pasadena, California, 1.934. TABLE OF CONT:FlfrS Int.roduetion ----.- - - - - - - - - - - - - - - - - - - - - - -·- 1 The Data .and Me thod o:r Analysis The Derivation of the Empirical Ilormula Figure 1. ___________,___,..~.~ -- -~----~....,_...._..____ 11 :l!'i gure 2. Comparison of Computed wtth Aetual Settlements ------•.. -· •·- --...----------· 15 Conclusion - - -·- - - - - - - - - - - - - - - - - - - - - 16 INTRODUCTION Sinee the formulation, by Coulomb and Rankine, of their widely known theories o:r earth pressures~ the science of soil mechanics has made littl e progress . T.b.es e theories are now reali zed to be of classical int erest r ather than practical s i gnificance, f or the eo nd i tiorts deecribed by t l:l.e•n a re ra r ely, if ever, encountered i n actual pra ct i se . Withi n recent years, widespread inter est has been evinced in inves tigat i on. of S J. ) i l pheno'!l.ena. However, due to t he complexity ot the f ac tors enteri ng into the action and reaction of soilB, t he advance o:r knowledge has been r el atively slow. lt i s not to be expect ed t hat the theory of' soil mechanics shall be put upon a thoro ughly scientifie basis for many yea.rs. t xperience and invest igation have s hom1. certain gene r a l charaeteri sties of' soil action under load. . The mot1t obvious , and of most pra ctical int !Dr es t,, are; tha t the s ettlement of a benring area wit h conatant l oad per unit area va ries as so:ne function of the a rea; that t ile settlement varies inversely as the depth of t he beari ng plate beneath the surface of t he ground; and t hat the s ettlen1ent varies as t he moi s ture content of t he soil. The investigations now under way , 'both experimental and theoretical, are pointing towar d t he ultimat e s ol ution of t he exact effects of t hes e char a ct eris t ics. However, until the exact soluti ons have been determined it is des irable to have some means of analyzing 2,. the effect of bearing area, depth, and moisture content upon the settlemeut of foundations. Since the effect of moisture con.tent is so clo::: ely dependent upon the physi~al proper ties of the soil, such as, density, si ze of particles and shape of pa rticles, beyond a general conclusion t hat an increase in moisture content means an increase in settlement, more e xa ct determinations cannot be made until t he comple- tion of exhaustive experimentation. This, fortunately is not the case with effect of bearing area and depth. It is the purpose of this paper to derive certain empirical relations which shall make possible the de t ermination of the effect upon settlements of .foundati ons of the bearing area and depth of f_9oting . A knu1'1le dge of t hese i'a.etors is essential in the design or foundations as it is of prime importance t hat the footings be so deai.gned as to secure a. uniform settlement of the structure as a whole. .lfailur·o to do so will r es ult in over- stres sing and perhaps cracking certain portions o:f' the structure, with the resulting danger of ultimate failure. The results of this investigati on are based upon a s tudy of data furni shed by the firm of Lo.barre and Gonverse, Consulting Engineers . Thla data was in the fonn of 97 load-settlement t ests. 'l'h e tests eove r·ed a. wide range o f' s oils, including decompos ed rock, ela.y, sandy loam, loam , sandy silt, and silt. 