coeficiente de reação do solo terzaghi

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E V A L UA T I O N O F CO E F F IC IE NTS OF SUBG RA DE RE A CT IO NbY

KARL TERZAGHI, %I.I.C.E., Hon.M.ASCE.SYi iOPSIS

T he theori es of verti cal and horizontal subgradereaction are based on the simpl ifyi ng assumptionsthat the subgrade obeys H ookes law, and that thesubgrade reaction on the base of a ri gid centr all yloaded plate resting on the hori zontal sur face of thesubgrade has the same value at every point of thebase. A lthough these assumptions are not ri gor-ously corr ect, the theories of subgrade reaction canbe used for obtain ing approximate solut ions ofmany practical problems, such as the computationof the stresses in conti nuous footings acted upon byconcentrated loads, or in pil es that ar e in tended totransfer horizontal load on to the subgrade. How-ever, in order to get reasonably accurate results, thecoefficients of subgrade reaction must be assignedvalues compatible with the deformation character-istics of the subgrade and the dimensions of theloaded area.T he Paper contai ns a discussion of the factorswhi ch determi ne the value of the coefficients ofboth vert ical an d hori zontal subgrade reaction ofcohesionless sand and stiff clay, and numeri calvalues are proposed for the constants whi ch appearin the equations defini ng these coefficients. I tcontai ns also bri ef reviews of the practical appli ca-tion of the theories of subgrade reaction.

L es theories de reaction verticale et hori zontaledes soubassement de terrassement sont fond& s surdes hypotheses de simpl ifi cation selon lesquelles laplateforme obeit a la loi de H ooke et que la reactionde soubassement sur la base dune plaque r igidechargee centralement et reposant sur la surfacehori zontale de la soubassemcnt a la m& me valeur entout point de cette base. Bien que ces hypothesesne soient pas stri ctement corr ectes, les theories surla reaction de soubassement peuvent & tre uti li seespour obtenir des solutions approximat ives de nom-breux pr oblemes prati ques tels que le calcul deseffort s dans des empattements conti nus sur lesquelsagissent des charges concentr ees ou dans des pieuxqui ont pour but de transfercr la charge horizontalesur la soubassement. T outefois, afin dobtenir dessemelles de fondation raisonnablement p& is, lescoefficients de reaction de soubassement doiventrecevoir des valeurs compatibl es avec les carac-teristi ques de deformation de la soubassement etavec les dimensions de la surface chargee.Ce document compr end un examen des facteursdeterminant la valeur des coefficients a la fois dereaction verticale et de reaction hori zontale desoubassement pour du sable non-coherent et deI argile raide. Y figurent en outre des valeurssuggerees pour les constantes apparaissant dans lesequations qui defin issent ces coefficients. L e Docu-ment donne aussi de brefs exposes sur lapplicati onprat ique des theories de reaction de soubassement.

A

B (ft)cD (ft)E (tons ft-2)E , (tons ft-2)H (ft)h (ft)I (f@)KK A&IKllkd (tons ft-3)k h (tons ft-3)k , (tons ft-3)

4*

N O T A T I O Ndenotes ratio between modulus of elasticity E , of cohesionless sand and over-.

burden pressure pv = yz.width of a surface of contact.constant of integration.depth, vertical distance.modulus of elasticity of a beam or slab.modulus of elasticity of cohesionless sand under confining pressure p,.thickness of a stratum, length of embedded portion of a pile.thickness of a beam or a slab.moment of inertia of a beam.coefficient of earth pressure.coefficient of active earth pressure.coefficient of earth pressure at rest.coefficient of earth pressure corresponding to lateral displacement of

wall with height H over distance ye = 0.0002H.coefficient of subgrade reaction on diaphragm in an earth dam.coefficient of horizontal subgrade reaction.coefficient of vertical subgrade reaction.

297


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298 K. TERZAGHIk,, (tons ft-a) denotes basic value of coefficient of vertical subgrade reaction (value of

L (W1 = L / BLd (tons ft-3)

coefficient for square area with width B z 1 ft.length.

I] (tons ft-3)

M

AI,,, (tons ft)rnh (tons ft-4)Xnh (tons ft-3)$J (tons ft-2)9, (tons ft-2)PO (tons ft-2)p, (tons ft-2)$, (tons ft-2)~5, (tons ft-2)Q (tons)R (ft)yo (ft)Y (ft)Yo (ft), (ft)y (tons ft-a)/J

ratio between length and width of a rectangular area.constant of horizontal subgrade reaction on diaphragm in earth dam

with height H value for z/H = 1).constant of horizontal subgrade reaction for anchored bulkhead with

free earth support, depth of sheet-pile penetration D (value forz/ D = 1 ).

ratio between settlement of rectangular area with width B ft, oncohesionless sand and that of an area with width of 1 ft, at equalunit load.

maximum bending moment in a beam or slab.ratio between coefficient of horizontal subgrade reaction and depth

below surface.dimensionless quantity.constant of horizontal subgrade reaction on narrow, vertical, or hori-

zontal strips (value for z/ B = I ).increase of contact pressure due to displacement y.contact pressure on area acted upon by active earth pressure.contact pressure corresponding to earth pressure at rest on vertical

face.contact pressure on vertical face with height H , corresponding to

horizontal displacement y. = 0 . OOO Z H .9 0 + $ = contact pressure on vertical surface.yz = effective overburden pressure.concentrated load.radius of equivalent circular footing.radius of stiffness of a concrete mat.displacement.initial displacement, required for increasing the coefficient of earth

pressure on a vertical wall from K. to K,.depth below ground surface or dredge line.effective unit weight of sand.Poissons ratio for concrete.

INTRODUCTIONThe term subgrade veaction indicates the pressure p per unit of area of the surface of con-tact between a loaded beam or slab and the subgrade on which it rests and on to which it

transfers the loads. The coeficient of mbg rade reaction k, is the ratio between this pressure atany given point of the surface of contact and the settlement y produced by the load applicationat that point :

k,=$ . . . . . . . . . . (1)The value of k , depends on the elastic properties of the subgrade and on the dimensions ofthe area acted upon by the subgrade reaction.

The concept of subgrade reaction was introduced into applied mechanics by Winkler (1867))and was used by Zimmermann (1888) for the purpose of computing the stresses in railroadties which rest on ballast over their full length. During the following decades the theory


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EVALUATION OF COEFFICIEKTS OF SUBGRADE REACTIOK 299was expanded to include the computation of the stresses in flexible foundations, such ascontinuous footings or rafts and in concrete pavements acted upon by wheel loads. Thefundamental principle of the theory is illustrated by Fig. 1, representing a beam with lengthL , height H , and width B , which rests on the subgrade. At mid-length, it is acted upon bya line-load, q per unit of width. Let :

B = width of the beam,E = m od u l u s of elasticity of the beam,

H 3BI = = = moment of inertia of the beam,S = the vertical shearing force at a distance x from the midpoint of the length of the

beam, per unit of width,~5 = the subgrade reaction at distance x from the midpoint of the length of the beam,

per unit of area,y = the settlement of the base of the beam at distance x from the midpoint, andE = the base of natural logarithms.

The rate at which the shearing force S changes with the distance x from the mid-pointof the length of the beam is :

dS- , ,=p=k ,y , . . . . . . _ .According to the theory of bending of beams :

d S--.=...._. d x E ; ; ;Hence, in accordance with equation (2) :

k , y= -E ; g . . . . . . . . .A solution of this well-known equation is :

y = C, cash # cos # + C, sinh # sin # + Ca cash # sin 1+4 C, sinh I+%os $wherein :

is a pure number and C, to C4 are constants of integration. The value of these constants isdetermined by the boundary and continuity conditions. For the simple case illustrated byFig. 1, the computation furnishes for the value M,,, of the bending moment in the beam theequation :

M maX=&I(l -D) . . . . . . . .wherein :

D = COS tJ 1 + sin *r - e-*1sinh #r + sin #rand :

$1 =L ygand Q =q .BSince about 1920 the theory of subgrade reaction has also been used by several investi-

gators for computing the stresses in piles and sheet-piles which are acted upon by horizontal


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300 K. TERZAGHIforces above the ground surface. In this case, the seats of the subgrade reaction acts in ahorizontal direction. Therefore, the ratio between unit pressure and displacement will bereferred to as coefl cient of hori zontal subgrade reacti on k h,

If the subgrade consists of cohesionless material, such as clean sand, the unit pressure 9on a vertical face, required to produce a given horizontal displacement y increases approxim-ately in simple proportion to depth Z, whence :

Pk,, = - = ml Lz . . . . . . . .Y

. (5)

In this equation rnlL s a factor, the value of which depends on the relative density of the sandand the dimensions of the area acted upon by the subgrade reaction. If the value kl,, equa-tion (5), is introduced into equation (3), the equation :

I d4ymnzy=-E-- . . . . . . .B da+is obtained.

