9-6 MODULUS OF SUBGRADE REACTION

The modulus of subgrade reaction is a conceptual relationship between soil pressure anddeflection that is widely used in the structural analysis of foundation members. It is used forcontinuous footings, mats, and various types of pilings to be taken up in later chapters. Thisratio was defined on Fig. 2-43c, and the basic equation when using plate-load test data is

k - ±ks~ 8

with terms identified on both Fig. 2-43c and Fig. 9-9b. Plots of q versus S from load testsgive curves of the type qualitatively shown in Fig. 9-9b. If this type of curve is used to obtainks in the preceding equation, it is evident that the value depends on whether it is a tangent orsecant modulus and on the location of the coordinates of q and S.

It is difficult to make a plate-load test except for very small plates because of the reactionload required. Even with small plates of, say, 450-, 600-, and 750-mm diameter it is difficultto obtain 8 since the plate tends to be less than rigid so that a constant deflection across theplate (and definition of ks) is difficult to obtain. Stacking the smaller plates concentric withthe larger ones tends to increase the rigidity, but in any case the plot is of load divided byplate contact area (nominal P/A) and the average measured deflection.

Figure 9-9c is a representation of ks used by the author where ks is taken as a constant upto a deflection Xmax. Beyond Xmax the soil pressure is a constant value defined by

#con = ^sv^max)

Obviously one could divide the q-8 curve into several regions so that ks takes on val-ues of the slope in the several regions; however, this approach tends to incorporate too much

Figure 9-9 Determination of modulus of subgrade reaction ks.

Load block

Stacked plates

q' # q and 6' ^ 6if plate not rigid

(a)

k depends on curvecoordinates used and isgenerally nonlinear

(*»

Linear Nonlinear

(c)

Previous Page

refinement into the problem since most analyses proceed on the basis of estimated values orat best an approximate load test.

A number of persons do not like to use the concept of a modulus of subgrade reaction;rather, they prefer to use Es (and fx) in some kind of finite-element analysis. The author'sexperience using both the finite element (of the elastic continuum) and the concept of themodulus of subgrade reaction is that, until the state of the art improves so that accurate val-ues of Es can be obtained, the modulus of subgrade reaction method is preferable owing toits greater ease of use and to the substantial savings in computer computation time. In thefollowing paragraphs we will see a direct relationship between Es and ks.

A major problem is to estimate the numerical value of ks. One of the early contributionswas that of Terzaghi (1955), who proposed that ks for full-sized footings could be obtainedfrom plate-load tests using the following equations:

For footings on clay3

For footings on sand (and including size effects)

In these two equations use B\ = side dimension of the square base used in the load test toproduce k\. In most cases B\ — 0.3 m (or 1 ft), but whatever B\ dimension was used shouldbe input. Also this equation deteriorates when B/B\ ~ > 3.

For a rectangular footing on stiff clay or medium dense sand with dimensions of B X Lwith m = L/B,

where ks = desired value of modulus of subgrade reaction for the full-size (or prototype)foundation

k\ = value obtained from a plate-load test using a 0.3 X 0.3 m (1 X 1 ft) or othersize load plate

Equations (9-3), (9-4), and (9-5) are presented primarily for historical purposes and arenot recommended by the author for general use.

Vesic (1961«, 196Ib) proposed that the modulus of subgrade reaction could be computedusing the stress-strain modulus Es as

k's = 0.65 n ^ r — ^ (units of E5) (9-6)Y EfIf 1 — JXZ

3The Bx is not usually seen in this equation, since at the time it was proposed by Terzaghi (1955) only Fps unitswere used, and with Bx — 1 ft it did not need to be shown. The equation is dimensionally incorrect, however,without including B1. Equation (9-3) is not correct in any case, as ks using a 3.0 m footing would not be -^ thevalue obtained from a Bx = 0.3 m plate.

where E5, Ef = modulus of soil and footing, respectively, in consistent unitsB, If = footing width and its moment of inertia based on cross section (not plan)

in consistent units

One can obtain ks from k's as

ks~ ~B

Since the twelfth root of any value X 0.65 will be close to 1, for all practical purposes theVesic equation reduces to

One may rearrange Eq. (5-16«) and, using E's = (1 - /JL2)/ES as in Eqs. (5-18) and (5-19)and m = 1, obtain

