Settlements - Steinbrenner Method

Settlement of Foundations -- GEOTECHNICAL ENGINEERING-1997 -- Prof. G.P. Raymond© 113

SETTLEMENT OF FOUNDATIONS

SYMBOLSNotation Dimensional Analysis

A = Area of loaded area L2

Pore pressure coefficient A -B = Breadth of loaded area L

Pore pressure coefficient B -C = Compression index per log cyclecv = Coefficient of consolidation L T-1

(vertical drainage)Df = Buried depth of footing's contact area LE = Deformation modulus or Young's modulus M L-1 T-1

e = Void ratio -F = Constant or coefficient or function -H = Depth or Height of Layer Lh = H/B -h = Height of wall LI = Influence factor -L = Length of loaded area LL = Span length between adjacent footings LR = L/B -mv = Modulus of volume change M-1 L TN = Standard Penetration Number -Q = Load of footing or area M L T-1

q = Load per unit area M L-1 T-1

S = Settlement or compression of layer LT = Time factor -t = Time Tu = Pore pressure or excess pore pressure M L-1 T-1

z = Depth Lα = Coefficient -β = Coefficient -µ = Coefficient or constant -δ = Differential settlement between adjacent columns Lν = Poisson's ratio -σ = stress M L-1 T-1

Subscriptsav = averagec = compression, critical or consolidationcr = recompressioncone= conei = general numbern = general valueo = overburdenoed= oedometeru = undrainedv = verticalx = in the x directiony = in the y directionz = in the z direction

1. SETTLEMENT OF COHESIVE (FINE-GRAINED) SOILIf a footing on a saturated clay is loaded quite rapidly, then

during the load application the clay will be deformed and pore pressureswill be set up in the clay. Owing to the extremely low permeability ofclays little if any water will be squeezed out of the clay during the loadapplication, and the deformations therefore take place without change involume. The deformations have both lateral and vertical components, andthe vertical component constitutes what is known as the "immediatesettlement."

In the course of time some of the pore-water drains out of the

clay, leading to a volume decrease, and the vertical component of thisvolume change is known as the "consolidation settlement", where theprocess of consolidation is defined as that involving a decrease of themoisture content of a saturated soil without replacement of the water byair.

Ideally the settlement below a footing is composed of"immediate settlement" involving no volume change plus "consolidationsettlement" which is ideally dependent on the dissipation of pore waterpressure (Skempton and Bjerrum, 1957). In actual fact both undrainedcreep, where the soil maintains a constant moisture content or volume, andconsolidation creep, known as secondary consolidation, can occur. Thesecondary consolidation occurs after the apparent dissipation of all porepressure although, clearly, a small pore pressure difference must exist tocause a moisture change.

In most foundation analysis, particularly where the factor ofsafety against general failure is greater than 2, undrained creep andsecondary consolidation settlements are neglected. The total settlementis then divided into two components

S ' Su % Sc (1)

whereSu = the immediate or undrained settlementSc = the consolidation settlement

(A) Immediate SettlementBy definition the immediate settlement takes place without

dissipation of the pore pressures. While some consolidation settlementmay occur during construction the settlement due to undrained movementsis obtained from

Su ' mz

o

1Eu

(σz & νu σx & νu σy )dz (2)

whereEu = the undrained deformation modulus or pseudo Young's

Modulusσ = the change in stress due to the increase in loadνu = the undrained pseudo Poisson's Ratio normally assumed equal

to 0.5.Equation (2) may be solved numerically be dividing the bed of clay intoa number of layers. The immediate settlement is then

Su ' j 1Eu

(σz & νu σx & νu σy )Hi

(3)

In some deposits the soil deformation modulus does not varymuch and may be regarded as constant. For these deposits solutions toequation (2) based on elastic theory may be obtained for direct use. Thebest known of these is that obtained by Steinbrenner (1934). Hecomputed the settlements at any depth below the corner of a uniformlyloaded rectangular area located on the horizontal surface of a semi-infinitehomogeneous isotropic elastic mass of constant elastic properties. Heassumed that the settlement of the corner on a soil layer of depth H wasequal to the settlement of the surface point minus the settlement of thepoint at depth H so that

S ' Ssurface & SdepthH '

qBE

[ (1 & ν2) F1 % (1 & ν & 2ν2 ) F2 ] (4)


Figure 1. Graph for estimating settlement below corner ofuniform rectangular loaded area (e.g. Steinbrenner, 1934).

Figure 2. Method of estimating settlement of point locatedwithin (or outside) rectangular loaded area.

Figure 3. Adaption of Steinbrenner's solution toapproximate analysis of layer soil (e.g. Butler, 1975).