In any event the application of the theory of subgrade reaction to the computation of thebending moments in flexible beams, slabs or piles leads to a differential equation of the fourthorder, and the solution of such an equation is beyond the capacity of the average practisingengineer. This fact led to the following peculiar situation. Most of the Papers dealing withproblems of subgrade reaction have been written by investigators who are primarily inter-ested in the theoretical aspects of the problem. They published the solution of the differentialequation, taking it for granted that the value of the coefficient of subgrade reaction, k, or k,,,be known. Hayashi (1921), in his comprehensive treatise on the subject, notified the readerthat the value of k, should be determined by a loading test, but he did not mention the factthat the results of a loading test depend on the size of the loaded area. The book by Hetdnyi(1946) on beams on elastic foundations does not contain any statement regarding the factorswhich determine the numerical value of the coefficient of subgrade reaction.

This condition led to the erroneous conception, widespread among engineers, that thenumerical value of the coefficient of subgrade reaction depends exclusively on the nature ofthe subgrade. In other words, it became customary to assume that this coefficient has adefinite value for any given subgrade. Half a century ago Engesser (1893) pointed out thatthe value k, in equation (1) decreases with increasing width B of the beam, Fig. 1, approx-imately in accordance with the equation :

Fig. 1. F lexible beam acted upon at mid-length by load Q


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EVALUATION OF COEFFICIEKTS OF SUBGRADE REACTION 301wherein a and b are empirical constants. Terzaghi (1932b) published a Paper which dealtwith the factors that determined the value of k , for flexible raft foundations, acted upon byline loads. Yet these Papers received little attention, and even in recent years Papers werepublished in which most unrealistic values were assigned to the coefficientsof subgrade reaction.The results obtained on the basis of such values can be very misleading. It is hoped that thisPaper will assist in remedying the situation.

The Paper deals with the factors which determine the numerical value of the coefficientsof vertical and horizontal subgrade reaction for sand and stiff clay under simple and frequentlyencountered conditions. It also contains a description of the procedures by means of whichreasonable values for the coefficients of vertical and horizontal subgrade reaction can besecured. It will be concluded by a brief discussion of the errors due to the simplifying assump-tions on which the theories of subgrade reaction are based and of the means for reducing theerrors.

FACTORS DETERMIXNG THE VALUES OF k, AND khFwidmnedal assumj5tionsThe theories of subgrade reaction are based on the following simplifying assumptions :

(A) The ratio k between the contact pressure $ and the corresponding displacement yis independent of the pressure p.

(B) The coefficient of vertical subgrade reaction k , has the same value for every pointof the surface acted upon by the contact pressure. If the subgrade consists of stiff clay,the coefficient of horizontal subgrade reaction also has the same value, k j , , for every pointof the surface of contact. For cohesionless subgrades the value of the coefficient of hori-zontal subgrade reaction is determined by equation (5) :

k , , = m h z . . . . . . . . . 151and the value mh is assumed to be the same for every point of the surface of contact.

The errors contained in these simplifying assumptions are discussed on pp. 322-325 ofthis Paper.H or i z on t a l b eam s

If a flexible beam, Fig. 2(a), acted upon by loads Qi to Qn, rests on a compressible sub-grade, the settlement y of the beam will be greater at the points of load application than it isbetween these points. According to assumption (B), equation (l), the ratio between thesubgrade reaction p and the corresponding settlement y has the same value at every point :

P- = k , = constant . . . . . . . .Y

In order to illustrate the influence of the width B of the beam on the value k ,, the conceptof the bu l b o f p r essu r e can conveniently be used. The bulb of pressure is arbitrarily definedas the space within which the vertical normal stress in the subgrade is greater than one-fourthof the normal pressure on the surface of load application. The value of one-fourth has beenselected because the major portion of the settlement of a loaded plate resting on a fairlyhomogeneous subgrade is due to the compression and deformation of the soil located withinthe space defined by this value. Replacing this value by another one, such as one-third orone-sixth, would have no influence on the conclusions, because the concept of the bu l b o fpyesszwe merely serves the purpose of assisting the reader to visualize the stress conditionsin the loaded subgrade.

Fig. 2(b) shows the bulb of pressure for a beam with width B I and Fig. 2(c) that for a beamwith width ? tB , . The depth of these bulbs is D and D I D , respectively. The influence of the


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302 R . TEKZAGHIdepth of the bulb on the settlement of the loaded area depends on the deformation character-istics of the subgrade.

If the deformation characteristics are more or less independent of depth, like those of stiffclay, it can be assumed that the settlement y increases in simple proportion to the depth of thebulb of pressure :

Q, Q* Q. l

Fig. 2. (a) Flexible beam acted upon by loads Q1 to Qn; (6) and (c)influence of width of beam on depth of bulbs ofpressure ; (d) influence of width of beam on value KS, /KEfor beams on sand

Substituting k,, , = k , , B , = 1 ft, n B , = B and $1~~ = k, , , the equation :k , = K S1 . . . . . . . . . . (7)

is obtained, where k sl is the coefficient of vertical subgrade reaction for a beam with a widthof 1 ft, resting on stiff clay. The settlement y1 for the beam increases with time on accountof progressive consolidation of the clay under load, whereby k sl decreases and, like yl, ap-proaches an ultimate value. In this Paper the symbol k , always indicates the ultimate value.

If the beams rest on clean sand, the settlement y assumes almost instantaneously itsultimate value, but the deformation characteristics of the sand are a function of depth,because the modulus of elasticity of sand* increases with increasing depth (Terzaghi 1932a).On account of this fact the lower portion of the bulb of pressure of a beam with width n Bresting on sand is much less compressible than that of the bulb of a beam with width Bresting on the same sand, and the settlement yfl of the wider beam has a value intermediatebetween y and ny :

$ = (n) < 12 .* Ratio bet\vccn increase of de\-iator stress and corresponding increase of linear strain at unaltered con-fining pressure.


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EVALUATION OF COEFFICIE KTS OF SLJ BGRADE REXCTIOSExperimental investigations have shown that :

Y-= =:M. . , . . .Y lwherein yr is the settlement of a beam with a width of 1 ft at a given pressure 9 per unitof area of contact and y the settlement of a beam with a width B (feet) at equal contact pres-sure (Terzaghi and Peck, 1948). Equation (8a) is represented in Fig. 2(d) by a curve. Theabscissas indicate the width of the beam in feet and the ordinates the values of M.

The coefficient of subgrade reaction for a beam with width B is :. . . . . (8b)

wherein K S1 s the coefficient of subgrade reaction for a beam with a width of 1 ft.Com en t r a t ed l oad s on conc r e t e pavemen t s and concrete m a t s

If a concrete pavement is acted upon by a concentrated load Q such as a wheel load,the settlement Y of the pavement decreases from the point of load application in radial direc-tions, as shown in Fig. 3(a). The problem of computing the bending moments produced byload Q has been solved by Westergaard (1926) by means of the theory of suhgrade reaction.

Fig. 3. (u) Concrete pavement supporting concentratedload Q ; (b) relation between distance to (abscissas)and N, wh ich is proportional to both the verticaldeflexion y and the contact pressure p (ordinates)=Iccording to \Vestergaards findings the bending moments in the loaded slab and the

vertical displacements of the base of the slab are a function of a value :4

I0 =J

Eh312(1 - / G )k S

which is called the yndizls of sii$lzess of the slab. In this equation E represents the modulus


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3 0 4 K . TERZAGHIof elasticity, p Poissons ratio of the concrete and h the thickness of the slab. The value p isroughly equal to 0.15 and r,-, has the dimensions of a length.