'I'h e t ests wore made with b earing area s of 1, 2, 3 , 4, and g squt1re f eet, and at elevations va r y ing from 3 to 20 f eet below the surface of the ground. The load was applied by means of a jack and after the bearing pla te had come to res t , the settlement was read to t he nearest hundredth of an inch. In anal yzing the dota, it was not cons idered suitable to derive a psuedo-theoretice.1 equation and attempt to evaluate constants to give the desired results. arrived a t by pure induction. The final relati ons were No attemp t ·. was made to embody any of t he theory of eoil me chanics ., it be ing t hought t hat strict a dherence t o inchlct ive reasoni ng would result in a s olut i on of' t he g r eat est s i mplicit y . In the inves tigation only that portion of each t e s t up to the s o-called " yi eld point" of the soil was consid ered;. t he "y i eld point" of t he soil bei ng that point at which a".1". a.11 increas e i.n load produces r api d increas e i n s ettl ement; it being t hat bear ing capaci t y wh ich in prectiee i s considered the ulU t1ate s a f e ce.pa ci ty of the soil. The :portto-ns of the test loaded beyond this point , being of no practical signifi cance, were , t herefore, disregarded. CB.AP'TiiB II . The first step undertaken was t .t10 det ermination of t he general form of t he curves es described b y the load-settlement data. A careful consid erati on disclos ed that the data was best fitted by cur v es of the genoral fonn * Y = a Xb; where ~,y -, is tlle s e ttlement and "X" is the load in pounds p-er square foot of bear ing area .. Ha.Ying de termined th.is general rel ation the nE>xt step was to evaluate the co n1: ta.nts "a 01 and "b". The proeedure was as f'ollo·,\'s. Ji'irst, the nume r ical values of "a" and "b'' were dete.:m1ined for a.pproxhria tely 30 typical t e sts. nxamples of these are:• # 69 #41-B 1t54 Sandy Loam Silt Bearing a rea = 2 sq .ft. Bearing area = 4 sq;ft. Bearing area = 2 sq.ft. Loam :Elevat ion =· - ? ft. y.., 11.6 106 1.213 X U.evation = - 20 ft. 1.10 y - 16.6 X io6 J-i.1 evati.on • - Y• 3.55 X 7 17 :ft. 1 .. 25 In further discussions the slope of a curve will be taken to mean the slope of the tangent to the curve at the ori gin, and the curvature wtll mean the rate of curvature. * tt:&Rpirical Formulas", by Theodore Running, p. 42 . Sons, 1917. .John Wiley & 6. In studying these Z-0 typical equations, eonsidering a group of t 1::,s ts on one type oi' soi .l sueh as loam, certain characteristics were noted., It was seen t hat as the bes.ring area was ine:reased t he slope of the curve (constant "a") was increased. the bearing area was increased the curvature (constant increased. Also, as 11 b") was 'I'hese facts we re evident both :from the curves and from the equations. This t hen made the gen.eral equation of the form:Y = m • f' (A) • (X)kf' (A) A• bearing area k anc5 m = constants In the sa111e manner, considering tlle effeet of change of elevation, it was :found that both slope and 01.u~vature were decreased as the depth was incr eased. With "t" representing the depth the genc;ral form of the equation then was:- k f' {A) y • m f {A) f ( t) f' ( t) (X) I t was then necessary to determine more exact values of f(A), f(t)t f'(A}, and f'(t). Considering f'irat the exponent of "X'' , it was noted that in every case it rang e d within the limits 1 and 2. This meant that the effect of change of beering area or depth was small and the relationship, therefore, waa something other than k f;#t' . ested that the f f .A t.. .• . An e:i-::a.rnination of' the values sn.<?g ""i:;:> t variation was caused by a relati onship of the form k {A:• A study of data wherein two tests were made at the sa'1l.e location, for e 2:a.mple: #29-A #29-B Bearing area• 2 sq.ft. Bearing area= 2 sq.ft. I~J.evation = -15.5 ft. Elevati on"" -10 ft. 1.387 X 1.35 X showed that this relationship was probable. The general equation then was or th e form:- 'l'he next step then was to proceed in the same manner for m !i!l . Examination of the data. showed that the value of this f(t) factor varied markedly as the bearing area and depth varied. This suggested that the relf1tionship could be e xpressed as k A . t Testing this on data, ,.such as //29-A and. #29-B, led t-0 the con- clusion tha t it was be1?t expressed as m ~ . then we.sin the fonn:Y = m~ The general equation w t X t A further study, however, showed that the exponent k ;A was not correct. .l!'or, wh ile with small boari ng areas and large depths the results seemed correct, it was seen that large bearing areas and small depths would give impossible v alues . It was evident tha t the effect of' depth ., -t 11 was too marked. 