Figure 4. Graph to estimate immediate settlement belowsurface of semi-infinite elastic solid (e.g. Janbu et al. 1956).

where

F1'1πR ln {R% R2&1 } R2%h 2

R [1% R2%h 2 ]% ln R% R2%1 1%h 2

R% R2%h 2%1(5)

F2 'h

2πtan&1 R

h (R 2 % h 2 % 1)(6)

I ' (1 & ν2 ) F1 % (1 & ν & 2 ν2 ) F2 (7)

whereL = the length of a rectangular area,B = the width of the areaR = L/B = the length factorh = H/B = the depth factorq = the unit loadE = the modulus of elasticity of the layer, andν = Poisson's ratio

The values of F1 and F2 are shown in Figure 1. The settlement at apoint N within (or outside) the rectangular area shown in Figure 2 maybe calculated by addition (or subtraction) of the corresponding settlementsdue to areas 1, 2, 3 and 4

S 'qE

(I1 B1 % I2 B2 % I3 B3 % I4 B4 ) (8)

In practice the deformation modulus of a deposit commonlyvaries with depth. Steinbrenner's solutions may be used as anapproximation to find the settlement in a soil of variable modulus. Thisis achieved by dividing the soil deposit into a multi-layered system as inFigure 3. Assuming the deposit is uniform Steinbrenner's method givesthe relative movement of any of the points, B, C etc. below A relative toA on the surface. By subtraction of the relative movement from A of twoadjoining points the compression of each layer may be obtained. Theactual settlement may then be approximated by

S %E1

E1

∆S1 %E1

E2

∆S2 %E1

E3

∆S3 etc. (9)

Some solutions using the above approach for a linearly increasingmodulus with depth have been derived and published for rectangular loadsby Butler (1975).

Few charts for estimating the immediate settlement of buriedfootings exists. One of the most useful charts has been prepared by Janbu,Bjerrum and Kjaerinsli (1956) and is shown is Figure 4. The chart


Figure 5. Ratio of consolidation settlement to oedometersettlement (e.g. Skempton and Bjerrum, 1957)

provides estimates of the average immediate settlement (as opposed to themaximum) of a uniformly loaded flexible area. The average settlementsare obtained from (all calculations are for ν = 0.5)

Su 'q B IE (10)

I ' µo µ1 (11)

If E varies with depth the same approach as used in obtaining equation (9)may be followed.

(B) Consolidation Settlements:The consolidation of a clay results from the dissipation of pore

pressure, with an accompanying increase in effective pressures. A givenset of stresses will in general cause different pore pressures in differentclays. Thus two identical footings carrying identical loads on twodifferent soils with identical compressibility would set up different porepressures and result in different amounts of settlement despite havingidentical consolidometer test results.

This may seem paradoxical, but the explanation is that in theconsolidometer test no lateral strains are permitted and, under this specialcondition, the pore pressure set up in a saturated clay by an appliedpressure is always equal precisely to that applied pressure irrespective ofthe type of clay; provided only that it is fully saturated.

It also follows that if, in practice, the conditions are such thatno lateral strains can take place during the load application then, otherthings being equal, the pore pressures will be the same in all saturatedclays, and the consolidation settlement (there will be no immediatesettlement if there can be no lateral strains unless air or gas is present inthe pore fluid) will be directly proportional to the compressibility of theclays. The conditions of no lateral strain is approximately true for at leasttwo practical cases; (a) that of a thin layer of clay lying between beds of sand or between sand

and rock, and(b) that of a loaded area of horizontal extent which is great compared with

the thickness of the underlying clay, when the lateral strain will benegligible expect near the edges of the loaded area.

In cases such as these the consolidation settlement can be estimated withreasonable accuracy by a direct application of the consolidometer testresults, and it is exactly for such cases that Terzaghi (1925) developed hisone-dimensional theory of consolidation, the essential data for which arederived from the consolidometer.

In the more general case where lateral deformations can occurthen pore pressures are set up by the stresses depend upon the type of clayas well as on the stresses themselves; and the consolidation settlementstherefore also depend on the type of clay. Any method of calculatingconsolidation settlements which does not enable some allowance to bemade for this effect is bound to be unsatisfactory in principle.

If ∆σ1 and ∆σ3 are the increases in the principal stresses at anypoint, caused by loading the footing, then the excess pore pressure set upin the clay at this point may be represented by the expression

u ' B {∆σ3 % A (∆σ1 & ∆σ3)} (12)

where A and B are the pore-pressure coefficients (Skempton, 1954).

Now, in the one-dimensional consolidation test the verticalcompression of the clay is measured under the condition of no lateralstrain and if a vertical compression Soed is caused by consolidation, in thistest, under an increase in effective pressure ∆σv

/, then:

Soed ' mv ∆σ)

v H (13)

where H is the thickness of the sample and mv is defined as the modulusof compressibility in the one-dimensional consolidation test.

Since the consolidation of an element of clay beneath afoundation takes place without appreciable lateral strain, the verticalcompression of an element during consolidation can be expressedapproximately by the analogous equation:

dSc ' mv u dz (14)

where dz is the thickness of the element.

The consolidation settlement Sc of the centre of a foundation,resting on a bed of clay of thickness z is therefore:

SC ' mz

o

mv u dz (15)

But, from Equation (12), with B = 1 for a saturated clay,

u ' ∆σ1 A %∆σ3

∆σ1(1 & A ) (16)

Hence:

Sc ' mz

o

mv ∆σl A %∆σ3

∆σ1

(1 & A) dz (17)

Below the centre line of a loaded area where the settlementsare usually greatest and ∆σ1 = ∆σv equation (17) for a uniform deposit orsoil layer, may be written as

Su ' µ Soed (18)

where

µ ' A %∆σ3

∆σ1

(1 & A) (19)

andSoed is the one-dimensional consolidation settlement. In a homogeneousmass where A and mv are constant with depth an average value of µ maybe obtained by integration of the ratio ∆σ1/∆σ3 and this is shown in Figure5.


Figure 6. Settlement-effective stress pressure relationship.