In Fig. 3(b) the abscissas represent the radius of stiffness ~a and the ordinates thequantity :

&+~~wherein y is the settlement of the base of the slab at any point and p the corresponding contactpressure. Since N is proportional to the contact pressure p, the figure shows that the majorportion of the load is transferred on to the subgrade within a distance 2.5~~ from the point ofload application. Beyond this distance the settlement of the base of the slab is very small andthe removal of the mat beyond this distance would have little influence on the maximumbending moment in the slab. Therefore, the distance :

4 I 1cIc1.2

will be referred to as the range of ing%cence of the concentrated load and that portion of themat which is located within a distance R from the point of load application is the equivalentcircular footing.For concrete slabs on sand, R is very roughly equal to seven times the thickness h of the

slab. The stress distribution in the subgrade beneath the slab is practically the same as ifthe load were transferred on to the subgrade by the equivalent footing. Therefore, thecoefficient of subgrade reaction is approximately equal to that for the equivalent circularfooting, and the diameter of the footing depends on the thickness of the slab.

It has been shown before that the coefficient of subgrade reaction for a spread footingdecreases with increasing width of the footing and, as a consequence, with increasing radius ofstiffness ye, whereas Westergaard assumed that k , is independent of ye. However, if thevalue of k , for a footing covering an area of 1 ft x 1 ft is known, the value of k , for theslab can be estimated by means of the following indirect procedure.

The range of influence R is assigned an estimated value R I equal to seven times the thick-ness of the slab, and the corresponding coefficient of subgrade reaction k , is evaluated as shownin the last part of this Paper. This value of k , is introduced into the equation for R . If thedifference between R , and R is smaller,than about 50% of R ,, the value k , can be com-puted on the assumption that R is equal to the greater of the two values. Otherwise the com-putation should be repeated on the assumption that the range of influence is equal to the valueR furnished by the preceding computation. This should be continued until the differencebetween assumed and computed value of R becomes smaller than 50%.

Fig. 4(a) is a vertical section through a concrete mat covering an area mB by n B on thesurface of a deposit of stiff clay. The mat carries concentrated loads Q, such as column loadsspaced B both ways. The spacing B is assumed to be greater than twice the range R ofinfluence of the individual loads. If the loads were applied successively, one by one, thedistribution of the pressures in the bulb of pressure of the first load and the bending momentsin the mat, beneath the load, would remain practically unaltered. Therefore it is assumedthat the coefficient of k , for the mat is equal to that for a circular disk with radius R ,(equation (9)) and independent of the number of loads. The average load on each disk is :

QP=J22yrand the average settlement of the disk is :

y=pwhich is independent of the dimensions of the mit.


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EVALUATION OF COEFFICIE NTS OF SUBGRADE REACTION 305

Fig 4. (a) and (6) bulbs of pressure beneath concentratedloads Q , equally spaced both ways, acting on rec-tangular concrete mat ; (c) concentrated loads, and(d) line loads acting on mat foundation

In reality the application of each new load on to the mat increases the average settlementof the mat and, as a consequence, that of each point of load application. This is due to thefact that each new load increases the pressure on the subgrade located below the level of thebase of all the other bulbs of pressure. At the level of the base of the bulbs this supplementarypressure is almost uniformly distributed. The supplementary settlement, due to such apressure, has little influence on the bending moments in the raft beneath and at the mid-points between the individual loads. Therefore, in the theory of subgrade reaction theexistence of these pressures and of the corresponding settlement is commonly ignored. How-ever, the theory of elasticity as well as experience indicate that the pressures produce a dish-shaped deformation of the entire raft, superimposed on the deformations associated with theindividual bulbs of pressure, shown in Fig. 4(a). The errors resulting from this fact arediscussed under the heading, Corrections for the errors involved in the fundamentalassumptions.


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306 K. TERZAGHIIf the spacing B between the loads Q is reduced to values smaller than 2R, the bulbs of

pressure, with a top diameter 2R, overlap, as shown in Fig. 4(b). As a consequence the levelI-I, at which the distribution of the pressure due to the loads becomes practically uniform,is located high above the bottom of the bulbs. According to the basic assumptions of thetheory of subgrade reaction the compression of the subgrade located below this level has noinfluence on the deformation of the raft. Therefore, it seems reasonable to compute thecoefficient of subgrade reaction k , on the assumption that the range of influence of each load isBz and not R . The corresponding bulb of pressure is indicated in Fig. 4(b) below one of theloads by a dash line.

It has already been shown that the real settlement of the mat is much greater than thesettlement computed by means of k ,. It multiplies with increasing overall dimensions of theconcrete mat, and is associated with a dish-shaped deformation of the mat superimposedon the deformations produced by the individual loads, in accordance with the theory of sub-grade reaction. The bending moments produced by the unequal compression of the sub-grade between the level I-I (Fig. 4(b)) and the base of the mat will be referred to as thel o ca l bending moments and those due to the compression of the subgrade below the planeI-I as the gene ra l bending moments. The theory of subgrade reaction furnishes only thevalues of the local bending moments.

If the spacing of the loads is very different in two different directions, as shown in Fig.4(c), the range of influence of the loads is also different in the two principal directions. There-fore, two different values must be assigned to the coefficient of subgrade reaction. For thecomputation of the local bending moments in vertical planes interconnecting the closelyspaced columns, a lower limiting value of k , can be obtained on the assumption that the effectivearea of contact is a rectangle with width B , and length B , (cross-shaded area in Fig. 4(c)).For computing the local bending moments in vertical planes through the widely spacedcolumns (spacing Bz) , the value of k , should be selected on the assumption that the effectivearea of contact is a strip with width B, (shaded area in Fig. 4(c)).* If the loads are trans-mitted on to the mat by a set of parallel walls spaced B , Fig. 4(d), k , should be estimated onthe assumption that the effective contact area is a strip with width B (shaded area), or withwidth 2R, whichever is smaller.

In connexion with these simplifying assumptions it should be remembered that it issufficient, for practical purposes, to know the order of magnitude of the coefficient of sub-grade reaction k ,. The errors in the evaluation of the stresses in the mat due to an error of& 50% in the evaluation of k , are negligible. The real difficulties reside in the computationof the stresses, as shown below.

Pr a c t i c a l app l i c a t i o n s o f t h e t h eo r y o f ver t i c a l subg vade yeac t i o nThe most common practical application of the theory of vertical subgrade reaction is

the computation of the distribution of the contact pressures over the base of rigid footingsor mats. The results of the computation are independent of the numerical value of thecoefficient of subgrade reaction, but due to the simplifying assumptions involved in thetheory, the difference between the computed and the real pressure distribution can be import-ant. It is discussed in pp. 322-325 of this Paper.

The theory is also used for estimating the bending moments in continuous footings sup-porting rows of columns or heavy, movable concentrated loads, such as the wheel loads ofore bridges and cranes, in the concrete floor of shipways or locks and in rigid pavementsacted upon by concentrated loads. The solution of the equations which determine the local

* I f R, is greater than 2R (equation (9)), R , sl~oultl be replaced by 2X.


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EVALUATION OF COEFFICIESTS OF SUBGRADE REACTIOX 307bending moments in most of these structures can be found in the book by Hayashi (1921)and various other publications.

The most difficult problem is the design of flexible mats acted upon by column loads whichare not equally spaced. The computation of the bending moments in such mats is a cumber-some operation even on the simplifying assumption that the mat is perfectly rigid. If themat is flexible, an accurate computation of thelocal bending moments is almost impracticable.Therefore, the following procedure is indicated.

The raft is first designed on the assumption that the distribution of the soil reactions onthe base of the raft is identical with that on the base of a rigid raft, and the deflexions areestimated which would be produced by these reactions. The next step is to ascertain theinfluence of these fictitious deflexions on the distribution of the soil reactions over the base ofthe raft. If the influence of the deflexions on the pressure distribution is found to be unim-portant, a modification of the design is unwarranted. On the other hand, if the difference isimportant, either one of two procedures may be used. The thickness of the raft is leftunchanged, but the reinforcement is reduced on the basis of the results of a rough estimate ofthe difference between the bending moments in the rigid and in the flexible raft. In con-nexion with this operation it must be considered that the real detlexions of the raft will besmaller than those which have been computed on aforementioned assumptions, because thesoil reactions on the base of a flexible raft decrease from the points of load application towardsthe areas located between these points. As an alternative procedure the rigid raft could beredesigned on the basis of higher stresses in concrete and steel, and an estimate is then made tofind out whether or not the influence of the flexibility of the raft on the pressure distributionreduces the stresses to values close to the allowable ones. In any event the approach to theproblem is an indirect one.Vertical piles acted upon by horizontal loads

Fig. 5 represents vertical beams with width B, which are buried in or have been driveninto the subgrade. Before any horizontal forces have been applied to the piles, the surfaces ofcontact between piles and subgrade are acted upon at any depth .z below the surface by apressure p, which is equal to the earth pressure at rest (buried piles) or greater than thispressure (driven piles),