'l'Ms e.ffect could be reduced to the pro per amount by the inclus ion of a eonstant "\ctr::Testing this upon the data showed that the "c" , thus k VA. assumption of this constant was justified. then was of the form:- The generai equation '\c~- k VA X Considering then the coefficient m~, and t esting it thoroughly upon the data, it was discovered that the f unction 1 2 was not Jl. / , but was somet h ing slir,b.tly di:fferent, depending upon t he soil coasidered.. n m t- The eoe.f f:J.cient was t t e:rei'ore written as where ttn" was a constant to be determined. then was: ya The C!)Uation "\ctr:k VA (X} with Y being the settlement in i nches, X being pounds per square foot of b earing 'area, and m, n, k, a nd c being constants thn t mu.st be determined . It should 'be understood that the constants must be determined each time the equation is us ed . 'i'ho values of the co nstants det er mined for sandy loam at one loea tion ar e not necessarily the same for s~.d) ,loam. a t another locati on. This fact is i mportant 9. and should be noted, for failure to do so will lead to erroneous results. To det ermine the constants m, n, k and a, two t ests are required. Having the COil;Stants determined the e quation then holds true for different values of bearing area (A) and depth (t). It should further be noted that the rela tionship of the depth does not hold for extreme projections. T'nat b, if the conBtants a.re de termi ned for a depth of 2', considerable error may be expected if the equa- tion is applied to a depth of 30'. However, for changes of depth from 5' to 10' or from 15' ts 25' the erro.r introduced will be relatively small. To make t he equation easier to handle and l ess s ensitive to inaccuracies the final form wa.s written;- Y = s ettlement in inches. x • pounds per s quare rt. of bearing area. A = bearing area. (l--i \i-- •· t m depth of bearing plate. lt ~ 'I'he following is an example illustrating the use and accuracy of the equation. 10. Willowbrook Junior Hi@ School. Comnton District Elevation = -5' Load Lb./sg.ft. l s9 .rt. Settlement in Inches 3 so.ft. 9 sg.ft. 500 0.,030 0.040 0.058 1000 0.060 0.081 0.118 1500 0.087 0.120 0.1?5 2000 0.120 0.168 0.236 2500 0.150 0.217 0.292 3000 0.18 0 0.267 0.357 5500 o.22z 0.335 0.435 4000 o.265 0. 412 0.540 11. 60 .,50 " • . ,,4() "' " ";'\ "' ~ "\ '- - ' I'\.. "\ !Sf· ~~ "\. "\. --~ ).... ~tFt. "' " "' " ~~ "' '\. . ~ .07 06 ~ ~ ' 04 .03 f/2. 4.0 3.6 3.0 2-S Z.0 /.5 Values t>f £ /.O 12. First plot the values of s ttlement against load of the tests on l sq. ft. and 5 sq. rt. one logarith~ie scale, n ~~ log Y = log { !....) + (k y A ) log x .. Then th& slope o:f the t ct 1000 line, Fi~ure Ho. l , ~q~.ls k\[A and tho intercept of the line ~ . : t - ~]) with the Y ax1aA'd:et~!:nea the value of log {m !f:. ) . The equot tions as deter. ined from this proce ure are:-_ l so . ft. area lh2!?. ( Y= ~ 100 ( Th.~etore vr Y = }l.022 X 100 1000 )1.050 X 1000 5o k • l . 022 5c = 1.050 k~ Therefore k = 1 . 022 and - 5c log 1 l e ::i, .!.. l • lo~11. 022 •05 log 3 = 8 ., 6"/ 5 log 1 . 025 Also m • 1n = 5 . 95 5 Ioo'" m . ':}J,c~ 5 100 Therefore m • 5 • 5 . 95 100 a 29 . 78 100 13. and log m + n log 3 - log 5 = log 8.35 - log 100 log m + n log 1 - log 5 = lgg 5.,95 - log 100 n log 5 = log 8.35 - log 5.95 n • + Theref'or~, m • log 3 • 0.31 29.78 100 n = 0.31 k = 1.022 C '"' 8.67 The equation then is of the form Y• 29.78 100 {0.31) A . --t ~.6,!:L_ 1.022 VA ( Then substituing values of A* 9 sq.ft-. t ... 