AH = AqBE'slsIF

and, since ks is defined as Aq/AH, obtain

k ~ Aq ~ l (9 1)

but carefully note the definition of E's. Now one can correctly incorporate the size effects thatare a major concern—particularly for the mat foundations of the next chapter. As for Eqs.(5-18) and (5-19), we can write a ks ratio from Eq. (9-7) as follows:

ks)_ = B2E'S2IS2IF2 (9 g .

ks2 B1E^1Ip1

Equation (9-8) should be used instead of Eqs. (9-3) through (9-5), and Eq. (9-7) is at least astheoretically founded as Eq. (9-6). Carefully note in using these equations that their basis isin the settlement equation [Eq. (5-16a)] of Chap. 5, and use B, Is, and Ip as defined there.

Equations (9-7) and (9-8) show a direct relationship between ks and Es. Since one doesnot often have values of Es, other approximations are useful and often quite satisfactory if thecomputed deflection (directly dependent on ks) can be tolerated for any reasonable value. Ithas been found that bending moments and the computed soil pressure are not very sensitiveto what is used for ks because the structural member stiffness is usually 10 or more timesas great as the soil stiffness as defined by ks. Recognizing this, the author has suggestedthe following for approximating ks from the allowable bearing capacity qa furnished by thegeotechnical consultant:

SI: ks = 40(SF)^ kN/m3

Fps: ks = l2(S¥)qa k/ft3 (9-9)

where qa is furnished in ksf or kPa. This equation is based on qa = qu\t/SF and the ultimatesoil pressure is at a settlement A// = 0.0254 m or 1 in. (1/12 ft) and ks is q^AH. ForAH = 6, 12, 20 mm, etc., the factor 40 (or 12) can be adjusted to 160 (or 48), 83 (or 24), 50(or 16), etc.; 40 is reasonably conservative but smaller assumed displacements can alwaysbe used.

The most general form for either a horizontal or lateral modulus of subgrade reaction is

ks = As + BsZn (9-10)

where A5 = constant for either horizontal or vertical membersBs = coefficient for depth variationZ = depth of interest below groundn = exponent to give ks the best fit (if load test or other data are available)

Either As or Bs in this equation may be zero; at the ground surface As is zero for a lateral ks

but at any small depth As > 0. For footings and mats (plates in general), As > 0 and Bs = 0.Equation (9-10) can be used with the proper interpretation of the bearing-capacity equa-

tions of Table 4-1 (with the dt factors dropped) to give

tfuit = cNcsc + yZNqsq + 0.5y BN7S7) (9-10«)

Observing that

As = C(cNcsc + 0.5yBNySy) and BSZX = C(yNqsq)Z

l

we obtain a ready means to estimate ks. In these equations the Terzaghi or Hansen bearing-capacity factors can be used. The C factor is 40 for SI units and 12 for Fps, using the samereasoning that qu\t occurs at a 0.0254-m and 1-in. settlement but with no SF, since this equa-tion directly gives qu\t. Where there is concern that ks does not increase without bound withdepth Z, we may adjust the B8Z term by one of two simple methods:

Method 1: S^ tan"1 —

Method 2: ^-Zn = B'sZn

Dn

where D = maximum depth of interest, say, the length of a pileZ = current depth of interestn = your best estimate of the exponent

Table 9-1 may be used to estimate a value of ks to determine the correct order of magnitudeof the subgrade modulus obtained using one of the approximations given here. Obviously if acomputed value is two or three times larger than the table range indicates, the computationsshould be rechecked for a possible gross error. Note, however, if you use a reduced value ofdisplacement (say, 6 mm or 12 mm) instead of 0.0254 m you may well exceed the table range.Other than this, if no computational error (or a poor assumption) is found then use judgmentin what value to use. The table values are intended as guides. The reader should not use, say,an average of the range given as a "good" estimate.