Y,'(b) li- (c )

-!--1-icl..!--.i-(4;-I= I

fig. 5. Vertical beam embedded (a) in stiff clay, and (b) i n sand ;(c) influence of width of beam on dimensions of bulb of pressure


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308 K. TERZAGHIIf the pile is moved to the right, at right angles to its width Br, the pressure on the left-

hand face of the pile drops down to a very small value. On account of arch action com-parable to that above a yielding trap door, this value is smaller than the active earth pressure,and can therefore be disregarded. At the same time, and as the result of the same displace-ment, the pressure fiP on the right-hand face increases from its initial value $+, to a value $awhich is somewhat greater than the earth pressure at rest, p,,. The lateral displacement yerequired to produce this change is so small that it can be disregarded. Hence, at the outsetof the movement of the pile towards the right, yr = 0 and the pressures on the two faces of thepiles are, at any depth z :

9, = 0 (left-hand side)and

p, = p, > PO (right-hand side).After the pile has moved through a distanceyr to the right,

9, = 0 (left-hand side),and

fiP = p, + p = p, + khyr (right-hand side) . . (10)wherein p = k / & y 1s the increase of the pressure on the right-hand face due to the displacementyr of the pile.

The value of k w and the change of k ,, with depth depend on the deformation characteristicsof the subgrade.

The deformation characteristics of stiff clay are more or lessindependent of depth. There-fore, at any time, the subgrade reaction p is almost uniformly distributed over the right-handface of the pile shown in Fig. 5(a) and the coefficient of subgrade reaction is :

However, on account of the progressive consolidation of clay under constant load the valuey1 increases and the value k h decreases with time, and both approach ultimate values. Inconnexion with design the ultimate value of k l , must be used.

In cohesionless sand, the values yr and k h are practically independent of time. However,the modulus of eIasticity of the sand increases approximately in simple proportion to depth.Therefore, it can be assumed without serious error that the pressure $ required to producea given horizontal displacement yr increases in direct proportion to depth x as shown in Fig.5(b) and:

Fig. 5(c) shows the bulb of pressure for the beam with width B, and Fig. 5(d) that for abeam with width nB,. The lengths of these bulbs, measured in the direction of the move-ment of the pile, are L and n L respectively. Furthermore, in both clay and sand the modulusof elasticity is constant in horizontal directions. Hence in clay, as well as in sand, thehorizontal displacement y increases in simple proportion to the width B , , yn = ny , .

For beams surrounded by clay :P P PBIkhn c - = - = --YS flYI Yl n*1

substituting khla = kI,, B , = 1 ft, n B , = B and p/y1 = k n l in this equation, we obtain :k,, = 2 .t = L kkl~lB * . . . . . . , (11)


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EVALUATION OF COEFFICIENTS OF SUBGRADE REACTION 309wherein Khl is the coefficient of horizontal subgrade reaction for a vertical beam, with awidth of 1 ft, embedded in clay.

If the pile is embedded in sand :P P

k l & , w & z = - = -Y ,I W lSince

P- = mhlz,Yl1 zk/m= ; mJ$ = md, no

1substituting k J L j l . k k , B , = 1 ft, n B , = B and rnh,B, = ??J,,

Zk J , = %J6- . . . . . _ . . .B . (12)

wherein ?tJL (tons ft-3) is the con s t a d o f ho r i zon t a l subg r ade r ea ct i o l z for piles embedded in sand.For vertical piles or beams embedded in the subgrade, the pressure p in equation (9)

can assume values which are many times greater than p,. Hence it is commonly assumed thatfi,, = 0, whence :

p , = p = k J ~Y . . . . . . . (13)

A~ l cho ved bu l k heads , f r e e ea r t h suppo r tIf the depth D l of penetration of the sheet piles forming the anchored bulkhead shown in

Fig. 6(a) is relatively small and the flexural rigidity of the sheet piles is great, the earthpressure acting on the inner face of the bulkhead causes the entire buried portion of the sheetpiles to yield in an outward direction, and produces a horizontal subgrade reaction on theentire outer face of the buried portion of the sheet piles.

Before the sheet piles move, both faces of the piles are acted upon at any depth z by apressure 9, which is approximately equal to the earth pressure at rest. However, a veryslight horizontal displacement ya towards the right reduces the earth pressure on the left-handface to the active earth pressure pa, whereas the pressure on the right-hand face assumes avalue $a which is greater than the earth pressure at rest, $a. Since y0 is very small com-pared to the displacements required to increase the contact pressure p, on the right-hand sideof the sheet piles to values which are considerably greater than p,, it is assumed that ye = 0.In the following discussions the symbol p indicates the difference between the total pressurep, on the left-hand face and the pressure p, :

In order to evaluate the coefficient of subgrade reaction k h for the buried portion of thesheet piles, the sheet piles are advanced over a distance y1 in an outward direction, parallelto their original position, as shown in Fig. 6(a).

If the subgrade consists of stiff clay, the value $a at any depth z below the dredge line canbe assumed to be equal to the effective overburden pressure 72 at that depth,

Po = YZwherein y is the effective unit weight of the clay. The error involved in this assumption is onthe safe side. The lateral displacement yl required to increase the contact pressure on the


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310 K. TERZAGHIright-hand face of the sheet piles by p from p, to p, increases in simple proportion to thedepth of sheet pile penetration D , and y,, = NY,, whence :

Substituting D , = 1 ft, nD , = D , k J L k f , l , a n d k f ,, = k l , ,1k , , = kn l z , . . . . . . . . .

wherein k h l is the coefficient of horizontal subgrade reaction for a depth of sheet pile penetra-tion of one foot into stiff clay.

If the subgrade consists of cohesionless sand the subgrade reaction produced by a move-ment of the sheet piles through a horizontal distance yr increases approximately in simpleproportion to depth as indicated by the triangle a b c in Fig. 6(a) and the coefficient of sub-grade reaction is :

Pk , , = - = r n l L z . . . . . . . . .Y l

Let y = the effective unit weight of the sand located below the dredge line,z = depth below the dredge line,

K , = coefficient of active earth pressure,K , = coefficient of earth pressure at rest, andK p = coefficient of passive earth pressure.An imperceptible movement y,, of the buried portion of the sheet piles reduces the pressure

on the left-hand face of the sheet piles to a value equal or close to the active earth pressureK , yz , whereas the pressure on the right-hand side increases to :

p , = K ,yzwhich is greater than the earth pressure at rest, Koyz . The value y. is of the order of magni-tude of D x IO-4 (see Fig. 7, Terzaghi 1934) and can be disregarded. Hence, after the sheetpiles have moved through a distance yr to the right, the unit pressure on the right-hand faceof the sheet piles, at depth z is :

p, = K l oy z + r n l L z y, . . . . . . . . . (15)In order to get some information concerning the influence of the depth D , of sheet pile

penetration on the value rnh in equation (15) it is assumed that the outward movement of thesheet piles shown in Fig. 6(a) has increased the contact pressure at any depth z from Kopto K , yz . The corresponding bulb of pressure, with width L , is shown in Fig. 6(b).

If the contact pressure on sheet piles with depth n D , is increased, at any depth, fromKeyz to K I y z , the corresponding bulb of pressure, shown in Fig. 6(c) has a width n L . Themodulus of elasticity of the sand does not change in horizontal directions. Hence the hori-zontal displacement required to establish the contact pressure K , y z increases in simple pro-portion to the depth of sheet-pile penetration :

Yn = fiY1According to this reasoning, the lateral displacement yP required to mobilize the full

passive resistance of a mass of sand adjacent to an abutment with a vertical contact face withheight D , should increase with increasing height of the face. The validity of this conclusionis obvious.

Since y increases in simple proportion to D ,, the coefficient of horizontal subgrade reactionk h n for the bulkhead with a sheet pile penetration n D l is :

P ml Zkhla = - = z z = mh,D, . z12Yl 1


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El.ALUATION OF COEF FICI ESTS OF SUBGRADE RE ACTION 311If D , is assigned a value equal to one foot, n D , = D , and :

w z w ,D , = m / t 1 tons ft-4) x unit length = It, (tons ft-3),the value of the coefficient of subgrade reaction k,, for sheet piles with any depth D of pene-tration becomes equal to :

zk h = Q , . . . . . . . . . . (16)The value of the coefficient ZJ, n this equation depends only on the relative density of the sandin contact with the sheet piles.