Figure 7. Time-settlement relation, (a) root time plot; (b)logarithmic time plot.

In practice the consolidation settlement of cohesive soils isnormally computed on the basis of a semi-logarithmic relation betweensettlement and effective vertical pressure, as shown in Figure 6, ratherthan a linear relationship.

For loadings less than the preconsolidation pressure, σc,settlement will be computed using a value of a compression indexrepresenting recompression, Ccr. For loadings greater than thepreconsolidation pressure, settlement will be computed using the virgincompression index, Cc. Where the increase in pressure represents bothrecompression and loading in excess of the precompression load, thesettlement equation for a uniform soil layer may be written:

Soed ' HCcr

1 % eolog

σ)cσo;

%CC

1 % eolog

σ)o % ∆σ

σ)c(20)

whereSoed= consolidation settlement for one-dimensional loadingH = original thickness of stratumCcr = recompression indexCc = compression indexeo = initial voidσo' = initial effective pressureσc' = preconsolidation pressure∆σ = the average change in pressure in the compressible stratum

considered

The estimation of the preconsolidation pressure is complex dueto its sensitivity to sampling disturbance and its variability with smallchanges in depth. Originally it was thought of as the maximum pastvertical effective stress but in practice many factors can cause an increasein this value and it is now often referred to as the apparent (sometimesquasi) preconsolidation or critical pressure. Its estimation along with thatof secondary consolidation are admirably discussed in the TransportationResearch Board's Special Report 163 (1976) "Estimation of ConsolidationSettlement".

(C) Settlement-Time Relationships:Consolidation is a time-dependent process which is

theoretically predicted by consolidation theory and, typically under aparticular load will plot as shown in Figure 7.

Three significant portions of the measured settlement may beconsidered.(a) An initial compression which occurs immediately owing at least in

part to the compression of gas in the pore space but which should notoccur if the soil is fully saturated and loaded one-dimensionally (it isnot shown in Figure 7).

(b) The compression indicated by the solid lines of Figure 6 known as

primary consolidation, which is accompanied by a corresponding dropin pore water pressure.

The time at which primary consolidation will take place can becalculated from the equation:

t ' Tvh 2

cv(21)

wheret = time elapsed to reach the percentage degree of settlement (S%)h = length of drainage path (for the usual case of double drainage,

2h equals the thickness H of the consolidating stratum)cv = coefficient of consolidation for the appropriate range of

pressures, andTv = time factor, which for instantaneous loading and one-

dimensional consolidation of a thin layer of soil is as follows(other cases have been solved are available in the literature)

An approximate value of cv can be obtained from the relationship:

cv 'k

γw mv(22)

whereK = permeabilitymv = modulus of compressibility or inverse of modulus of

deformationγw = unit weight of waterApproximately:

Tv 'π4

S %100

2from S ' 0% to 60 (23)

Tv '1.781&0.933 log10(100&S%) from S ' 4 (24)

S% 10 20 30 40 50 60 70 80 90Tn .003 .031 .071 .126 .197 .287 .403 .557 .848

(c) The compression indicated by the difference between the solid anddashed lines of Figure 7, known as secondary compression, or theconsolidation resulting from the secondary time effects. This takesplace for practical purposes at constant effective stress with no changein pore water pressure and is related to the portion of the curve inwhich excess pore water pressures are negligible.

2. SETTLEMENT OF NON-COHESIVE (COARSE-GRAINED)SOIL

The permeability of sands and gravels is sufficiently great thatconsolidation normally takes place during the construction period.Settlement of sands and gravels is largely the result of rearrangement ofthe particles and may be significant, particularly in loose deposits.Settlement, even when very low soil pressures are used in design, is likelyto follow submergence, soaking or vibration from blasting, machineoperations, or earthquake.


Figure 8. Allowable pressure for footing - cohesionless soiland 25 mm max. settlement (e.g. Terzaghi and Peck, 1985).

(A) Standard Penetration Test:The settlement of shallow footings may be roughly related to

the N value obtained from the Standard Penetration Test. From theserelationships one can obtain allowable settlements to meet givensettlement criteria. Terzaghi and Peck (1948) have suggested therelationship shown in Figure 8. The allowable bearing pressure obtainedfrom this relationship is such that the resulting total settlement will beabout 25 mm and the differential settlement less than 3/4 the totalsettlement. These settlement limits can generally be tolerated by normalbuildings without causing structural damage as presented later.

Where other settlement criteria are to be used the curves shownin Figure 8 may be expressed mathematically by the approximateequations given below.

S '2σavN

for B # 1.2m (25)

S '3σavN

BB % 0.3

2for B > 1.2 m (26)

whereS = the settlement in mmσav = the average contact pressure in kPaB = the footing width in metersN = the average standard penetration number of blows per 300 mm

to a depth from the contact level to between B and 2B belowthe contact level depending on variability.

Using these equations the allowable pressure may be relatedto any given amount of settlement. The equations assume that the load-settlement relationship has not exceeded a linear one which is reasonablefor factors of safety greater than 3. It should be noted that the equationsare empirical and thus the correct units must be inserted.

According to theory, the submergence of the sand locatedbeneath the base of a footing should halve the confining or lateraleffective pressure in the sand deposit and also halve the deformationmodulus. This would approximately double the settlement provided thebase is located at or near the surface of the sand. The values obtainedfrom Figure 8 should theoretically be reduced by 50 percent as noted onthe figure and as originally suggested by Terzaghi and Peck.

This procedure leads to conservative and probably over-conservative results (Meyerhof, 1965). In practice, where the standardpenetration test was performed below the water table and furthersubmergence of the deposit is not likely to occur the use of the N values,determined in the field, inherently includes a correction for submergence.Therefore, it is not uncommon to neglect the effect of submergence andthis may quite properly be done where local experience supports theprocedure or where the possibility of greater settlement is not ofcontrolling importance in the design. Another factor which adds to theconservative nature of Figure 8 is the effect of footing depth. Thesurcharge, due to overburden, results in an increase of the lateral effectivepressure below the footing and thus the deformation modulus.Theoretically if the footing is buried to a depth of half the footing widththe settlement is approximately halved.

Where the N values are determined in dry soil which issubsequently submerged the reduction procedure given in Figure 8 shouldbe followed. Settlements calculated using this procedure are generallygreater than those actually observed. The method is of limited value forsoils containing gravel, cobbles, or boulders where single fragments mayaffect the blow count, and is not valid for cohesive or cemented soil.

(B) Static Cone Penetration TestSettlement may be estimated from the results of static cone

penetration tests by means of the relationship between the coefficient ofcompressibility, αc and the cone point resistance, qcone.

αc 'β qconeσ)o

(27)

whereαc = coefficient of compressibilityqcone= cone resistanceσo' = effective overburden pressure, andβ = a coefficient depending on soil density as follows:

Soil density βDense sand < 1Compact sand 1Loose sand 1.5

Settlement under a shallow foundation can then be estimatedby substituting the value of αc into the settlement equation.

S ' mDf%2B

Df

2.3 1αc

logσ)o % ∆σ

σ)odz '

j 2.3∆Hα

log10

σ)o % ∆σ

σ)o

(28)

where


Table 1. Guide to maximum allowable deflections.

Type of construction Maximum deflection

Members supporting walls orpartitions of(i) Masonry, glass or other

frangible material(ii) Metal cladding or similar non-

frangible finishesSteel or concrete framesTimber framesSteel or concrete shear walls

L/360

L/240L/150 to l/180

L/100by design

Table 2. Guide to maximum allowable slopes.

Type of construction Maximum slope*

High continuous brick wallsBrick dwellingsBrick cladding between columnsReinforced concrete building framesReinforced concrete curtain wallsContinuous steel framesSimply supported steel frames

0.005 to 0.0010.0030.001

0.002 to 0.0040.0030.0020.005

* Maximum slope of deflected configuration from the line ofthe as-built configuration.

Figure 9. Suggested relationship between angular distortionand building performance (e.g. Bjerrum, 1963)

S = settlementH = thickness of the compressible soil layer or twice the width of

the foundation (measured below the footing base) whicheveris less

∆σ = pressure change applied to the soil layerB = the footing breadthDf = the buried depth of the footing

The static cone test was developed for use in loose, uniform,fine-grained cohesionless soils and field verification has been restrictedlargely to deposits of these materials. The equipment normally used iseffective in such soils, but may give trouble in dense or mixed-graineddeposits. Experience indicates that the αc calculated by this method isusually low, giving an upper limit to estimated settlements. The staticcone penetration test should be supplemented with subsurface data fromconventional boreholes.

(C) Plate Bearing Test:A plate bearing test may be carried out on non-cohesive soils

in which the settlement of a 300 mm square test plate is measured andrelated to the expected settlement of a footing. The relationship suggestedby Terzaghi and Peck (1948, 1967), who also describe the conditionsrequired for a Standard Load Test is:

S ' S12B

B % 0.3

2(29)

whereS = settlement of footing with width of B metresS1 = settlement of a 300 mm square loading plate under the

pressure expected to be applied by the footing Because the equation is empirical the units for the width ** must be inmetres

The method is only considered suitable for use in non-cohesivesoils where time-dependent settlement relationships are negligible. Unless a series of tests at different depths are performed, usually in a shaftat considerable expense, the tests only give data on a shallow depth of soilwhich, to be of value, must be representative of the stratum affected by thefooting. Extrapolation to large footings should be carried out withcaution. From an inspection of the bulbs of equal pressure in thediscussion of stress distribution, it will be obvious that the prototype willstress an entirely different depth of material. It follows that the test willbe misleading if the material properties change within the depth affectedby the larger footing.

The test is cumbersome to perform and potentially misleading.It requires supplementary information from boreholes and, generally, thesewill yield sufficient information to allow satisfactory estimates to be madewithout the use of detailed load tests.

(D) Pressuremeter Test:The settlement of a footing on granular soil may also be

estimated from pressuremeter test results. As for the plate load test resultsshould be obtained from different depths to measure the variability of thedeposit.

3. ALLOWABLE SETTLEMENTFor any given structure there is a certain amount of settlement,

either differential or total, that can be tolerated without:- overstressing the structure;- creating an unacceptable maintenance or aesthetic problem.

The foundation must be designed so that anticipated settlements do notexceed the lesser of these amounts.

Allowable displacement criteria in common use are as follows:(a) Maximum differential or relative deflections between supports, where

L is the span length are given in Table 1.

(b) Limitation of slopes as given in Table 2.

Similar values have been presented by Bjerrum (1963), andthese are shown in Figure 9.