The value of Xmax used in Fig. 9-9c (and used in your diskette program FADBEMLP asXMAX) may be directly estimated at some small value of, say, 6 to 25 mm, or from inspectionof a load-settlement curve if a load test was done. It might also be estimated from a triaxialtest using the strain at "ultimate" or at the maximum pressure from the stress-strain plot.Using the selected strain emax compute

TABLE 9-1Range of modulus of subgradereaction ks

Use values as guide and for comparison whenusing approximate equations

Soil kS9 kN/m3

Loose sand 4800-16000Medium dense sand 9600-80 000Dense sand 64 000-128 000Clayey medium dense sand 32 000-80 000Silty medium dense sand 24 000-48 000Clayey soil:

qa< 20OkPa 12000-24000200 <qa< 800 kPa 24 000-48 000

4a>800kPa > 48 000

The 1.5 to 2B dimension is an approximation of the depth of significant stress-strain in-fluence (Boussinesq theory) for the structural member. The structural member may be eithera footing or a pile.

Example 9-5. Estimate the modulus of subgrade reaction ks for the following design parameters:

B = 1.22 m L = 1.83 m D = 0.610 m

qa = 200 kPa (clayey sand approximately 10 m deep)

Es = 11.72 MPa (average in depth 5B below base)

Solution. Estimate Poisson's ratio /x, = 0.30 so that

K = l-^ = l-^f = 0.077 65 mVMN

For center:

H/B' = 5B/(B/2) = 10 (taking H = 5B as recommended in Chap. 5)

LlB = 1.83/1.22 = 1.5

From these we may write

/, = 0.584 + 1~_2(()

Q3

3) 0.023 = 0.597

using Eq. (5-16) and Table 5-2 (or your program FFACTOR) for factors 0.584 and 0.023.At DlB = 0.61/1.22 = 0.5, we obtain IF = 0.80 from Fig. 5-7 (or when using FFACTOR for

the Is factors). Substitution into Eq. (9-7) with B' = 1.22/2 = 0.61, and m = 4 yields

K = 0.61(0.07765X4 x 0.597X0.8) = U - 0 5 M N / m 3

You should note that ks does not depend on the contact pressure of the base qo.For corner:

H/B' = 5B/B = 5(1.22)/1.22 = 5

[from Table 5-2 with L/B = 1.5 obtained for Eq. (5-16)]

04ls = 0.496 + 7^(0.045) = 0.522 IF = 0.8 (as before)

Again substituting into Eq. (9-7) but with B' = B = 1.22 m and one corner contribution, we have

k> = 1.22 X 0.0776^X0.522 X 0.8 = 2 5 - 2 8 M N / m 3

For an average value we will use weighting, consisting of four center contributions + one cornervalue, giving

4(11.5)+ 25.28 ^ O A i r w x T / 3fe(avg + — ~ = 13.896 MN/m3

We can also estimate ks based on SF = 2 for sand to obtain

ks = 40(SF)(^) = 40(2)(0.200) = 16 MN/m3

For practical usage and since these values of 13.896 and 16.0 are estimates (but reasonably close)we would use

ks = 15.0 MN/m3 (15000 kN/m3)

Comments. It is evident from this example that the "center" ks is softer (or less stiff) than a corner(or edge). The center being less stiff is consistent with the dishing of uniformly loaded plates. Onecan also zone the area beneath a footing by computing a series of ks values at, say, center, \, | , andedge points using for the \ and | point the contributions from four rectangles and for the edge thecontributions of two rectangles of the same size.

Note the use of H = 5B = 5 X 1.22 = 6.1 m for both center and corner.

9-7 CLASSICAL SOLUTION OF BEAMON ELASTIC FOUNDATION

When flexural rigidity of the footing is taken into account, a solution can be used that isbased on some form of a beam on an elastic foundation. This may be the classical Winklersolution of about 1867, in which the foundation is considered as a bed of springs ("Winklerfoundation"), or the finite-element procedure of the next section.

The classical solutions, being of closed form, are not so general in application as the finite-element method. The basic differential equation is (see Fig. 9-10)

E1ji = q = ~Ky (9"n)

where k's = ksB. In solving the equations, a variable is introduced:

A = V 4Ei or AL = V 4^7

Table 9-2 gives the closed-form solution of the basic differential equations for several load-ings shown in Fig. 9-10 utilizing the Winkler concept. It is convenient to express the trigono-metric portion of the solutions separately as in the bottom of Table 9-2.

Hetenyi (1946) developed equations for a load at any point along a beam (see Fig. 9-10Z?)measured from the left end as follows: