Chaoyang University of Technology -- General KT Theory --

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1 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- NIT 3 APPLICATION OF TERZAGHI S THEORY OF ONE DIMENSIONAL CONSOLIDATION TO PROBLEMS INVOLVING VARIOS STRESS SRFACES AND DRAINAGE CONDITIONS, TIME DEPENDENT LOADING, AND LAYERED SYSTEMS Prepared by Dr. Roy E. Olso o Sprig 989 Modified by Jiure Lai o Fall 3. Itrodutio Previously, Terzaghi's theory of oe dimesioal osolidatio was applied to the ase of istataeous loadig, uiform distributio of iitial exess pore water pressure with depth (retagular stress surfae ad freely draiig boudaries. Although this solutio is the oe of most pratial usefuless there are problems i foudatio egieerig where other solutios should be used. As a first step i geeralizig previous solutios, this set of otes will otai solutios for ases i whih the stress surfae is ot retagular, ad where the boudaries may ot be freely draiig, the loadig may be time depedet, the soil is stratified, ad a few other ases. Several ases i whih the earlier solutios do ot apply will be preseted to illustrate the types of problems that may be ivestigated usig the solutios give i this hapter. I all ases the magitude ad rate of settlemet are to be determied. Case : At various, ad ukow, times i the past, fill was plaed over the surfae of a deposit of soft lay. It is ow desired to add a fial layer of ompated fill ad to ostrut idustrial buildigs o the site. Piezometers istalled i the lay layer idiated that exess pore water pressures remai i the lay from earlier loadigs. Therefore these remaiig exess pore water pressures, whih are ot uiform through the lay, have to be added to the uiform exess pore water pressure idued by the ew fill to obtai the total iitial exess pore water pressure to be used i the settlemet aalysis (Fig. 3.a. Case a: A ompressible lay layer has bee subjeted to steady state seepage from a uderlyig sad layer that is uder artesia pressure. The the pressure i the sad layer is suddely redued to the stati value (Fig. 3.b. Case b: A layer of soft orgai silt is overlai ad uderlai by sad. As part of ostrutio work at the site, deep well pumps are used to dewater the lower sad without iflueig the water level i the upper sad (Fig /9/5

2 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- Fig. 3. No-Retagular Stress Surfaes Eoutered i Egieerig Pratie Case 3: A approah embakmet for a large bridge is ostruted over a deep deposit of lay. A period of several years is ivolved i the plaemet of the fill. der the eter of the embakmet the osolidatio is essetially oe dimesioal but the loadig is time depedet. Case 4: A large grai-elevator struture with a mat foudatio is to be ostruted at a site where there is a desiated rust overlyig a soft lay overlyig a freely draiig sad. The thikess ad oeffiiet of permeability of the rust are suh that a sigifiat amout of pore water esapig from the soft lay will pass through the rust. I this ase, oe boudary of the ompressible soft lay is freely draiig but the other is either freely draiig or impervious. Solutios for these ad a variety of other similar osolidatio problems will be preseted i these otes. 34 4/9/5

3 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory Compressible Layer of Thikess H Loated Betwee Two Freely Draiig Boudaries with Iitial Exess Pore Water Pressure a Futio of Depth Beause the boudary oditios are the same as those used previously, the aalysis may proeed diretly from the followig equatio from the previous set of otes (valid for double draiage: u = ( H H = πz πz u( z,si( dz si( exp( π T H H 4 (3. where u(z,, represets the iitial distributio of exess pore water pressure, i.e., the stress surfae. Sie oly itegratio of simple futios is ivolved i fidig the solutios, oly the fial solutios will be preseted. Equatios will be preseted for the isohroes ad for the average degree of osolidatio. I all ases the average degree of osolidatio is defied i suh a way that: S = S u ( Triagular Stress Surfae The term "triagular stress surfae" is used to desigate a stress surfae of triagular shape that has its base at oe boudary ad its apex at the other (Cases a ad b. To avoid ueessary ompliatios i the aalysis, the depth oordiate, z, will always be measured ito the ompressible layer from the apex of the triagular stress surfae. The iitial pore water pressure at the base of the triagular stress surfae will be desigated as u b. The solutios for the isohroes ad average degree of osolidatio are: u = =,,3K + ub ( π πz si( exp( π T H 4 (3.3 ad: = m=,,, K exp( M T M (3.4 t i whih M = π (m + = π ad T = v. The isohroes are plotted i Fig. 3.. H The T- relatioship is exatly the same as that obtaied i the ase of a retagular stress surfae. 3.. Trapezoidal Stress Surfae The stress surfae has a trapezoidal shape i ases where the iitial exess pore water pressure ireases liearly from a fiite value at oe boudary to a greater value at the other boudary. I pratie this ase will arise if a uiform distributio of iitial exess pore pressure is superimposed o the oditios desribed i Cases a ad b. Hee the 35 4/9/5