Fi g. 6. (a) Pressure distribution on bulkhead if buri ed portion advances parallelto its origin al position ; (b) and (c) influence of depth of penetration ondimensions of bulb of pressure ; (d) real displacement of anchored bulk headwith free earth suppor t, and (e) corr esponding pressure distributionFig. 6(d) shows the deflexion of an anchored bulkhead with free earth support and Fig.

6(e) the distribution of the pressures on the two faces of the sheet-pile wall. The left-handface is acted upon by the active earth pressure, represented by the abscissas of line a ??a,The outward movement of the buried part of the sheet piles is resisted by the soil reactionrepresented by the area b c e d in Fig. 6(e). The abscissas of line b ICI0 are equal to the valuesK,,~.z in equation (15). The lateral displacement required to mobilize horizontal soil reactionsof less than Koyz are negligible, whereas the displacements produced by pressures in excessof Koyz are determined by equation (16).An cho red bu l k h eads w i t h @ted ea r t h su@o r t

Fig. 7(a) shows the deflexions of an anchored bulkhead with fixed earth support. Thebulkhead is assumed to be backfilled with sand and the sheet piles were driven into sand.The earth pressures which act on the two sides of the row of sheet piles are indicated inFig. 7(b). Between the surface of the backfill and a certain depth below the dredge line the


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312 K. TERZAGHIleft-hand face is acted upon by the active earth pressure represented by the abscissas of thestraight line aK_,. The pressure on the right-hand face is given by the abscissas of the curveb d e f, which intersects the straight line b K, in point e at depth zi. At any depth smallerthan zi the lateral displacement y of the pile is :

Y= Koyz p, - Ko~zZf . . .11~ zl h Da

(17)

Fig. 7. (a) Deflexion of anchored bulkhead with fixed earthsupport wi th sheet piles driven into sand ; and (b ) corres-ponding pressure distribution ; (c) deflexion of bulkheadwith sheet piles driven into stiff clay

Below depth zi :KOPy=- Z~'JLD

In connexion with vertical piles and beams and with bulkheads with free earth support thedisplacements produced by pressures smaller than K , y z have been disregarded. In otherwords, it has been assumed that lh = co. In connexion with bulkheads with fixed earthsupport this assumption may or may not be indicated, depending on the nature of the investi-gation.If the sheet piles are driven into stiff clay, the bulkhead deflects, as shownWithin depth D the piles deflect outward and the corresponding coefficientreaction is obtained by substituting D for D in equation (14), whence :

in Fig. 7(c).of subgrade

Between depths D and D the sheet piles deflect to the left. For this range of depth thecoefficient of horizontal subgrade reaction is approximately equal to the coefficient of verticalsubgrade reaction k , for a beam with width D , resting on clay :


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EVALUATION OF COEFFICIENTS OF SUBGRADE REACTIONFlexible diaphragms in earth dams

313

If an earth dam, Fig. 8, must be made entirely of pervious material, such as clean sand,the flow of water from the reservoir through the dam can be intercepted by an imperviousdiaphragm, such as a reinforced concrete wall. The upstream face of such a diaphragm isacted upon by both the active sand pressure and the full water pressure. Its downstreamdisplacement is resisted by the passive earth pressure of the downstream section of theembankment.

The deformation of the diaphragm, due to the unbalanced water pressure and the intensityof the shearing forces and bending moments in the diaphragm, depends on the flexural rigidityof the diaphragm and on the coefficient of horizontal subgrade reaction kd of the downstreamportion of the embankment. In order to get at least a crude conception of the factors whichdetermine the value kd , the diaphragm is assumed to be perfectly rigid. It is further assumedthat the surface of the fill material on the downstream side of the diaphragm is horizontal and

b_-_____-----Fig. 8. F icti tious displacement of impervious diaphragm insand dike, parallel to its original position

located at the elevation of the crest of the dam, as indicated in Fig. 8, by line ab. On theseassumptions the value of the coefficient of subgrade reaction is determined by equation (16),

kk =Zk; . . . . . . . . . _ [16]In reality the surface of the sand on the downstream side of the diaphragm descends at

a slope of 2 (horizontal) on 1 (vertical). The passive earth pressure of such a body of sandis roughly equal to 40% of that of a mass of sand with equal height but with a horizontalsurface. As a consequence the pressure required to produce a displacement y of the rigiddiaphragm hardly exceeds 0.4 times the pressure required to produce the same displacementin an otherwise identical body of sand with a horizontal surface. The corresponding co-efficient of subgrade reaction, kl,, is equal to 0.4 times the value determined by equation(16) : 2 2kd =0.41h- =ld-H H . . . . . . , - (18)The value of ld = 0.411, depends only on the physical properties of the downstream section ofthe dam. At any depth z below the crest of the dam the contact pressure on the downstreamface of the diaphragm is :

p, = 0.4 K, + yld $

Practical applications of he theories of horizontal subgrad e reactionSome practical problems can only be solved at a reasonable expenditure of time and labour

by means of the theories of horizontal subgrade reaction. These include the computation of


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314 K. TERZAGHIthe bending moments in piles which are acted upon by horizontal forces above the groundsurface (Cummings, 1937) and of those in core-walls of earth- and rock-fill dams (Ldfquist,1951).

Attempts have also been made to apply the theories to the solution of bulkhead problems(Rifaat, 1935). Baumann (1935) used them for estimating the stresses in an anchored bulk-head which had failed. Quite recently Blum (1951) proposed a procedure for the design ofanchored bulkheads by means of the theory of horizontal subgrade reaction. All theseinvestigations and design procedures were based on the tacit assumption that I?,-, in equation(15) is identical with the coefficient of active earth pressure K,. The error due to this assump-tion may be quite important.

General fvocedweThe numerical values of the coefficients of subgrade reaction k, and k~ required for the

solution of engineering problems can either be estimated on the basis of published observa-tional data or else they can be derived from the results of field tests to be performed on thesubgrade of the proposed structure. For practical purposes, rough estimates of these valuesfully serve their purpose.

I;ertical subgrade reactionAs a basis for estimating the coefficient of subgrade reaction k , for beams and slabs, the

value k,r for a square plate with a width of 1 ft has been selected, because this value can, ifnecessary, be determined by averaging the results of several loading tests in the field, at thesite of the structure.

If the subgrade consists of cohesionless or slightly cohesive sand, k , can be estimated onthe basis of the empirical values of R,r given in Table 1. The density-category of the sand canbe ascertained by means of a standard penetration test or other convenient means. Thegreatest error on the unsafe side results from using the proposed value in the case of mediumsand if its real value is equal to the lower limiting value of 60 tons,cu. ft.

Table 1.Values of ES, in tonsjcu. ft for square plates, 1 ft X 1 ft, or

beams 1 ft wide, resting on sand

Relative density of sand / Loose / Medium / lknscDry or moist sand, limiting values for i;,, 20-60 60-300 300-I ,000J hy or moist sand, proposed values

~ 40 130 .500Submerged sand. proposed values 3 5 80 300

In order to investigate the influence of such an error on the results of the computationof the bending moments in a beam, the maximum bending moment M,,, in the beam shownin Fig. 1 was computed on the basis of both the assumed and the real value of &r for thesupporting sand. The value of M,,,, for this beam is determined by equation (4). It wasfound that the moment computed by means of the proposed value exceeds the actual bendingmoment by not more than about 5%.

Once the value &i has been selected, the value of k, to be used in the solution of a given


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EVALUATIOS OF COEFFICIESTS OF SUBGRADE REACTION 313problem can be computed by means of the equations derived under the preceding sub-headings. Experience has shown that the value k sl for a beam with a width of 1 ft resting onsand is roughly equal to that of a square plate, 1 ft wide. For a beam with a width B feet(Fig. 3) or for a mat acted upon by line loads spaced B feet (Fig. 4(d)), K,Y s determined byequation (8) :

k,=& & . . . . . . . . .t )If applied to spread footings, continuous footings or beams, this equation is valid for

contact pressures smaller than one-half of the ultimate bearing capacity of the subgrade perunit of area of the base of the supporting structure. In connexion with slabs or mats, sup-porting concentrated loads, it is valid, if none of the concentrated loads is greater than one-half of the ultimate bearing capacity of the equivalent circular footing with radius K,equation (9).