Differential settlement will occur in all foundations because ofthe natural variability of soils even where total settlements are calculatedto be uniform. The magnitude of these differential settlements are


Table 3. Guide to maximum total settlements.

Type of soil below structure Total settlement

Structures on clayStructures on sand

80 mm40 mm

calculated to be uniform. The magnitude of these differential settlementsmay be related to the magnitude of the total settlements, for example, seeD'Appolonia et al 1968. Consequently, limiting the total settlement of astructure is frequently used as an indirect means of controlling the amountof differential settlement unless the structure is specifically designed totake or prevent large differential settlements.

The values given in Table 3 are suggested.

Design limits on differential settlement are frequently set intotally unrealistic terms. In fact, each structure should be consideredindividually with the tabulated values providing only a guide.

Design limits on total settlement are simple criteria to applyand are commonly used. Many successful structures may be seen,however, with total settlement greatly in excess of the values quoted.

4. DESIGN FOR UNIFORM SETTLEMENTFor a structure in which the columns are supported by separate

footings resting on an ideal soil it is possible, at least in theory, to soproportion the footings that the structure settles uniformly with zerodifferential settlement between columns. The method of approach woulddepend on the theoretical model used for the foundation soil. If the soildeformation modulus was relatively constant then elastic theory withuniform elastic parameters would be appropriate. Such conditions wouldprobably occur for deposits of over consolidated clay. On the other hadfor deposits of sand the elastic modulus would be expected to increaselinearly with depth from zero at the surface and give a settlement responseclosely proportional to the applied pressure characteristic of a Winklermodel. Design for equal settlements of separate footings would dependon the foundation modelling.

(A) Uniform Elastic ModulusThe settlement of a footing on an elastic media with uniform

elastic modulus may be expressed by (see Equations (4) and less exactlyEquation (10))

S ' q BEI (30)

Whereq = the average footing pressureB = the breadth of the footingE = the deformation modulus (based on either undrained

conditions or effective stresses depending on the problem).I = an influence factor depending on the shape of the footing, the

depth of soil and Poisson's ratio of the soil.If, as a first approximation, the depth effect is neglected, Poisson's ratiois assumed constant and the settlement of any footing due to the loads onany neighbouring footings is ignored, then for uniform settlements ofsimilar shaped footings (length/breadth ratio constant)

q B ' a constant µe (31)

QB

' a constant µ4 (32)

whereQ = the total load on the footing.

Many building codes specify the maximum allowable pressurepermitted on a certain soil so that the logical procedure is to design thefooting with the least load first. The size of this footing is determinedfrom either the maximum allowable pressure permitted or the ultimatebearing capacity theory with an adequate factor of safety. The value maybe increased if necessary if the total settlement is undesirably large. Theremaining footings can then be designed using Equation (32) to form thefollowing relationship.

Bn 'QnA1

B1 (33)

where the subscript 1 denotes the footing with the least load and thesubscript n denotes any other footing.

The use of Equation (33) will not, in fact, avoid all differentialsettlements, for the following reasons:(a) Central footings will settle more than peripheral ones due to the

influence of neighbouring footings, i.e. the site will become slightlydish-shaped.

(b) The "fair average" stress range for the soil may vary appreciably fromfooting to footing thus producing changes in E.

(c) Random variations in the soil may exist. This is particularly likelywith sandy clays.

(d) The depth of soil may not be large compared with the maximum sizeof footing and hence the influence coefficient, I, may not be the samefor all footings.

(e) On clay soils, the rate of settlement may vary between footings.

Nevertheless, design on the basis of Equation (33) shouldproduce considerably smaller differential settlements than the usual basisof designing all footings for the same bearing pressure for the one givensoil. Furthermore, corrections to footing size to take into account effectsa, b and d could be made if thought desirable.

(B) Elastic Modulus Increasing Linearly with Depth:Where the elastic modulus increases linearly with depth from

a value close to zero at the surface, such as a sand deposit, the settlementsbelow loaded areas approximate those of a Winkler model resulting insettlement proportional to average pressure. The design is therefore basedon the following procedure:(a) The footing with the largest load is selected for design first. The size

of the footing is determined from either the maximum allowablepressure permitted or the desirable total settlement.

(b) The smaller footings are proportioned on the basis of the allowablepressure determined for the largest footing.

(c) Check to insure an adequate factor of safety against bearing capacityfailure of the smallest footing.

(C) Other DepositsIn the general design of footing sizes to give equal or near

equal settlements it is likely that either the heaviest or lightest loadedfooting would give a critical condition. A logical starting point is thedesign of these two footings. The other footings may then beproportioned accordingly. Even though all footings have been designedto settle equally it must always be remembered that soils are notmanufactured and their behaviour cannot be guaranteed. For this reasonwhen settlements are large the structure should be designed to permit thelikelihood of differential deformations between footings.

4. FOOTING LOAD-DIFFERENTIAL SETTLEMENTSThe normal method of designing a statically indeterminate

structure is to calculate the column loads on the assumption that there isno differential settlement of the footings. The footings are then designedto carry these column loads. This procedure is satisfactory if the resultingsettlements are equal and involve negligible differential settlements for theparticular case being studied. However, in practice, it is frequently


impossible to avoid some differential settlement. This must involveredistribution of the column loads leading to alteration in the differentialsettlements and further redistribution of the loads and so on. Themagnitude of this redistribution will depend on the stiffness of thestructure. Generally, such structural calculations are much harder toperform accurately a the soil mechanics calculations of settlement. Notonly do they involve structural frame theory of greater sophistication thanhas in the past been generally employed in ordinary design practice, butthey also involve estimation of the stiffening effects of the cladding orshear walls about which little is known except that it is frequently veryimportant.

