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The Foundation Engineering Handbook

Manjriker Gunaratne

Spread Footings: Analysis and Design

Publication details https://www.routledgehandbooks.com/doi/10.1201/b15592-4 Manjriker Gunaratne Published online on: 26 Nov 2013

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The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The publisher shall not be liable for an loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Manjriker Gunaratne Spread Footings: Analysis and Design 3 3.4 3.3 3.2 3.1 CONTENTS 3.4.3 3.4.2 3.4.1 Factor Criteria Resistance Load and Design 3.3.3 3.3.2 3.3.1 Analysis Settlement 3.2.8 3.2.7 3.2.6 3.2.5 3.2.4 3.2.3 3.2.2 3.2.1 Evaluation Capacity of Bearing 3.1.2 3.1.1 Criteria Design 3.4.3.10 3.4.3.9 3.4.3.8 3.4.3.7 3.4.3.6 3.4.3.5 3.4.3.4 3.4.3.3 3.4.3.2 3.4.3.1 Theory Calibration by Reliability to ASDCalibration by Fitting Philosophy LRFD Soils Computation Organic in of Settlement ComputationSettlement on Plate Based Load Test Data 3.3.1.2 3.3.1.1 due Subsurface to Soils Foundation in Loading Distribution Stress CapacityPresumptive Load-Bearing 3.2.7.3 3.2.7.2 3.2.7.1 Capacity Using Bearing Capacity Footings of Eccentric Bearing Layers in Mixed Capacity Soils in Bearing Stratum Overlying aHard Soil Foundations on Soft Clay Overlying aSoft Stratum Soil Foundations on Stiff Capacity Bearing Net Ultimate 3.2.1.1 Capacity EvaluationBearing Homogeneous in Soil Criterion Settlement Capacity Criterion Bearing Determination of the Simplified Resistance Factor Simplified ofResistance the Determination Factors of Resistance Determination Load Statistics Statistics Resistance Index Reliability of Design Reliability of Probabilities Estimation Distribution Lognormal (Gaussian)Normal Distribution Variability Data of Soil Method Approximate Distribution Stress Methods Analytical Plate Load Test Data PenetrationStandard Test Data CPT Data Factors Capacity Bearing for Meyerhof’s Expressions Mathematical Hansen’s and ...... In Situ In ...... Test Data Test ...... 125 123 123 128 128 128 129 120 150 135 138 136 124 124 106 106 121 139 139 144 144 143 143 107 107 107 148 148 140 140 146 112 147 117 119 116 105 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 3.6 3.5 106 overloading would failure, which shear lead condition to either can aglobal or apunching foundation’s the deformation plasticcause shear within zone (Figure influence 3.1). This would capacity, ground the induced in bearing load ultimate the stresses shear exceeds the structural by the imposed stress capacity contact foundation. of ground the the If bearing ultimate the foundation by termed borne the be is can that contact stress maximum The 3.1.1 Criteria Design 3.1 the foundation.as to referred is generally the footing by influenced ground the and footing excessive the system encompasses causing that The settlements. and capacity ground of the bearing the without load exceeding ground to the structural the order in to safely transmit column for designed abuilding is spreadA shallow footing References Influence zone of a shallow footing. shallow of a zone Influence FIGURE 3.1