Table 2.Values of ES, in tonsjcu. ft for squar e plates, 1 ft x 1 ft andfor long strips, 1 ft wide, resting on pm-compr essed clay

Consistency of clayValues of I_u. ons/sq. ftRange for ksI, square platesProposed values, square plates

IStiff Hardl-2 >150-100 > 20075 300*

-- I + 0.5For rectangular plates with width lft and length 1 ft : ksl = k,, 1 .I

* Higher rnlues should bo uswl only if they mere estimated on the basis of adequate, est results.

If the subgrade consists of heavily pre-compressed clay, the value of KS1 ncreases approxim-ately in simple proportion to the unconfined compressive strength of the clay qU. On thebasis of our present knowledge of the deformation characteristics of precompressed clays thenumerical values of K,i contained in Table 2 are proposed. These values are valid for contactpressures which are smaller than one half of the ultimate unit bearing capacity of the clay.The latter is independent of the dimensions of the loaded area.

The proposed values of K,i for stiff clay are of the same order of magnitude as those formedium sand, Table 1, but the k , values for clay decrease in inverse proportion to the widthof the loaded area, whereas those for sand approach a limiting value equal to 0.25 ,&r. Fornormally consolidated clays, the values of AS1are so small that the bending moments in loadedbeams and rafts should be computed on the assumption that the load-supporting structure isperfectly rigid.The As1values for stiff clay can also be determined by loading tests. However, the resultsof the tests can be misleading, because the time which is commonly allotted to the performanceof such tests is too short to permit complete consolidation of the loaded clay. Furthermorethe test results cannot be relied upon unless the loaded block is practically rigid. The heightof the block should be at least equal to its width.

If the contact area has the shape of a rectangle with width B and length ZB , k , sl assumesthe value :

1 + o-5ks, = 41 151For I = 00, k , , = 0.67 k,,. This equation is based on the fact that the deformations of theloaded subgrade below a depth of more than about 3B has no significant influence on the


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316 Ii. TERZAGHIlocal bending moments. For long beams or continuous footings resting on clay, 1 can beassumed to be equal to infinity, whence :

For concrete mats acted upon by concentrated loads such as column loads the value of k , isdetermined by &, and the dimensions of the areas of effective contact are shown in Fig. 4(c).

H o r i zo n t a l subg vade r ea ct i o n on ver t i c a l p i l es an d $ i e r sIf a vertical pile or pier is surrounded by sand, the coefficient of horizontal subgrade

reaction k j , at a given depth z below the surface depends on the width B of the pile measuredat right angles to the direction of the displacement, the effective unit weight y of the sand,and the relative density of the sand. At any depth z below the surface the modulus ofelasticity E , of the sand is roughly equal to :

Es = AP,wherein A is a coefficient which depends only on the density of the sand, and 9, is the effectiveoverburden pressure at depth z. If y is the effective unit weight of the sand, pV = yz and :

E , = y zA .The coefficient of horizontal subgrade reaction for a vertical beam with width B is deter-mined by the relation between the contact pressure p and the corresponding lateral displace-ment of the contact face. The displacement is due to the deformation of the adjacent mediumwith a modulus of elasticity E ,. Displacements beyond a distance of about 3B have practi-cally no influence on the local bending moments. Hence the displacement y can be computedon the assumption that the pressure p acts on an elastic layer with thickness 3B. On thisassumption the theory of elasticity (Terzaghi, 1943, p. 424) leads to the equation :

ES Ay zP=YmB= --1_35B . . . . . . .whence :

P AY zk I L - = - - = m , h zY 1.35 BThe factor :

is identical with nh in equation (12), which representsreaction for vertical beams or piles.

the constant of horizontal subgradeThe value A ranges between about 100 for very loose sand and 2,000 for dense sand. For

loose sand the value A is fairly constant for confining pressures up to at least 50 kg/sq. cm.For dense sand it starts to decrease as soon as $+, becomes greater than about 3 kg cm-s(Terzaghi, 1932a ) . The unit weight of dry or moist sand ranges between 1.9 (dense) and l-3gm/cu. cm (loose), average :

y = 1.6 gm/cu. cm = 0.05 *tons/cu. ft

Z= n/L -B . * * . (19b)

and the effective unit weight of submerged sand between 1.2 and 0.8 gm/cu. cm, average :y = I . 0 gm/cu. cm = 0.03 tons/cu. ft.

On the basis of these values Table 3 was prepared, containing the values 9th of the constant* 1 ton = 2,000 lb.


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EVALUATIOH OF COEFFICIENTS OF SUBGRADE REACTIONTable 3.

Values of the constant of horizontal subgrade reaction rah intons/cu. ft for a pile 1 ft wide, embedded in moist and in sub-merged sand, and for a horizontal strip, 1 ft wide

317

Relative density of sand I LOOSC Medium / IknseRange of values of A 100-300 I 300-l ,000 1ooo-2,000.\dopted values of r1 200 I 600 1,500lhy or moist sand, \-alucs of ~26 7 21 .56Submerged sand, values of ~1~ 4 14 34

of horizontal subgrade reaction for a pile with a width of 1 ft, embedded in moist sand and insubmerged sand.

For piles embedded in stiff clay, the values for key,can be assumed to be roughly identicalwith the values of k,, for beams resting on the horizontal surface of the same clay. Becausethe horizontal displacement of the contact face between a pile with width of 1 ft is con-siderably smaller than that of a strip w&h the same width on the surface of the clay, at equalpressure per unit of area, the error involved in this assumption is on the safe side. The valuek n for a pile with width B in ft is given by the equation :

1 1k I L - k I L 1 - . k , , = 1B B - & I1.5B1 The values of & are given in Table 2.The value of nh for piles driven into sand or of k j , for piles embedded in clay can also be

determined experimentally. One method consists in driving a very rigid pile, preferably witha square cross-section, into the ground and measuring the tilt and the horizontal displacementof the upper end of the pile produced by a horizontal force acting on the upper end.

According to the second method a steel tube with closed lower end is driven into the groundand a horizontal force is applied to its upper end. The lateral displacement of the centre-line of the tube is then computed for different elevations below the ground surface on the basisof the results of the readings on strain gauges attached to the inner walls of the tube, or of tilt-meter observations. If the lateral displacements are known, the corresponding values of thecoefficient of subgrade reaction can be estimated. The strain-gauge method has been usedby Loos and Breth (1949). A suitable tilt meter has recently been constructed and success-fully employed by S. D. Wilson.*

Figs 9(a) and 9(b) illustrate a test of the first category, to be carried out on a rigid pilewith width B driven to a depth D into stiff clay. The displacement y1 of the upper end ofthe pile has been measured and the relation between the horizontal subgrade reaction p at anydepth z is determined by the equation :

P = by = AB&y1 HH2+-; . . . . .1 2In this equation the values K,, and H z are unknown. They can be computed on the basis ofthe condition that the sum of all horizontal forces and the sum of all moments about anarbitrarily chosen point must be equal to zero. The distribution of the horizontal pressuresover the vertical contact face is shown in Fig. 9(b).

If the pile is embedded in cohesionless sand :Z Z H , - zp = k l , y = yn l , - = y l n h - . -.-z---B B H , +H , . . .

*Shannon and \Vilson, Consulting Engineers, Seattle, \Yashington.


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31s K . TERZAGHIThe unknown quantities nl, and Hz can also be determined on the basis of the two conditionsfor the equilibrium of the pile. However, the computation is somewhat more cumbersome.The distribution of the horizontal pressures is shown in Fig. 9(c)

In any event the computations furnish the value H, of the depth at which point a of zero-displacement is located. If the pile is practically rigid, H, can also be obtained from theresults of an accurate tilt meter and displacement observation on the exposed portion of thepile. If the computation is performed by means of equation (20a) and it is found that H2 is

_ay Sand(4 (b) (4

Fig. 9. Diagrams illustrating experimentalprocedur e for determining value of coefficientof subgrade reaction by measuring displace-ment of upper end of ri gid pile acted upon,at upper end, by horizontal force Q

greater than the computed value, it can be concluded that the elastic properties of the sub-grade are intermediate between those of a cohesionless sand (E, = Ayz) and of a stiff clay(E, = constant). In such doubtful cases it is indicated to interpret the test results on bothassumptions, compute the bending moments on the basis of both results and design on thebasis of the more unfavourable findings.

Equations (20a) and (20b) are based on equation (13). This equation was obtained fromequation (10) by the simplifying assumption that p, = 0. The error due to this assumptiondecreases with increasing displacement yi of the upper end of the test pile. Therefore it isindicated to repeat the test for different values of yl.