5 f t. Y• 11.74 100 ( X 1000 1.078 ) X 1000 ) The values given by this equation are:9 sq.f't. bearing area. Kl.evation y • 1.1..74 100 = -5' I X )1.0?8 \ 1000 Settlf;lllant in. Inches 500 0.000 0.057 1000 0.118 0.118 1500 0.175 0.182 2000 0.236 0.248 2500 0.292 0.315 3000 0.357 0.383 3500 0.435 0.452 4000 0.540 0.622 This data is plotted in Figure 2. IS. Il COMPARISON OF COMPUTED WITH ACTUAL SETTLEMENTS ao----~------,------,------, ()./ \ \ \ 0.2 \ ~ ., v-1\,, 0 ', v(\ 0.3 \ I,.. ~ !.., ........ Iii ., T£ST -~ b ~ ~ ~ I \ 0,4, COMPUTE. \ \ \ ~ \ o.s . 1,~ ll•[ \ ~6J------+-----+-,-----;-------"-1 \ ~1'-----......L.----....i-----'---....i..---0 /1100 20110 LO/ID IN POl/N DS f£R 3()00 S9 UAR£ F'tlOT 4()00 16 .. CONCLUSION While the equation developed in this pa.per gives a close correlation with test datat its i-esults for pra ctical applications are not final. . This limitation is due to the neglect of certain factors 'l'l1e dif:fi cul ty which eanoot well be embodied in such an eq_uati on. entailed . i:n an attempt to formulat-e an expression embracing such variable eonditiona as moisture content t effeet of' proximity or footings and non-homogeneity of soil. is obvious. However, the effects of theae conditions do not make t he results any less valid. In every instance, (exee_pt in the oase of pure sand} under uniform load· ~ inerease in bearing area will result in an inerease in settlement encl a:n increase in depth results in a decrease o.f settlement. 'l'herefore, the use of the equation is a s tep closer to the _,exa ct solution. In an application of' the equation it should be noted that the ac.curm.ey of the results depends for the most part upon the accuracy of the test data. This does not mean accuracy in reading the settlement to the thousandth of an inch, f'or slight ina.ceuracies of this sort can be eliminated by drawing a smooth curve through the plotted points of the test data. However, an error of the sort in which one tneth of an inch is added to eaeh moo.sured settleinent will lead to entirely erroneous results. 17. If the test data do-es not contain serious errors the results obtained by the use of the equation will closely approximate the actual settlements, The value of the equation is demonstrated in the design of footings for a building. In the usual oos e a typical interior footing is subjected to a load approximately twice the load on a typical exterior footing,. If the interior footing then is designed with twice the bearing area, it is .o bvi ,:ms that the settlement will be much larger than tha t of the exterior tooting, thus causing undesira ble secondary stresses. This condition of unequal settlements would be to a great extent elimina ted by the use .o f the equation developed in the body of this paper. 18. BIBLIOGRAPHY "Prineipl.e.s of Boil Mechc·1.nicsn ~ Dr. Charles 'i'erza.ghi, 99 PP• McGraw...Hill Pub. Co., 1926. "The Sc.i enee o~ Foundation s " , Dr. Charles Terzaghi, PP• 272-380, Trans. Am-. Soc.C .E., 192$) . 0 Ji. :rdbaumechanik au.f.' BodenphysikaH:sher Gru.nalage" 1 Dr. Karl Terzagh1., pp. 184-385.* .l.i'r anz Deutike, 1925. "Soil Mechanics Reeearch" , F. K6glers pp .. 1560-1565, Proceedings Am.soc.O.l;~ * Nov.• 1932. "Besearch in Soil .Engineering"~ Glennon Gilbgy, Proceedings Am.Soc.C. E., Oct., 1931. "A Practical Method tor the Selection o-f Foundations,, Based upon Fundamental Research i n Soil ~fochanies", w. s. Housel, 88 pp., Univ. of tl1ichi gan .Engrg. Research Bulletin No. 13, Oet., 1929. Iowa State College l!:ngi neering Reaeareh Bulletin No. 101, 54 pp., J. Griffith,. June, 1931. "The Effect of a Varying Moisture Content on the- Settlement of Soil Subjected to a. Constant Loadtt, Masters •1'1lesis by Laverne Leeper._ Cal-Tec-h, 1933. nn:npirical Formulas", Theodore Running, John Wiley & Sons. 1917.