4 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory --. Elevatio Relative to the Clay Surfae (z/h % 9% 8% 7% 6% 5% 4% 3% % % % Dimesioless Exess Pore Water Pressure. Fig. Isohroes for Triagular Stress Surfae solutios for isohroes are obtaied by dividig the trapezoidal stress surfae ito its retagular ad triagular ompoets ad obtaiig two separate solutios whih are the added to obtai the isohroes. The T- relatioship is the same as for the retagular stress surfae Siusoidal Stress Surfae As osolidatio of a homogeeous layer of ompressible soil otiues, the pore pressure isohroes ted toward a siusoidal shape. Thus, if a areal fill is applied at some time, or times, i the past, there may be a residual siusoidal distributio of exess pore water pressure at the time you beome ivolved i the projet. I the ase i whih both boudaries are freely draiig, the exess pore water pressures are zero at the two boudaries ad irease siusoidally to a value of u s at the mid-depth of the ompressible layer, Case. Whe this distributio of exess pore water pressure is iserted ito Eq. 3., all terms disappear exept the first oe ad the exess pore water pressure ad average degree of osolidatio are give by: πz u = u si( exp( s π T (3.5 H 4 ad: = exp( π T (3.6 4 The T- urve is plotted i Fig. 3.3 where it may be ompared with the T- urve for a liear stress surfae. Values of T are preseted i Table /9/5

5 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory Degree of Cosolidatio (% liear siusoidal Time Fator Fig. 3 T- Curves for Liear ad Siusoidal Stress Surfaes Table 3. T- Relatioships for the Two Most seful Stress Surfaes Average Degree of Cosolidatio, % Time Fator for Trapezoidal Stress Surfae Time Fator for Siusoidal Stress Surfae 3..4 T- Relatioship for Composite Stress Surfaes As oted earlier, settlemet aalyses for ases with omplex stress surfaes, suh as trapezoidal, a be simplified by dividig the stress surfae ito its simple ompoets, suh as a retagle ad triagle, obtaiig separate solutios for eah ompoet stress surfae, 37 4/9/5

6 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- ad the addig the solutios. For pratial purposes the T- relatioship is of mai iterest. It is oveiet to develop a equatio for the alulatio of a omposite T- relatioship i terms of the separate solutios for eah of the ompoet stress surfaes. The average degree of osolidatio is defied usig the equatio: H u dz i = H i H u dz udz (3.7 where i desigates the iitial oditio. Equatio 7 a be rewritte i terms of ompoet stress surfaes as: N H ij = j= H u dz u dz i H u dz j (3.8 where j desigates the j-th ompoet of the stress surfae. Equatio 8 a be simplified to: = N j= H j H u dz ij (3.9 u dz i where j deotes the average degree of osolidatio for the j-th ompoet stress surfae. The omposite average degree of osolidatio is thus a weighted average. The most ommo example of the formatio of a omposite stress surfae i egieerig pratie was give as Case at the begiig of this hapter Cosolidatio Resultig from Seepage Pressures The lay layer show i Fig. d is iitially i equilibrium with the water table at the surfae of the overlyig sad layer. The site uder ostrutio is loated i a valley. Dowstream of this site a dam is ostruted ad the reservoir is filled with water. For simpliity it is assumed that the time eeded to fill the reservoir is very small ompared with the time eeded for the lay to osolidate. If the water surfae at the site is raised by a distae L, the the pore water pressures throughout the sad ad lay layers is raised by a amout Lγ w. However, the lower sad has a distat draiage outlet suh that its pore water pressures are maitaied at their origial value. The higher pore water pressures i the lay layer tha ause water to flow from the lay ito the lower sad ad osolidatio ours. The resultig settlemet of the surfae is easily alulated by dividig the lay layer ito sublayers of suitable thikesses, alulatig the iitial ad fial values of effetive stress at their mid-depths, determiig the hage of thikess of eah usig the proedures outlied i Chapter, ad summig to obtai the settlemet. 38 4/9/5