If the horizontal loading tests are made on flexible tubes or piles, the assumption thatpo in equation (10) is zero can be eliminated and the values of $a and k h n this equation canbe estimated for any depth if the tube or pile is equipped with fairly closely spaced straingauges and if, in addition, provisions are made for measuring the deflexions by means of anaccurate deflectometer. The strain-gauge readings determine the intensity and distribu-tion of the bending moments over the deflected portion of the tube or the pile, and on thebasis of the moment diagram the intensity and distribution of the horizontal loads can beascertained by an analytical or graphic procedure. The load diagram and the deflexiondiagram combined furnish the value of p, + khy, in equation (lo), in which p, and kh areunknown. If the test is repeated for different horizontal loads acting on the upper end of


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EVALUATIOK OF COEFFICIENTS OF SUBGRADE REACTIOS 319the pile, a curve can be plotted showing for different depths the relation between $, and Y1in equation (10) and this curve is then replaced by a broken line with the equation :

P, = Po + khYlTheoretically the same result could be obtained on the basis of the results of the strain

gauge readings alone, because these readings determine the shape of the elastic line. However,the errors involved in the computation of the deflexions are so important that the procedurecannot be recommended.

An ch or e d b u l k h ea d s a n d J l ex i b l e d i a $& r a gm sThe coefficient of horizontal subgrade reaction for the outer face of an anchored bulkhead

with free earth support (Fig. 6) in contact with sand is determined by equation (16) :k , , = I , , _i i

wherein D is the depth of sheet-pile penetration.The outward movement of the buried portion of the sheet piles is resisted by the passive

earth pressure of the sand into which the piles were driven. The few available experimentaldata (Terzaghi, 1934 ; Rifaat, 1935) indicate that the lateral displacement increases withincreasing contact pressure, as shown in Fig. 10. In this figure the abscissas represent thehorizontal displacement and the ordinates the coefficient of lateral earth pressure. Thiscoefficient increases from K, for the initial displacement y,,, which is very small, tothe value KP of the coefficient of passive earth pressure.

In connexion with anchored bulkheads the contact pressure may assume values as highas those corresponding to two-thirds of the coefficient of passive earth pressure K p . For thisrange the relationship between contact pressure and displacement y can be representedapproximately by the chain-dotted line Oab in Fig. 10. For valuesof K = K , , , y = 0 , and forvalues between I(, and 2/3 1C, the contact pressure is determined by equation (16) :

k l , = l , , - . . . . . . . . . .D WAt the present state of our knowledge K , for sand can be assigned the following values :

Loose MediumK, = 0.4 0.8

Dense1.2

Table 4.Values of the subgrade constant lh in tonsjcu. ft for verticalwalls, embedded in moist and in submerged sand

/ IHelative density- j LOOse / J lcdiurn Lknsc

The values of 11 ,n equation (16) are given in Table 4. They were derived from experimentaldata presented by Terzaghi (1934) and they are in satisfactory agreement with the results ofestimates based on the theory of elasticity.than K,Yz can, as a rule, be disregarded.

The displacements produced by pressures smallerIf a theoretical investigation does not permit this

simplification, equation (17) must be used and the value II~,which appears in this equation,


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320 K. TERZAGHI

Fig. 10. Curve C = real relationship between hori-zontal displacement y of vertical wall with heightD buried in sand, and coefficient of earth pressureK ; straight lines Oab, the assumed relationship

can be assigned the following values which were derived from the results of large-scale earth-pressure tests (Terzaghi, 1934) :

Relative density Loose Medium DenseDry or moist sand, value of I* (tons ft-3) 125 200 300Submerged sand, value of 11~ tons ft-3) 75 120 180

If the lowest part of the sheet piles of the bulkhead shown in Fig. 7(a) moves to the leftwith reference to its original position, the corresponding value of the constant of horizontalsubgrade reaction is even greater than ln, because the contact pressure acts only on a narrowstrip. Hence, if it is assumed that Zh = 14, the computed displacements will be greater thanthe real ones.

If the sheet piles have been driven into heavily precompressed clay :1k , , = ks l -D . . . . . . . . . Cl 41

A procedure for estimating the value of k l , , is illustrated by Fig. 11. It represents avertical section through a bulkhead driven to depth D into stiff clay, with surface ac. Theclay is assumed to be weightless. On this assumption it makes no difference whether theloaded area ab is vertical or horizontal. The sheet piles ab are moved through a distance 3to the right.

If the space above ac were also filled with weightless clay, acted upon over width ab by

Fig. 11. Diagram illustrating in-fluence of heave of free claysurface on horizontal displace-ment of vertical wall withheight D on intensity of contactpressure


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EVALUATION OF COEFFICIENTS OF SUBGRADE REACTION 32 1an advancing strip, the bulb of pressure would assume the shape shown in the figure and thecoefficient of subgrade reaction would be :

wherein k, , is the coefficient of subgrade reaction for a strip with width 1 ft, resting on clay(Table 2). During the advancement of bb, the surface ac would remain plane.

In reality the space above ac is empty and an advance of ab to the right causes the surfaceac to bulge in an upward direction as shown in Fig. 11. Therefore, the horizontal pressurerequired to produce the displacement y of ab against the mass with horizonta1 surface bc issomewhat smaller than that required to move bb through distance y and the correspondingcoefficient of subgrade reaction is smaller. On account ofassign to kh the value :

1 1 1kh = k,, - = - k,, -30 3 D .If the sheet piles were driven into stiff clay, Fig. 7(c), the

between the dredge line and depth D is obtained by replacing

this -fact it is proposed to

. . . . . . . (21)value kh for the clay locatedD in equation (21) by D :

k,, = ; k,, &.Below depth D the coefficient is equal to that for a loaded strip with width D on the horizontalsurface of stiff clay :

For vertical diaphragms embedded in sand fills, Fig. 8, the values of ,?h,Table 4, should bemultiplied, as indicated by equation (18), with reduction factor 0.4, because the surface ofthe sand in contact with the diaphragm slopes away from the crest of the diaphragm, and, asa consequence, its coefficient of subgrade reaction kdis smaller than that of a sand stratum witha horizontal surface, everything else being equal. Therefore :

wherein, H is the height of the sand embankment.

Concersion to other amitsThe values of coefficients of subgrade reaction are commonly given in kilogrammes andcentimetres, metric tons and metres, tons and feet or pounds and inches. The conversion of

one system into another can be performed by means of the following conversion-factors :1 kg cm-2 = 10 *tons, m-2 = 1.03 ?-tons ft-s = 14.2 lb in.-21 kg cm-3 = 1,000 tons, m-3 = 31.3 tons ft-3 = 36.1 lb in.-31 kg cm-4 = 100,000 tons, m-4 = 950 tons ft-4 = 91.2 lb in.-4

* tonm = 1,000 kg.t ton = 2,000 lb.

In this Paper (Terzaghi 1934) all the numerical values were given in tons and feet.In connexion with the conversion of numerical data into another unit system, it mustalways be remembered that the values of k and m are not subgrade constants. They are


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322 I ( . TERZAGHIvalues referring to the subgrade acted upon by a unit load or a unit pressure over an area withspecified dimensions. The following example illustrates the procedure for taking this factinto consideration.

Loos and Breth (1949) obtained from horizontal loading tests on a hollow steel pile witha diameter of 0.062 metres = 0.2 ft, embedded in dry, compacted sand the value :

950mh = 30,000 tons me4 = - .100,000 30,000 = 285 tons ft-4

The corresponding value ?&h, quation (19) and Table 3, is equal to :s Z h m n B

wherein B is the width of the strip-shaped contact area. Since B = 0 .2 ftnh = 58 tons ftta

This value is slightly higher than the average value of 56 tons ft-3 for dense sand, given inthe Table. It is interesting to remember that the Table values have not been derived frompile tests. They were computed by means of the theory of elasticity on the basis of whatis known about the elastic properties of sand,

CORRECTIOSS FOR THE ERRORS INVOLVIX~ IX THE hYJKDAMl3T:iL _lSSUhIPTIOKSUnder the sub-heading Fundamental assumptions (p. 301) it is stated that the

theories of subgrade reaction are based on two simplifying assumptions : (A) The coefficientof subgrade reaction k is, at every point, independent of the contact pressure 9, and (B) Thecoefficients k, for any subgrade, kw for stiff clay and rn~ for sand have the same value at ever!point of the contact face. Both assumptions involve crude approximations as shown below.