Meyerhof (1947 and 1953) and Chamecki (1956) have givenmethods of combining the structural and soil mechanics calculations. Themethods are similar in that they both start by calculating the settlementsof the footings under column loads given by structural theory in which itis assumed that there is no differential settlement and then proceed toreach the final solution by successive approximation. The steps inChamecki's method are:(a) Calculate the reactions at the foundation assuming no settlement but

allow for frame stiffness.(b) Calculate the settlements resulting from these reactions, treating the

structure as perfectly flexible.(c) Calculate the coefficients of load transfer which depend on frame

stiffness and use these coefficients to estimate the transfer of loadsbetween the columns, because of settlements given by (2) above.

(d) With the new foundation loads form (3) calculate revised settlements.(e) Calculate new coefficients of load transfer and so on until the desired

accuracy is attained. Frequently it is sufficient to average thesettlements of steps 2 and 4. It is also accurate enough in many casesto assume that load transfer only takes place between a column and itsimmediate neighbours.

Programs for sophisticated structural analysis on computers arebeing developed and are sometimes employed for design. These advancedcomputer design programs should be possible to adapt to the analysis ofsoil-structure interaction.

5. REFERENCES BY TOPIC

(A) Settlement of Cohesive Soils

Bjerrum, L. 1967. Engineering Ecology of Norwegian Normally-Consolidated Marine Clays as Related to Settlements of Buildings.Geotechnique, Vol , pp. 83-117. (TA1.G3).

Crawford, C.B., 1964. Interpretation of the Consolidation Test. Journalof Soil Mechanics and Foundation Division, Proceedings AmericanSociety of Civil Engineers, Volume 90, No. SM5, pp. 87-102. (TA710.A1A57).

Estimation of Consolidation Settlement. Transportation Research Board,Report 163, p. 26. (TE7N2S).

Janbu, N., Bjerrum, L. and Kjaernsli, B., 1956. "Veiledning ved Losningav Fndamenteringsoppgavaer. Norwegian Geotechnical Institute,Publication No. 6, p 93.

Schmertmann, J.H., 1953. Estimating the true consolidation behaviour ofclay from laboratory test results. Proceedings American Society of CivilEngineers, Separate 311.

Skempton, A.W., 1954. The pore pressure coefficients A and B.Geotechnique, Volume 4, No. 4, pp 143-147. (TA1.G3).

Steinbrenner, G., 1927. Der Zeitliche Verlauf einerGrundwasserabsenkung. Wasserwirtsch. u. Tecknik, volume 4, pp. 27-33.

Terzaghi, K., 1943. Theoretical Soil Mechanics. J. Wiley & Sons, N.Y.(TA710.T3).

(B) Standard Penetration Test

D'appolonia, D.J., D'applolonia, E., and Brissette, K.F., 1968. Settlementof spread footings on Sand. Journal of the Soil Mechanics andFoundation Division, Proceedings American Society of Civil Engineers,Volume 94, pp. 735-760. (TA710.A1 A57).

Fletcher, G.F.A., 1965. Standard penetration test: Its uses and abuses.Journal of the Soil Mechanics and Foundation Division, ProceedingsAmerican Society of Civil Engineers, Volume 91, No. SM4, pp. 67-75.(TA710.A1 A57).

Meyerhof, G.F., 1965. Shallow foundations. Journal of the SoilMechanics and Foundation Division, Proceedings American Society ofCivil Engineers, Volume 91, No. SM2,pp 21-31. (TA710.A1 A57).

Peck, R.B., Hanson,W.E. and Thornburn, T.H., 1953. FoundationEngineering. J. Wiley & Sons, N.Y. (TA775.P36).

Terzaghi, K. and Peck, R.B., 1948. (1st edition), 1967 (2nd edition). SoilMechanics in Engineering Practice. John Wiley. (TA710.T33).

(C) Static Cone Penetration Test

Sanglerat, G., 1972. The penetrometer and soil exploration, Elsevier Publ.Co., Amsterdam. (TA710.5 S2513).

Schmertmann, J. H., 1970. Static cone to compute static settlement oversand. Journal of the Soil Mechanics and Foundation Division,Proceedings American Society of Civil Engineers, Volume 96. No. SM#,pp 1011-1043. (TA710.A1 A57).

(D) Plate Bearing Test

D'Appolonia, D.J., D'Appolonia, E., and Brissette, R.F., 1968. Settlementof spread footings on sand. Journal of the Soil Mechanics and FoundationDivision, Proceedings American Society of Civil Engineers, Volume 94,No. SM3, pp. 735-760. (TA710.A1 A57).

(E) Pressuremeter Test

Baguelin, F., Jezequel, J.F., and Shield, D.H., 1979. The pressuremeterand foundation engineering. Trans Tech Publications, Germany.(TA775.B22 1978t).

Menard, L., 1965. Regle pour le calcul de la force portante et dufassement des foundations en fonction des resultats pressiometriques.Proceedings of the Sixth International Conference Soil Mechanics andFoundation Engineering, Montreal, Volume 2, pp. 295-299. (TA710.I6t).