7 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- The remaiig problem is to alulate the T- relatioship ad the pore water pressure isohroes. The exess pore water pressures, u, are here defied as those i exess of the iitial pore water pressures, before the water surfae was raised. The thikess of the ompressible layer is H. The boudary oditios are:. u (, t = ub. u ( H, t = ad the iitial oditio is: 3. u ( z, = ub The exess pore water pressure for this problem osists of two parts oly oe of whih dissipates: u = ub ( z / H + u (3. where u is the part that dissipates. Equatio 3. satisfies the differetial equatio of oe-dimesioal osolidatio. The aalysis proeeds, as i the previous set of otes, by substitutig F(zG(t for u ad applyig the boudary ad iitial oditios. The solutio for the isohroes is: u z ub + πz = ub( + ( si( exp( π T (3. H π H 4 The terms withi the summatio sig are idetial to those for the full triagular stress surfae with double draiage (Eq. 3. Thus, the pore water pressure isohroes are obtaied diretly from previous aalysis (Fig.. Sie all terms ivolvig time fator are the same as for the triagular stress surfae with double draiage, the T- relatioship is also idetial ad the time-settlemet urve is easily obtaied. 3.3 Time Depedat Loadig 3.3. Itrodutio Previous aalyses have applied to ases i whih the loadig time was so small, ompared to the times required to dissipate the exess pore water pressures, that the loadig ould be assumed istataeous. I some problems of pratial iterest this assumptio aot be made (Case 3. For suh problems the usual pratie is to divide the load-time diagram ito a suitable umber of subdivisios, assume that the load applied durig eah subdivisio is applied istataeously, alulate the settlemet-time urve for eah suh loadig, ad the add these settlemet-time urves to obtai a estimate of the atual time rate of settlemet. Although the foregoig proedure yields solutios of satisfatory auray, it is ofte simpler to utilize solutios that have bee developed by dividig the load-time diagram ito differetial elemets ad itegratig (Terzaghi ad Frohlih, 936; Olso, 977. Although 39 4/9/5

8 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- oe-dimesioal solutios for time-depedet loadig a be obtaied without great diffiulty for a variety of stress surfaes ad shapes of load-time urves, it appears that the ase of most pratial iterest is a retagular stress surfae, sigle or double draiage, ad a loadig that ireases (or dereases liearly with time up to the ed of ostrutio ad the remais ostat; this will be the oly ase osidered i this setio. Other solutios will be foud i Terzaghi ad Frohlih ( Cosolidatio of a Doubly Draied Clay Layer Subjeted to a Retagular Stress Surfae Produed by a iform Surfae Pressure that Ireases Liearly with Time A lay layer of thikess H is elosed betwee two freely draiig boudaries. The applied pressures are uiform aross the horizotal surfae so that the problem is oe dimesioal ad the stress surfae is retagular. The surfae pressure is ireased at a uiform rate for a period of time t to a value of q whih is the maitaied ostat idefiitely. The loadig diagram is show i Fig It is assumed that every differetial q q dq dt(i t(i t t Fig. 3.4 Time-Depedat Loadig Diagram iremet of loadig auses a uiform iremet of exess pore water pressure through the lay layer. Thus: q d u = dt i = dq i (3. t As a matter of oveiee, a dimesioless time, ν, defied by the followig equatio: is used. ν = t t (3.3 It is assumed that the properties of the lay remai ostat durig loadig ad osolidatio so that the priiple of superpositio a be applied. The exess pore water pressure, du, 4 4/9/5