In Fig. 12, showing the relation between subgrade reaction p (abscissas) and correspondingsettlement y (ordinates), assumption (A) is represented by a straight line OA through theorigin 0. Yet if a loading test is performed on a subgrade of any kind it will be found that thesettlement y increases with increasing pressure 9, as shown by curve C which approaches avertical tangent. On account of the relationships represented by the diagram, assumption(A) is approximately valid only for values of 9 which are smaller than about one-half of theultimate bearing capacity 9%. For stiff clay the value of 9, is practically independent of thesize of the loaded area, whereas for sand it increases with increasing size. In connexion withproblems involving coefficients of subgrade reaction these limits of the validity of assumption(A) should always be taken into consideration.

Hayashi (1921) and Freund (1927) have solved various problems involving vertical sub-grade reaction on the assumption that the value k , decreases with increasing pressure $, as itdoes according to curve C in Fig. 12. However, the practical value of these solutions isdoubtful, because the errors due to assumption (A) represented by the difference between lineOA and curve C in Fig. 12 within the range of pressure zero to PJs are commonly unimportantcompared to those due to assumption (B).

For loaded beams on a perfectly elastic subgrade, Biot (1937) has shown that the coeflicientof subgrade reaction depends not only on the width of the beam, but also to some extent onthe flexural rigidity of the beam. He derived equations which make it possible to take thisinfluence into consideration, provided the nature of the problem justifies such a refinement.

According to assumption (B) the subgrade reaction p on the base of a centrally loadedperfectly rigid beam or slab has the same value pi everywhere as indicated in Figs 13(a) to13(d) by rectangular areas with height 9,. In reality the pressure p at the rim of the surfaceof contact is either greater or smaller than at the centre, depending on the elastic propertiesof the subgrade. In Fig. 13(a) the plain curve represents the pressure distribution on thebase of a rigid beam resting on stiff clay. In a similar manner the real pressure distribution


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EVALUATION OF COEFFIC IENTS OF SUBGRADE REhCTION 323

0 PUP

1..I-,Y !Fig. 12. Relation between sub-grade reactionp and displace-ment y : Line OA assumedand curve OC real relation-

ship

Q Q

Fig. 13. Di fference between theoretical distributionof subgrade reaction on the base of rigid beams(a) and(c), and rigid slabs (b) and ( d ) , epresentedby dash-dot lines, and the real distribution(plain lines)


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324 K. TERZAGHIis shown in Fig. 13(b) for a rigid circular plate resting on stiff clay, in Fig. 13(c) for a rigidbeam on cohesionless sand, and in Fig. 13(d) for a rigid circular slab on sand.

The difference between the uniform and the real pressure distribution in Figs 13(a) to13(d) represents the error associated with assumption (B). If the subgrade consists of stiffclay the real bending moments beneath the points or lines of load application are somewhatgreater than those computed on the basis of assumption (B) and if the subgrade consists ofsand; they are smaller. This conclusion applies to both rigid and flexible beams and slabs onthe surface of the ground.

For flexible beams and plates resting on a perfectly elastic subgrade, rigorous or semi-rigorous methods of computing the bending moments can be obtained on the basis of thetheory of elasticity. This has been done for uniformly loaded strips and circular footingsby Borovicka (1936), and for beams by Habel (1938), who derived approximate equations forthe contact pressure on the base of elastic beams which transmit an arbitrary system ofloads on to the surface of a semi-infinite, elastic solid. Similar investigations have beencarried out by De Beer (1948) and De Beer and Krasmanovitch (1952). Schultze (1953) dis-cussed the problem of computing the stresses in loaded concrete mats on an elastic subbase.The literature dealing with this subject is still expanding.

In connexion with practical problems, the errors resulting from assumption (B) can inmany cases be disregarded. If it is doubtful whether or not the errors are inconsequential, acorrection can be made on the basis of our general knowledge concerning the distribution ofthe real soil reaction on the base of beams, rafts, and buildings, resting on the subgrade underconsideration. For this purpose a diagram is constructed which shows approximately thereal distribution of the contact pressure on the base of the load-supporting structure. Thenthe general component of bending moments are computed which would develop in the struc-ture, if all the loads acting in any portion of the structure were uniformly distributed in thatportion over the top surface of the members acted upon by the loads, and these bending mo-ments are added to the local moments computed by means of the theory of subgrade reaction.

The theories of horizontal subgrade reaction are based on assumptions similar to assump-tions (A) and (B) in the theories of vertical subgrade reaction.

According to assumption (A) the horizontal displacement of a vertical contact face shouldincrease in simple proportion to the horizontal pressure $, as shown by a straight line in Fig.12. In reality it increases with increasing pressure in the manner indicated by curve C.

The errors associated with assumption (B) are different for different subgrades. Forinstance, if the subgrade consists of stiff clay, it is assumed that :

kh = $ = constant,because the modulus of elasticity of stiff clay is practically independent of depth. Hence, if

Fig. 14. Difference between theo-retical and probable real pressuredistribution over face of rigid,vertical , bur ied plate, advanci ngparallel to its original position


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EVALUATION OF COEFFICIENTS OF SUBGRADE REACTION 325a rigid vertical plate, Fig. 14(a), embedded in stiff clay, is advanced over a horizontal distanceyi the corresponding subgrade reaction pi should be independent of depth, as indicated inFig. 14(a) by a rectangle with width pi. In reality the subgrade reaction is, at the loweredge of the plate, greater than it is at the upper end, as indicated by curve C.

If the subgrade consists of clean sand it is assumed that :k h = $ = w & z = COIIStant . 2

because the modulus of elasticity of clean sand increases approximately in simple proportionto depth. Hence, if a vertical, rectangular, rigid plate embedded in sand is advanced througha horizontal distance yi the subgrade reaction should increase in simple proportion to depthas indicated in Fig. 14(b) by a triangular area. In reality the subgrade reaction on theupper part of the plate is smaller and on the lower one greater than the preceding equationcalls for, as indicated in Fig. 14(b) by curve C. Yet the errors in the computation of stressesin foundations and sheet piles due to these discrepancies are not important enough to requirecorrection.

CONCLUSIOSS(1) Both private and published opinions concerning the influence of the dimensions of

the area acted upon by the subgrade reaction on the value of the coefficient of subgrade reactionvary widely, and in many instances the existence of this influence has been ignored. There-fore, the errors involved in the application of the theory of subgrade reaction to the solutionof engineering problems have often been very great.

(2) This Paper contains rules for adapting the values of the coefficient of subgrade reactionto the elastic properties of the subgrade and to the dimensions of the area acted upon by thesubgrade reaction. If the numerical values of the coefficients are selected in accordancewith these rules, the results of the computation of stresses and bending moments in footingsor mats can be considered reasonably reliable. However, the theories of subgrade reactionshould not be used for the purpose of estimating settlement or displacements.

(3) Refinements in the evaluation of the coefficients of subgrade reaction are seldom justi-fied, because the errors in the results of the computations are very small compared to thoseinvolved in the estimate of the numerical values of the coefficients of subgrade reaction.

(4) At the present state of our knowledge the greatest uncertainties prevail in connexionwith the values K , , and lh which appear in the equations for the coefficients of horizontal sub-grade reaction for anchored bulkheads and flexible diaphragms in earth dams. More reliableinformation concerning these values could be obtained by investigating the relationship be-tween the horizontal displacement of rigid, vertical walls advancing against sand with differentrelative density and the corresponding contact pressure.

(5) The fundamental equations of the theories of subgrade reaction have been obtainedby a radical simplification of the real relations between contact pressure and correspondingdisplacement. However, under normal conditions the errors due to these simplifications areadequately taken care of by the customary factor of safety in design, provided the coefficientsof subgrade reaction have been assigned numerical values which are compatible with boththe elastic properties of the subgrade and the dimensions of the area acted upon by the sub-grade reaction. If necessary, the error can be reduced considerably, as shown in the last partof this Paper.

lCKSOWLEDGEMENTSThe Author wishes to express his gratitude to Messrs L. Casagrande, N. M. Newmark,

R. B. Peck, and F. E. Richart, Jr., for their valuable comments and suggestions during thepreparation of this Paper.


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326 K . TERZAGHI: EVALUATIOS OF COEFFICIENTS OF SUBGRADE REACTIOXR E F E R E N C E S

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coeficiente de reação do solo terzaghi

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