Menard, L., 1972. Rules for the calculation of bearing capacity andfoundations settlement based on pressuremeter tests. Draft Translation159. U.S. Army Corps of Engineers, Cold Regions Research andEngineering Laboratory. (GB2401.U58C).

(F) Allowable Settlement

Bjerrum, L., 1963. Allowable settlements of structures. Proceedings ofthe European Conference on Soil Mechanics and Foundation Engineering,Wiesbaden, Volume 2, pp. 135-137. (TA710.A1 E8).


Burland, J.B. and Wroth, C.P., 1974. Allowable and differentialsettlements of structures including damage and soil structure interaction.Proceedings of the British Geotechnical Conference on Settlement ofStructures, Pentech Press, pp 611-763. (TA775.S43).

B'Appolonia, D.J., D'Appolonia, E. and Brissette, R.F., 1968. Settlementof spread footings on sand. Journal of the Soil Mechanics and FoundationDivision, Proceedings American Society of Civil Engineers, Volume 94,No. SM3, pp. 735-760. (TA710.A1 A57).

Feld, J., 1965. Tolerance of structures to settlement. Journal of the SoilMechanics and Foundation Division, Proceedings American Society ofCivil Engineers, Volume 91, No. SM3, pp 63-77. (TA710.A1 A57).

Polshin, D.E. and Tokar, R.A., 1957. Maximum allowable non-uniformsettlement of structures. Proceedings of the Fourth InternationalConference on Soil Mechanics and Foundation Engineering, London,Volume 1, pp. 402-405. (TA710.I6t).

Skempton, A.W. and MacDonald, D.H., 1956. The Allowable settlementof buildings. Proceedings, Institution of Civil Engineers, Part III, Volume5, pp. 727-768. (TA1.I554t).

Sowers, G.F., 1962. Shallow Foundations. In: G.A. Leonards.Foundation Engineering, McGraw-Hill, N.Y. (TA710.I6t).

(G) Interaction of Footing Loads and Settlements

Chamecki, S., 1956. Structural rigidity in calculation settlements. Journalof the Soil Mechanics and Foundation Engineering Division, ProceedingsAmerican Society of Civil Engineers, Volume 82, Paper No. 865. p 19.(TA710.A1 A57).

Meyerhof, G.G., 1947. The settlement analysis of building frames. Structural Engineers, Volume 25, pp 369 - . (TA680.S2).

Meyerhof, G.G., 1953. Some recent foundation research and itsapplication to design. Structural Engineer, volume 31, pp 151- . (TA680.S2).


σv 'q (ψ % sinψ)

π

σy 'q (ψ & sinψ)

π

At mid clay depth

σv '200 (ψ % sinψ)

π

where ψ ' 2 tan&1 1.54

' 41.1E

σv ' 87.5 kPaσh ' 3.8 kPa

For plain strain

εy ' 0 'σy & ν (σv % σh)

Eˆ σy ' ν (σv % σh) ' 45.7 kPa

εv 'σv & ν (σh % σy )

E

'87.5 & 0.5 (3.8 % 45.7)

15000' 0.00418

Su ' 8.37 mm

Alternative method for immediate settlement using StienbrennerLB' 4 ; D

B' 3.33to bottom of deposit

DB

' 2.00 to top of deposit

S '4 q BE

Ib ' It ; Note I ' 34F1

'3 (200) 1.5

15000Fb & Ft

'3 (200) 1.5

15000(0.40 & 0.26)

' 0.0084 metre ' 8.4 mm

At centre of clay bedσ)o ' γgravel Hgravel % γclay Hclay & γwaterHwater

' 20 (3) % 17 (1) & 4 (9.81) ' 37.8 kPa

∆Soed ' HC

1%eolog10

σ)o % σvσ)o

' 2 0.41 % 0.7

log1037.8 % 87.5

37.8' 0.2449 metre ' 245 mm

µ ' A %∆σ3

∆σ1

(1 & A)

' 0.7 %3.8

87.5(1 & 0.7) ' 0.713

Scons ' µ Soed ' 0.713 (0.2449) ' 0.174 metre' 174 mm

EXAMPLE 1A strip footing 3 metres wide is built on the surface of a layer of

gravel 3 metres thick. The gravel overlays a 2.0 m thick layer of claywhich in turn overlays permeable rock. The water table is level withthe ground surface. The load on the footing may be assumed toincrease instantaneously form zero to 200 kPa and then remainsconstant. Calculate the immediate and consolidation settlement onthe centre line givenγ gravel = 20 kN/m3

γ clay = 17 kN/m3

Initial void ratio in clay = 0.7Compression index of clay = 0.4Coefficient of consolidation = 0.01 m2/dayPore pressure coefficient A = 0.7Deformation modulus = 15.0 MPaPseudo poisson's ratio = 0.5_________________________________________________________

IMMEDIATE OR UNDRAINED SETTLEMENTFrom theory of elasticity of centre line

CONSOLIDATION SETTLEMENT


Standard PenetrationTest. Blow count.