Additional Examples 3.5.4 3.5.3 3.5.2 3.5.1 Vibrations to Withstand of Footings Design Bearing Capacity Criterion Capacity Bearing ...... Foundation Vibrations due Masses to Rotating Oscillations Sliding Oscillations Rocking Vertical Steady-State Vibrations 3.5.4.3 3.5.4.2 3.5.4.1 Sliding VibrationsSliding Oscillations Rocking Translational Oscillations ...... d ...... 45 +φ/2 ...... P B ...... The Foundation Engineering Handbook Engineering Foundation The D ...... 154 156 164 164 164 173 163 165 160 162 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 q where capacity the following failure, frombearing condition must satisfied be forsafety without Therefore, following prior the footing of warning. the sinking immediate in result Footings Spread q Terzaghi’s capacity expression bearing 3.2.1 capacity provided are next. bearing ultimate for the expressions by load occasions on various (e.g., tests Plate load in test capacity verified of have that been foundations bearing also ultimate for the expressions ( footing of the (Figure area 3.1) influence the within theembedment subsoil and of the strength shear 3.1.1, Section in discussion on the Based afoundation capacity the derives from its bearing 3.2 ( settlements its estimated to limit or compressible sands loose must proportioned clays be in silts. Therefore, and footing the occur can settlement immediate significant hand, other the On distress. of building signs other and tilting, cracking, compressible saturated through in clays prior with warning occurs long Excessive the term. place settlement oring in immediately time-dependent of aresult irreversible as compressive foundationthe generally occurs deformation tak Excessive footing. the of settlement within settlement or differential aunit as settlement not does undergo either excessive footing the that total ensure must designer also The 3.1.2 Meyerhoff’s capacity expression bearing P q F A For loads vertical ult = = Evaluation of Bearing Capacity Evaluation of Bearing = eteet Criterion Settlement = Bearing Capacity Evaluation in Homogeneous Soil Evaluation in Homogeneous Capacity Bearing mination of the structural loads ( structural of the mination involved deter the appropriate factor in accounts that uncertainties for the safety (m area footing soil load) kips) level +refill or load(kN total at footing (structural the ultimate bearing capacity foundation of the (kN/m bearing ultimate D ). Over the years, many eminent geotechnical engineers have suggested engineers ). geotechnical years, Over the eminent many 2 or ft or 2 ult ) = ult cN δ = est c s cN ) within tolerable ( ) within settlements c d c c + s c P δ A + P qN ) and the ultimate bearing capacity ( bearing ultimate the ) and est ≤ qN ≤ q s q δ F q q ul d + 0.5 + tol t q + 0.5 + (3.1b) , (3.1a) B γ B N γ N γ Section 3.2.7.3 Section s γ γ (3.2) s 2 γ , kPa, or ksf) d γ (3.3) δ tol ). ). Several common q ult ) 107 - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 where q 108

b g i d s γ q N c Hansen’s capacity expression bearing Vesic’s capacity expression bearing For inclined loads For inclined For undrained conditions ϕ 25 20 15 10 5 0 ϕ Factors Capacity Bearing TABLE 3.1 45 40 35 30 ======i =

friction angle friction base tilt factors (Tables tilt base 3.3a 3.3b) and slopeground factors (Tables 3.3a 3.3b) and factors (Tablesinclination 3.2a, 3.2b, 3.2c, 3.3a, 3.2) 3.3b; and Figure also see factorsdepth (Tables 3.2a, 3.2b, 3.2c) and shape factors (Tables 3.2a, 3.2b, 3.2c) and weight soil unit of surcharge level base at footing the stress vertical effective strength cohesive bearing capacity factors (Tablebearing 3.1) 172 95.7 57.8 37.2 25.1 17.7 12.9 N 9.6 7.3 5.7 Terzaghi’s (1943) c Expression 173 81.3 41.4 22.5 12.7 N 7.4 4.4 2.7 1.6 1.0 q q q ult ult = = 298 100 42.4 19.7 N cN cN 9.7 5.0 2.5 1.2 0.5 0.0 γ qs ul c c s tu s ult c c =+ d d = and Vesic’s Expressions c c 134 51 i i c Hansen, Meyerhoff, c 75.3 46.4 30.1 20.1 14.8 11.0 .( cN

N g g 8.34 6.49 5.14 c c c 41 b b c c c d + + c i c qN qN + sd qN q q cc ′ s s 135 q q + 64.1 33.5 18.4 10.7 d d N q 6.4 3.9 2.5 1.6 1.0 d q q q i i q ′ q q i g g q − + 0.5 + q q b b ig cc ′ q q + 0.5 + + 0.5 + − The Foundation Engineering Handbook Engineering Foundation The B Hansen ′ (1970) γ 201 − N B B 79.4 34.4 15.1 N bq 6.8 2.9 1.2 0.4 0.1 0.0 γ γ γ γ c ′ N d N ) γ + γ γ i s γ s (3.4) γ γ d d (3.6) γ γ (1951, 1963) Meyerhoff i i γ γ g g 262.3 γ γ 93.6 37.6 15.7 b N b 6.8 2.9 1.1 0.4 0.1 0.0 γ γ γ (3.7) (3.5) Vesic (1973, 271.3 109.3 1975) 48.1 22.4 12.5 N 5.4 2.6 1.2 0.4 0.0 γ Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread given in Tablesgiven in 3.4a, 3.4b, 3.4c. and Finally, are situations appropriate for factors recommended construction various safety Note: Inclination factor Depth factor Shape factor Factors for Meyerhoff’s Expression Inclination and Shape, Depth, TABLE 3.2 Source: Shape Factors Factors for Hansen’s Depth Expression and Shape TABLE 3.2 s s s s

q c c γ ′ =+ == θ =+ =− istheloadinclinationtoverticaland 10 02 10 10 .. .f .s ..

b B L Hansen, J.B., A revised andextendedformulaforbearingcapacity, Design Reproduced withpermissionfrom Bowles,J.E., Danish Geotechnical Institute, Copenhagen, Bulletin No. 28, 1970. 04 N L N B or q c a B L in φ B L , McGraw-Hill,New York, 2002. φ 0 ° dd qP ss ii dK qP == cq cP sK == == i cP i γ =+ γ =0for γ =+ d s γ 10 =− q q 10 = =    10 1    10 k + d s . 1 (rad)=tan + γ 2 γ . =1 =1 2 − d φ θ . θ q 1 . ° ° =1+2tan >0 90 1 θ K    K ° K k 2 ° P B L = dk Depth Factors =tan D    B c ′ B 2 L d D == D B c =1.0+0.4 / 04 d −1 .f B 2 γ ( =1.00 (45 + for D ϕ Foundation Analysisand / or (1−sin B D ϕ φ ) for / /2). B k ≤1 0 ° D ϕ / ) B 2

ϕ ϕ >1 k ϕ ϕ ϕ ϕ >10° >10° =0° >0° =0° =0° 109 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 110 Note: Source: 2 ≤ Load InclinationFactors 3.2) Tilt Factors Slope, for Ground Hansen’s (Figure Base and Expression Inclination, TABLE 3.3 i ii i i q cq c γ ′ =− =− =− =− α 05         1

≤5 1 1 ..

Primed factorsare for gests attraction betweentwodifferent materials(e.g.,concrete andsoil).Hence, smooth then, N VA (. Hansen, J.B., A revised and extended formula for bearing capacity, Danish Geotechnical Institute, Analysis andDesign Copenhagen, BulletinNo.28,1970.Reproduced withpermissionfrom Bowles,J.E., 1 05 07 2 ≤ q VA − + − a i + C q 05 − 1 1 α . a fa θ 2 − C = (0.6–1.0) ≤5 H fa / C AC i co 450 Source: Shape Factors Factors for Vesic’s Depth and Shape Expression TABLE 3.2 H s s s fa q c γ co t i C =+ =+ =− φ t a ) 10 10     10 φ wouldbehigherthanthatofarough base. H α .. .t .. 1

i     Vesic, A.S., Analysis andDesign Reproduced withpermissionfrom Bowles,J.E., Fang, H.Y. (eds.),Van Nostrand-Reinhold, New York, 1975. 1973. α C 04 N B N L 2 , McGraw-Hill,New York, 2002. . Theactualvaluedependsonthe concrete finish.Ifconcrete foundationbaseis q c c an ϕ B L =0. B L Foundation EngineeringHandbook φ Factors forBaseonSlope( ( β ° measured clockwisefrom g C q J. SoilMech.Found.Eng.Div., ASCE

(cohesion)=attractionbetweenthesamematerial; = g g γ c horizontal) =(1−0.5tan g =− c ′ 10 , McGraw-Hill,New York, 2002. = . 147 β ° k 147 (rad)=tan ° β d ° q ° =1+2tan k β = dk ) c 5 ′ Depth Factors d D = β c =1.0+0.4 / 04 ) d −1 . B γ The Foundation Engineering Handbook Engineering Foundation The =1.00 ( for , 1sted.,Winterkorn, H.F., D ϕ for / (1−sin B ( D η ) for ϕ ismeasured counter-clockwise from / =0° B k , 99,SM1,45–763, ≤1 D Factors forTilted Base( b b ϕ / γ q ) =exp(−0.0349 =exp(−0.0471 B 2

>1 k Foundation b C horizontal) c b a =− c ′ < = 1 C 147 ; Bowles(2002)sug η 147 ° η ° ° η η ° C ° tan ° tan a (adhesion)= ϕ ϕ Foundation η ) ) ) - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread Guide for obtaining inclination factors. inclination obtaining for Guide FIGURE 3.2 i Note: Source: When When When Load InclinationFactors Tilt Factors Slope, for Ground Vesic’s Base and 3.2)Inclination, (Figure Expression TABLE 3.3 mm mm mm i i ii c ′ q cq parallel toboth γ =− =− =− =− 22 == == 1         =+

10 10 H . . H H

Primed factorsare for base issmooththen, suggests attraction betweentwodifferent materials(e.g.,concrete andsoil).Hence, AC L B N hascomponents isparallelto isparallelto 1 mH Vesic, A.S., McGraw-Hill, New York, 2002. York, 1975.Reproduced withpermissionfrom Bowles,J.E., Handbook BL fa q − VA VA − i b N 2 1 2 1 i q + + m 1 c + + + + 2 H C H LB BL fa LB fa BL C C B a / / / i / i , 1sted.,editedbyWinterkorn, H.F. andFang,H.Y., Van Nostrand-Reinhold,New and = (0.6–1.0) co co J. SoilMech.Found.Eng.Div., ASCE t t L B φ φ L         m m H + 1 i C R ϕ a C wouldbehigherthanthatofarough base. =0. . Theactualvaluedependsontheconcrete finish.Ifconcrete foundation Factors forBaseonSlope( C ( β (cohesion)=attractionbetweenthesamematerial; ° measured clockwisefrom g q = gi cq g =− horizontal) γ g =(1.0−tan θ <φ c ′ = 51 51 .t . β 1 4 4 − an i q V , 99,SM1,45–763,1973. i φ β ) 2 β ) ( Foundation AnalysisandDesign η Factors forTilted Base( measured counterclockwise bb q == from horizontal) b c =− γ Foundation Engineering C 1 bg a (. cc < ′ 10 = 51 C .t − C ; Bowles(2002) 4 ′ 2 ηφ a η (adhesion) = (adhesion)= an ta n) φ 2 η)

, 111 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Failure Condition Sliding Failure and Capacity for Bearing of Capacity Spread Footings Geotechnical Ultimate on Factors of Safety Source: Sliding resistance offootingonsoilorrock Bearing capacityoffootingonsoilorrock TABLE 3.4 Basis forSoilStrengthEstimate Soils on of Capacity Spread Footings Bearing Ultimate on Factors of Safety Source: Laboratory/field strength tests Standard penetrationtests TABLE 3.4 112 N factors wouldcapacity quite be useful. For spreadsheet applications, bearing of the expressions following mathematical the 3.2.1.1 Source: Category of Capacity Spread Footings Bearing Variable Ultimate on Factors of Safety TABLE 3.4 A C B

AASHTO, Standard Specifications for Highway Bridges, American Association for State Highway and Transportation Officials, Washington, DC,1996. AASHTO, Standard SpecificationsforHighway Bridges, American AssociationforStateHighwayand Transportation Officials, Washington, DC,1996. AASHTO, Standard SpecificationsforHighwayBridges, American AssociationforStateHighwayand Transportation Officials, Washington, DC,1996. Mathematical for Expressions Meyerhof’s Hansen’s and Bearing Capacity Factors a b c Railway bridges Retaining Walls Light industrial office buildings Apartment and Blast furnaces Warehouses and public Structures Hydraulic buildings Highway Typical bridges Silos Maximum design Maximum Maximum design occur load unlikelyto failure serious consequences of occasionally; may occur design load disastrous failure consequences of occur often; load likelyto Characteristics Category q = e π tan ϕ tan Exploration Complete Permanent Structures Soil 2 2.0 2.5 3.0 (45 + Required MinimumFactorofSafety(FS) Suggested MinimumFactorofSafety(FS) Required MinimumFactorofSafety(FS) The Foundation Engineering Handbook Engineering Foundation The ϕ /2) Limited Soil Exploration (3.8a) 3.0 3.5 4.0 Exploration 2.5 3.0 1.5 3.0 Complete Temporary Structures Soil 2.0 2.0 2.3 Exploration Limited Soil 2.3 2.6 3.0 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 N Footings Spread Illustration for Example 3.1. Example for Illustration FIGURE 3.3 For Hanson’s expression For Meyerhoff’s expression realistic from a construction point of view. point aconstruction from realistic is depth the that to verify criterion this reevaluate can one obtained, are parameters However, (diameter) footing. of the dimension design the once footing minimum to the approximately equal a depth assume could one Therefore, depth. footing the to select acriterion as used be can obviously none of case, these current the In soil. foundation the underlying to be table groundwater or layer a weak the known not reach does that footing. acircular assume can one design this Hence, in footing. or acircular asquare for not one to use no reason generally is there line, erty prop to the proximity close the as such spacing in limitations are there shape, unless footing of the case the In depth. and shape must asuitable one First, footing on decide Solution weight ( unit properties: following the clay with silty homogenous considered be can that asubsoil kN, in of 400 load acolumn to carry 3.3a, asuitable footing design Figure in shown column For the 3.1 Example As for the foundation depth, typically one would seek some significant embedment embedment significant some would one seek typically depth, foundation for the As Case (2)Case (1) Case a . Assume that the groundwater table is not in the vicinity. the not in table groundwater is the that . Assume . Assume that the groundwater table is 0.5 mabove table footing. groundwater is the the that . Assume d Silty cl γ ) =17 kN/m ay γ N =1.5 ( γ N =1.5 ( c =( 3 , internal friction ( friction , internal P =400kN N N q N −1)tan (1.4 q B −1)cot q −1)tan ϕ (3.8b) ϕ

ϕ) ϕ ) =15°, ( cohesion (3.9a) D C ) =20 kPa - (3.9b) 113 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 114 limits. practical within is obtained depth footing the cases, both of view. point in Furthermore, aconstruction from cant insignifi is disparity the that realizes one different, are solutions two the Although B 509.3/ B 509.3/ of factor 2.5. asafety with capacity bearing 3.3a). (Figure soil recompacted the and column level foundation =4×400/( at the stress Contact level base ( footing at the stress effective vertical The The tables indicate the following bearing capacity parameters: capacity bearing following the indicate tables The Using Hansen’s expression (Equation 3.5), (Equation Hansen’sUsing expression (2).Case water 0.5 mabove table the footing. is that the Assume expression, Hansen’s From Terzaghi’s expression, From ultimate the and stress contact applied the be to compare can criterion following The capacity: bearing ultimate for the written be can expressions following the Then, qB • • • • ul t =+ = Terzaghi’s (Table factors 3.1) Hansen method (Equation 3.5) (Equation method Hansen 3.2) (Equation method Tezarghi Hansen’s (Table factors 3.1) 20 3 qB 886 ul (. t 10 .. =+ =+ 27 20 417 97 + (. N )( N s 110 10 .. c s d c 1 =1.359 4 c c =10.97 c =12.9 =1.3 .) =1.4 359 97 qB 69 114 ul )( t B 1 (. =+ =+ 14 .) 22 359 335 20 )[ B (. 12 (. .. 48 14 4 ×400/( 17 91 B )( )( B 2 75 2 − + 17 + + 17 + .) 31 17 (. 5 98 B π d B () B N )( )( s N B s q q =(417.4 +114.22

γ =1.55 m =1.75 m = 1.294 =(335.4 +87.55

05 39 2 q 74 q = 1.26 =0.6 =4.4 ) +17 =3.9 . .) )]( (. (. 12 39 40 .) B ). 61 = )( + )( 12 π q .) .9 B ult 2 51 The Foundation Engineering Handbook Engineering Foundation The 61 2 94 () q /(2.5) ) +17 ) =(17)(depth) =17 (. 72 ). B + B 294 () )/2.5 )/2.5 B 05 B (. ). = stresses imposed by the by the imposed =stresses () +− 17 50 05 )( N N d s γ () γ γ γ =0.6 =1.0 B =1.2 =2.5 (. .. 17 6 (. ) 12 98 )( 06 B .) )( . B (. )( 10 12 .) ) (. 06 )( 10 - .) Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread Illustration for Example 3.2 Example for Illustration FIGURE 3.3 B 509.3/ A V H 3.3a) (Table 3.3b. Figure in shown as base footing the along acts also kN load 50 of a horizontal when conditions, soil same the for aflooring Design 3.2 Example table. due groundwater to the strength