9 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- remaiig at some depth z i the layer at some time t-t i after appliatio of a differetial exess pore water pressure, du i, at time t i, is: dui Mz M v ( t ti d u = si( exp (3.4 m= M H H To obtai the exess pore water pressures remaiig from all previous differetial loadigs, du i, is replaed by (q /t dt i ad Eq. 3.4 is itegrated. The itegratio is performed separately for < ad ν >. For ν <, Eq. 3. is writte: u d u = t m= q Mt Mz M v ( t ti si( exp dti H H (3.5 Term-by-term itegratio yields: m= [ exp( M T ] q Mz u = si( (3.6 3 M T H i whih M = π (m+, T = vt H ad T = vt H. The average degree of osolidatio,, is defied as: = Hνq Hq H udz = ν Hq H udz (3.7 This partiular defiitio was hose so that the settlemet of the surfae of the ompressible layer would equal the average degree of osolidatio times the ultimate settlemet uder the load q. Equatio 6 is iserted ito Eq 3.7 to obtai: [ ] exp( M T T = (3.8 4 T T m= M For ν, a similar aalysis leads to the followig equatios: u = = m q M Mz [ exp( M T ] si( exp( M T 3 T H (3.9 ad: m= [ exp( M T ] = exp( 4 M T T M (3. 4 4/9/5

10 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- The T- relatioship are plotted i Fig. 3.5 for a wide rage of values of the ostrutio time fator, T. It is apparet that time-depedet loadig should be take ito aout if preditios of the time-settlemet urve are eeded before times equal to about 5 t. Fig. 3.5 T- Relatioships for Various Values of the Costrutio Time Fator 4 4/9/5

11 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory Disotiuous Loadig A ommo problem i pratie is that of alulatig the time-settlemet urve for a embakmet that is ostruted at a disotiuous rate. For example, the loadig may irease liearly durig the first ostrutio seaso, remai ostat durig the witer, irease liearly durig the seod seaso, ad the remai ostat. The solutios are obtaied, for suh problems, by applyig the foregoig equatios to eah stage of timedepedet loadig separately ad addig the settlemets to obtai the total time-settlemet urve. A omposite T- urve a be alulated usig Eq. 9 with the weightig fator defied as the stress applied by ay oe ramp load divided by the applied stress used i alulatig the ultimate settlemet. 3.4 Partially Draiig Boudaries 34. Itrodutio I some problems of pratial iterest the mai soure of settlemet is a sigle more-or-less homogeeous layer whih is bouded above ad/or below by layers that otribute little to total settlemet but whih have oeffiiets of permeability ad thikesses suh that they a be osidered either freely draiig or impervious. I real ases, of this type, the properties of the partially draiig boudaries are likely to be too ill-defied to make a aalysis useful. However, the aalysis is relatively simple ad will be preseted for possible use i aswerig "what if" questios. The idealized ase to be osidered, Fig. 3.6, osists of a ompressible layer of thikess L bouded by two iompressible layers of thikess H ad H ad oeffiiets of permeability k ad k, respetively. The iompressible layers are, i tur, bouded by freely draiig layers. Cosolidatio is assumed to be oe dimesioal. A solutio will be developed for a geeralized stress surfae with istataeous loadig. Equatios will be preseted for the speial ase of a retagular stress surfae. These Freely draiig boudary Iompressible Layer No. H Compressible Layer z L Iompressible Layer No. H Freely draiig boudary Fig. 3.6 Soil Profile sed for Aalysis of a Sigle Compressible Layer with Partially Draiig Boudaries 43 4/9/5