Depth of Testmetre

8101012201828282630

0.751.502.253.003.754.505.256.006.757.50

N %10 % 10 % 12 % 0.5 (8 % 20)

4' 11.5

S '3 σN

BB % 0.3

2'

3 (100)11.5

33 % 0.3

2

' 21.7 mm

Cone ResistanceMPa

Depth of Testmetre

5 7 710 9121416

1.252.253.254.255.256.257.258.25

S ' mDf % 2B

Df

2.3 σ)oqcone β

log10

σ)fσ)o

Approx. ∆σ 'qπ

(ψ % sinψ) ; where ψ ' 2 tan&1

B2

z & Dfq ' footing load & overburden removed ' 100 kPa

EXAMPLE 2Calculate the settlement below a strip footing 3 metres wide

founded in a deposit of sand at a depth of 0.75 metres using the datagiven below. The load on the footing is 100 kPa.

_________________________________________________________

Average N from Df to Df + B below base

EXAMPLE 3Calculate the settlement below a strip footing 3 metres wide

founded in a deposit of sand at a depth of 0.75 metres using the datagiven below. The estimated increase in load on the footing above theoverburden removed is 100 kPa. The sand is compact (β = 1).Assume the water table remains deep and γ = 18 kN/m3.

_________________________________________________________

TABLE FOR EXAMPLE 3

Depth m

Cone RMPa

D - 0.75m

σ2'kPa

∆σkPa

∆S m

1.25 2.25 3.25 4.25 5.25 6.25

5 7 7 10 9 12

0.5 1.5 2.5 3.5 4.5 5.5

22.5 40.5 58.5 76.5 94.5112.5

99.681.862.548.839.633.1

0.00760.00640.00610.00380.00370.0024

0.0300Settlement = 30 mm


Ss & Sd 'q BE

(1 & ν2 )F1 % (1 & ν & 2ν2 )F2

since ν '12

; Ss & Sd 'q BE

34F1

(Su )centre ' 4 q B 34

' ∆FEtElayer

1Et

' 4 (50) 8 34

[0.231] 160000

' 0.0046 m ' 4.6 mm

(Su )corner ' 1 q B 34

' ∆FEtElayer

1Et

' 1 (50) 16 34

[0.1154] 160000

' 0.001154 m ' 1.2 mm

EXAMPLE 4A foundation 30 m x 16 m carries a uniform pressure of 50 kPa on

the surface of 1 16 m deposit of saturated clay. The undrainedmodulus of deformation increases with depth at a rate of 2 MPa permetre of depth and is 60 MPa at the surface. Dividing the deposit intofour 4 m layers calculate the undrained settlement caused by thebuilding at the centre and corner of the foundation. νu = one half._________________________________________________________ CENTRE

Divide into four equal areas L = 15 B = 8 Et = 60 MPa Calculations tabulated in table for centre settlement below.

CORNERDivide into one area L = 30 B = 16 Et = 60 MPa Calculations tabulated in table for corner settlement below.

Diff Settlement = 3.4 mm

TABLE FOR CENTRE SETTLEMENT OF EXAMPLE 4

H Elayer H/B L/B F1 ∆F

∆FEtElayer

4 8 12 16

64 72 80 88

0.5 1.0 1.5 2.0

1.875 1.875 1.875 1.875

0.050.150.230.29

0.050.100.080.06

0.04690.08330.06000.0409

0.231

TABLE FOR CORNER SETTLEMENT OF EXAMPLE 4

H Elayer H/B L/B F1 ∆F

∆FEtElayer

4 8 12 16

64 72 80 88

0.25 0.5

0.75 1.0

1.875 1.875 1.875 1.875

0.020.050.100.15

0.020.030.050.05

0.01880.02500.03750.0341

0.1154


EXAMPLE 4A continuous beam 40 metres long is supported in the centre and

at each end by footings resting on soil having a deformation modulusof 100 nm/m. Calculate the load on each footing assuming nodifferential settlement and the deformations resulting from such loads,the resulting transfer load, etc. for several iterations. The beam takesa uniformly distributed load of 20 kN/m and its stiffness 3MN/m2._________________________________________________________

FIRST APPROXIMATIONR1 = 0.38 wR = 0.38 x 20 x 103 x 20 N = 152 kN R2 = 1.24 wR = 1.24 x 20 x 103 x 20 N = 496 kN δ1 = R1 k = 152 x 103 x 100 x 10-9 = 0.0152 mδ2 = R2 k = 496 x 103 x 100 x 10-9 = 0.0496 m

_________γ1 = δ2 - δ1 = 0.0344 m

FIRST CORRECTION - CHANGE DUE TO FIRSTAPPROXIMATION∆R1 = 3 EI ∆/R3 = 38.7x100x106x0.0344/203 = 38.7 kN∆R2 =-6 EI ∆/R3 = -77.4 kN∆δ1 = ∆R1xk = 38.7x100x10-9 = 0.0039 m∆δ2 = ∆R2xk =-77.4x100x10-9 = 0.0077 m

___________ ∆2 =-0.0116 m

SECOND CORRECTION - CHANGE DUE TO FIRSTCORRECTION∆R1 = -0.0116x38.7/0.0344 = -13.1 kN∆R2 = = +26.1 kN∆δ1 = 0.0039x(-13.1)x38.7 =-0.00131 m∆δ2 = = 0.00259 m

-----------

FINAL TALLY:R1 = 152 + 38.7 - 13.1 = 177.6 kNδ1 = 15.2+ 3.9 - 1.3 = 17.8 mmR2 = 496 + 38.7 - 13.1 = 444.7 kNδ2 = 49.6- 7.7 + 2.6 = 44.5 mm


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