It is noted that a slightly larger area is needed to counteract the loss of foundation of foundation loss the to counteract needed is area larger aslightly It that noted is Case (1) Case C • a b == == Hansen’s (Table factors 3.1) 23 Adhe / . Groundwater table is not in the vicinity. the . Groundwater not in table is () 20 si on N 1 s be c d 333 c =1.359 =10.97 c .k =1.4 tw 3 een d Pa Silty cl so il ay an H =50kN dc B 2 + 17 + on cret B P =400kN d = 400 kN =400 = (386.27+110.69 = f s N =1.62 m i =( q ϕ e/ = 50 kN =50 q =1.294 =1.26 q = 15° = =3.9 B π 23 B 2 )/4 co he s iio n( B )/2.5 se et he D N d s γ γ f γ =0.6 =1.0 oot =1.2 no te in Ta bl e 3.3a) 115 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 116 3.1) to design the footing, then one has to strictly use the corresponding bearing capacity bearing 3.1) corresponding the use to strictly one has then footing, the to design capacity (Equation criterion (column) bearing the in load structural to used be the is If 3.2.2 B problem like in Example 3.1 with the above determined Example 3.1 above in the determined problem with like

509.3/ of factor 2.5. asafety with capacity bearing ( soil recompacted the and column level foundation =4×400/( at the stress Contact i tan Net Ultimate Capacity Bearing (Case 2) load. inclined an to support needed is footing alarger that It seen is From Hansen’s expression, Hansen’s From ultimate the and stress contact applied the be to compare can criterion following The qB • Since ul t i γ =+ =+ Hansen method (Equation 3.5) (Equation method Hansen solution. explicit that (assume sheet a having i =− q 354 20 =−    . When the groundwater table is in 0.5 m below the footing, one can solve the can one footing, 0.5 mbelow the in table groundwater is the . When 1 (.     10 .. B 1 8 is approximately 1.0 is (only the in (. 97 07 VA VA 101 B B )( is to be determined, we do not want to complicate the equations by we do not equations to want complicate the determined, to be is + + −° 2 1 term in the the in term .) θ 65 05 359 . f f / C B B C H 450 a values) and perform the design. In this case, one can assume assume can one case, this In design. the values) perform and a (. co i 14 co ° t t ) )( φ H φ 0.85     i 4 ×400/( α    i 1 c α factors. Therefore, it is convenient to set up aspread it convenient to set Therefore, factors. is (use 2 as the exponent without other information) without other exponent (use the 2as =0.85 B 2 )( 2 (use 2 as the exponent without other information) without other exponent (use the 2as + 17 + θ Figure 3.3b Figure 17 =50/400, then ii cq π B )( =− B =(354.8 +101.65 i =1.75 m 39 q 2 =0.89, ) +17 .) (. 12 N 1 ). q B − 661 − = i )( q i 1 factors expressions) to obtain an an to obtain expressions) factors π i q θ γ .) B ult =0.85 294 = 7.1°. = The Foundation Engineering Handbook Engineering Foundation The 2 /(2.5) ) +17 () B 0.89 i factors. )/2.5 B = stresses imposed by the by the imposed =stresses + 05 .( 17 )( B )( 12 .) (. 06 )( 10 .) - ( 0.85 ) Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 only, removed soil counteracts the concrete that to lay footing. it the assumed if is where as it expressed and capacity is ground of the bearing net ultimate the as known is overburden. soil of the excludesthat This effects the Footings Spread Illustration for Example 3.3. Example for Illustration FIGURE 3.4 computed be can following equations: byuted the stress soil/soft clay relatively must the meet capacity layer. soft of low the bearing distrib The 3.4) (Figure induced footing by the stress the that at stiff interface of is criterion the the layers, surface to the next most critical the capacity respect then with criterion bearing the (i.e.,stratum clay, founded satisfies stiff is sand) or where footing dense dense the medium stiff the that layer. one softer underlying assumes case, if the this than In stiffer and thin layer surface the relatively if is capacity failure of bearing type apunching expect can One 3.2.3 Then, EquationThen, 3.1as modified be can load would structural ground the be on the net load the hand, increase other the On For rectangular spread footings For rectangular Foundations on Stiff Soil Overlying a Soft Clay Soil Overlying aSoft Stratum Stiff on Foundations q denotes the total overburden total denotes the stress. Sandy Cla ∆ pq ye PA = y st q ruct n,ult    ur () 1.2 m Bd = al / ++ q ult cc BL ≤ − () Ld q n, q F , ul (3.10) t