12 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- equatios will the be itegrated to aout for time depedet loadig. Solutios for oretagular stress surfaes ad diverse load-time relatioships a be obtaied from the solutios preseted Istataeous Loadig Sie the assumptios regardig the osolidatio behavior of the ompressible layer are the same as those made previously, the aalysis may begi diretly from: u = [ C os( Az + C si( Az]exp( A ( vt The boudary oditios are: u(, t R = u(, t z L u ( L, t R = u( L, t z L (3.a (3.b ad the iitial oditio is: u ( z, = f ( z (3. i whih: k L R = ad kh k L R = (3.3 kh The dimesioless parameters R ad R defie the degree of perviousess of the iompressible layers; a value of zero idiates a impervious boudary whereas a value of ifiity idiates free draiage. se of these parameters was first suggested by Hamilto Gray (945. Appliatio of the boudary ad iitial oditios usig the geeralized Fourier series (Appedix 3-A leads to: u = L = L exp( α T D Z Z f ( z dz (3.4 i whih α represets suessive positive roots, other tha zero, of: ad: α ( R + R taα = (3.5 α R R D = ( α + R ( α + R ( α + R + ( α + R R ( R + R ( /9/5

13 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- Z z z = α os( α + R si( α (3.7 L L The simplest, ad ertaily most useful, solutio is for the ase of a ostat iitial exess pore water pressure, i.e., f(z = u i. Itegratio of Eq. 3.4 for this ase yields the followig solutio: i whih: i = u = u D E Z exp( α T (3.8 E = [ α si( α R os( α R ] α + (9 It is easily demostrated that the solutios preseted by Hamilto Gray (945 ad Bishop ad Gibso (964 are speial ases of Equatio 8. The average degree of osolidatio is defied so as to preserve the validity of Eq. 3.. Thus Eq. 3.8 is itegrated to obtai: = = D E exp( α T (3 Represetative T- urves for R ad R ragig from. to are preseted i Fig For pratial purposes the iompressible draiage layers may be osidered impervious for values of R less tha. ad freely draiig for R i exess of. The urves i Fig. 3.7 make it possible to alulate time-settlemet urves for most pratial problems i whih there is a sigle ompressible layer Time Depedet Loadig The solutios for the isohroes ad T- urves for istataeous loadig a easily be expaded to ilude ay desired variatio of applied surfae load with time by applyig the methods used earlier i this hapter. Agai, it is assumed that the load-time urve of most iterest is give i Fig The solutios are as follows: For ν q u = T = D E α Z [ exp( α T ] (3.3 DE = ν [ exp( α ] T (3.3 T α = For ν : 45 4/9/5

14 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- Fig. 3.7 T- Relatioships for a Compressible Layer with Partially Draiage Boudaries 46 4/9/5

15 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- q u = T = DEZ [exp( α T ]exp( T α α (3.33 = T = D E α [exp( α T ]exp( α T (3.34 I Eqs. 3.3 ad 3.34, was defied so as to maitai the validity of Eq. 3. for alulatig the time-settlemet urve. The additio of the variable T preludes oveiet presetatio of umerial results. However, solutios a easily be obtaied usig a digital omputer. 3.5 Other Cases Iludig Various Iitial ad Boudary Coditios Equatio 3.4 a be used to obtai solutios for oe dimesioal osolidatio of a ompressible layer with a variety of iitial ad boudary oditios. Solutios are easily obtaied for ay distributio of iitial exess pore water pressure that a be approximated by a itegrable futio, for ay degree of boudary draiage, ad for ay variatio of applied load with time Two-Layer Systems Solutios for a system of two otiguous layers with freely draiig or impervious exteral surfaes was preseted by Gray (945. The solutio is obtaied by applyig Terzaghi's differetial equatio withi eah layer ad, at the iterfae betwee the two layers, requirig that there be a sigle exess pore water pressure ad that the flow rates i the two layers be equal: u u ku ( = kl ( z z u l (3.35 For the ase of double draiage the T- relatioships are: siξα( α siξα + siα = ( osα exp( α α ( α si ξα + ξ si α T siα ( α siξα + siα = ( osα exp( α ξ α ( α si ξα + ξ si α T (3.36a (3.36b Where α represets suessive positive roots of: α os α si ξα + si α os ξα = ( /9/5