   (3.12a) 2.0 m 1.0 m (3.11) 117 - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 118 For square or circular spread footings For or circular square For strip footings For strip clay layerthe with Alternatively, 3.6 Equation from s q N since conditions, Under undrained clay layer, of the capacity mate bearing =20 cohesion clay layer kPa). the (undrained in rion crite capacity clay layer. layer bearing soft overlying sand the the the Check in founded to be designed well 3.4 been Figure has in shown footing square the that Assume 3.3 Example dk c ′ Hence, Also, From Table 3.2a, the to estimate 3.5) used (Equation is equation capacity Hansen’sIf bearing == 04 .. qs n, i c ′ ul = tu 04 =+ 0 51 ,   

.( g 12 30 41 q c ′ . . n = ,

ult    respect to the distributed load from the footing). the load from distributed to the respect 0 (since (since = and and cN sd qs cc ′ n, + c ul s b d tu c c =5.14 d / c ′ =+ ′ c n,ult b = − i 51 c =3.0/1.2 of capacity bearing the that considers when one g q 0 ig .( cc ′ c n, = c b =1.195 footing) (square 41 − ul c + t cN ∆ d ∆ = = O ′ c N pq q pq c () − 245 =+ ( u s q 20 = N = 0, = =1.0 = c bq d 10 c sd ′ q c .. s . ). − 1) − cc ′ (. i 69 c ′ c 51 +−    + g    = () c () b Bd 04 kP Bd 41 02 ′ c s (Equation 3.5) )( . q + − N B + B d q a    q ig γ .) = cc i ′ =0(Table 3.1) 12 30 195 q c c − g . . 51 q    b       2 The Foundation Engineering Handbook Engineering Foundation The q ′ (. 42 + 0.5 + (3.12c) 20 (3.12b) − (0 bq 01 ) c ′ )( ) +− B ++ γ N 02 γ .) q s γ d γ 1 i γ g = γ b 226 γ (3.5) . 16 kP a net ulti - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread Soft surface layer overlying a harder layer. aharder layer overlying surface Soft FIGURE 3.5 where 1995): Boorman, sions and (Tomlinson following expres the from capacity layer surface of obtained be the can bearing ultimate net the cases, such In layer in. soft the to squeeze tends away footing sinking the while which one into in transforms capacity failure of bearing 3.5),strata (Figure mechanism the clayey layers surface on thin overlying relatively constructed are foundations hard When 3.2.4 S d B

u = = = Strip footings Circular/square footings

Foundations on Soft Soil Overlying Stratum aHard Soft on Foundations The The undrained strength of the surface layer surface of the strength undrained thickness of the surface layer surface of the thickness dimension footing net stress applied on the soft clay can be estimated as estimated be clay can soft applied the on stress Factor of safety (FOS) =226.16/48.8Factor of safety =4.63 (satisfactory) q q Hard stratu So n, n, ul ul ft tu tu la ∆ =+ =+ pq yer       = = = 2 3 [( 48 B B 500 d d m    . () 88k Bd // B π π + B 12 Pa P + + .) c 12 16 22          ][( 2 S S 12 .. fo fo 32 r r )] d B B d d ≥ ≥

(3.14) (3.13) (3.12b) 119 - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 When the subsurface constitutes an alternating (sandwiched) mixture of two distinct soil soil of distinct two (sandwiched) alternating mixture an constitutes subsurface the When 3.2.5 3.2.1. Section the in discussed methods on the based formed claygeneous layer. per be can capacity estimation For homogeneous bearing the cases, respectively, footings, strip be foundation the can then 120 Bearing capacity of soils mixed in layers. in mixed of soils capacity Bearing FIGURE 3.6 Q would capacity estimate be bearing corresponding the and only sand asilty with deals of 3.6, presence clay silt the (CL/ML). and Figure in ignoring thus peaks by the one Then, capacity. upper alower bearing and capacity bound an estimates, actual for bound the bearing two obtain and at atime type one soil to assume is way only the to address this to average incorrect conceptually the is layers, CL SM (or and because tinct and properties, it ML) have engineering very different 3.2.7.1). zoneence (Section not physically are separated dis two into types soil the Because capacity, bearing the orderIn to estimate the (CPT) test etration 2.5): (Section results capacity. example, an As 3.6 following layers Figure the identifiedthe has as conepen by bearing judgment the to estimate 3.6, Figure engineering shown in use as one can types ult)2 ult)2 It must be noted that if the criteria It criteria the must if noted be that Step 1. Assume SM type only with a continuous linear acontinuous linear with only Step 1. SM type Assume 2. 1. (the upper bound). layers). friction, (if shaft the CL or (clay ML silt). and one would As the expect, cone resistance the expected, of portion silt.As SM a significant (silty with sand), contaminated sand is which Bearing Capacity in Soils Mixed in Soils in Layers Capacity Bearing De pt h f s , profile was provided, it wouldrelativelybe these high for q c profile peaks out profile peaks forsand. B d q ≥ c values across the entire influence zone. influence Hence, entire values the across 2 q and and c values have influ to averaged be the within C L/ML profile B d The Foundation Engineering Handbook Engineering Foundation The ≥ 6 q c