16 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- ad: H ξ = H v v (3.36d α = k k v v (3.36e T t = (3.36f H v ad v, k, ad H are the oeffiiets of osolidatio ad permeability, ad the total layer thikess, respetively, ad subsripts ad deote the two layers. Either layer ( or a be o top. The time fator, T, is defied usig the properties of layer, eve i the equatio for layer. For the ase of sigle draiage, layer must be ext to the impervious boudary. The solutios are: siα osα siξα = exp( α ( α ( α si ξα + ξ os α T os α( osξα = exp( α ξ ( α ( α si ξα + ξ os α T (3.37a (3.37b where α represets suessive positive roots of: α si α siξα osα osξα = (3.37 ad Eqs. 3.36d-3.36f remai valid. It is oveiet to defie a omposite suh that Eq. 3. remais valid, but with: S u = S ul + S u (3.38 ad S ul ad S u are the ultimate ompressios of the two layers. = ( + ζ (+ζ It may be show that: (3.39 where: ζ = S u S ul = ( H H ( vl v ( k k ( /9/5

17 Departmet of Costrutio Egieerig Advaed Soil Mehais Chaoyag iversity of Tehology -- Geeral KT Theory -- (Gray used a differet defiitio of whih ivalidated Eq. 3.. Gray's equatios have foud o kow appliatio i egieerig pratie beause they are too diffiult to solve i a speifi ase ad beause the umber of variables is too large to allow oveiet presetatio of geeral solutios. The equatios a be (ad have bee solved usig a digital omputer but the effort is exessive osiderig the remaiig approximatios (ostat properties, small strais, istataeous loadig.... Efforts have bee made to approximate layered systems usig a equivalet homogeeous soil, e.g., Davis ad Lee (969. Suh solutios are ot aeptably aurate ad serve little purpose i a age of umerial methods. Suh methods will ot be disussed. It may be oted that Gray's equatios a easily be exteded to over time depedet loadig problems followig the methods disussed earlier. There appear to be o published losed-form solutios for systems omposed of three or more layers but a "solutio" left i somewhat more geeral terms, was developed by Shiffma ad Stei (97 for multilayered systems Other Cases Aalyses have bee performed for a variety of other ases ivolvig oe-dimesioal primary osolidatio. However, the resultig equatios have geerally bee suh that losed form solutios ould ot be obtaied. The authors thus formulated umerial solutios (see later otes. 3.6 Referees Bishop, A. W. ad R. E. Gibso (964, "The Ifluee of the Provisios for Boudary Draiage o Stregth ad Cosolidatio Charater-istis of Soil Measured i the Triaxial Apparatus," pp , ASTM STP 36. Davis, E. H. ad I. K. Lee (969, "Oe Dimesioal Cosolidatio of Layered Soils," Pro. Seveth Iter. Cof. o Soil Meh. ad Foud. Egr., Mexio City, Vol., pp Glik, G. W. (945, disussio, Trasatios, ASCE, Vol., pp Gray, Hamilto (945, "Simultaeous Cosolidatio of Cotiguous Layers of like Compressible Soils," Trasatios, ASCE, Vol., pp Olso, R. E. (977, "Cosolidatio der Time Depedet Loadig," Jour., Geot. Egr. Div., ASCE, Vol. 3, No., pp Shiffma, R. L. ad J. R. Stei (97, "Oe-Dimesioal Cosolidatio of Layered Systems," Jour., Soil Meh. ad Foud. Div., ASCE, Vol. 96, No. SM4, pp Terzaghi, K. T. ad O. K. Frohlih (936, Theorie der Setzug vo Toshihte, Fraz Deutike, Leipzig, 66 pp. 49 4/9/5