are not satisfied for circular and not satisfied circular are for treated as one placed ahomo as treated in q C SM c profiledrops clay for or silt q L/ML c profile depth)(with defined SM profile - - - - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Q would capacitying estimate be clay/silt with one conditions. deals Then, undrained ing bear corresponding the and only troughs the defined (indentations),by thesand of assum andpresence (SM) thus ignoring Footings Spread Bottom pressure distribution on rigid eccentric footings. eccentric rigid on distribution pressure Bottom FIGURE 3.7 and interface)soil of view load in factoring: criteria (foreccentricity the no tension footing/ at maximum the madein are modifications satisfied. be must conditions following the order footing, to prevent the to uplift tends forces that at bottom tensile the 3.7. Figure shown in that, as in stresses, bending realizes and also One axial combined from determined be can footing eccentric of bottom an on the distribution pressure The 3.2.6 Then, the effective bearing capacity could be estimated from the following inequality: the from capacity could estimated be bearing effective the Then, CL/ML acontinuous linear with Step 2.Assume only type For the load and resistance factor (LRFD) 3.4),For design (Section load method resistance the and following the as abovefollows: The for modified rock are conditions Bearing Capacity of Eccentric Capacity Bearing Footings ( P/BL)[1−6e /B ] Q ult)2 ult)2 (the lower bound). ult)2 e e xy xy ≤≤ ≤≤ < e e B B 66 44 y x Q e ≤ ≤ ult B e e B L 4 4 < (3.17b) Q P L L ult)1 (3.16a) (3.15) (P/BL)[1 +6eB q c profile depth)(with ] (3.16b) (3.17a) 121 - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Because the contact pressure is nonuniform at the bottom of the footing (Figure 3.7), (Figure footing of at bottom the the nonuniform is contact pressure the Because 122 (a) Rectangular footings with eccentricity. (b) Circular footings with eccentricity. with footings (b) Circular eccentricity. with footings (a) Rectangular FIGURE 3.8 expression: following of dimensions footing angular equivalent rect an into transformed is this shape, typically regular not of ageometrically is area footing effective the Because area. footing effective the represents shaded area the centered at arc Then, F(EF footing. of to =CE) that aradius the with equal acircular struct 3.8b) (point Figure Ein eccentricity con to the then and corresponding locate diameter the P/A distinct two one would cases, such in obtain H and shapecases, factors (Tables 3.2a, 3.2b, 3.2c) and computed by are interchanging twice to components, two load has At that ahorizontal is, that times, L B both in Forfooting. eccentricities loaded order capacity rectangular eccentrically in to compute of an to used be bearing the Meyerhoff (1963) (1970) Hansen and dimensions footing following effective suggested the B (a) L The above conditions are modified for rock as abovefollows: The for modified rock are conditions In the case of circular footings having load eccentricity load having eccentricity footings of circular case the In and then with with then and L for the footing design. footing for the , can act on the column producing two eccentricities producing eccentricities two column act on the , can L ′ . Also, i factors (Tables 3.3a 3.3b) and computed by are replacing twice H B . Finally, the L Y O e y L' e x B B ′ and and X ′ term in the the in term and and X L ′ . The effective dimensions can be found from the the found from be can dimensions effective . The Y ′ ′ = e e = directions (Figure 3.8), (Figure directions y x B' L ≤ ≤ B −2 q −2 3 3 ult q 8 8 B L ult values. The lesser of them is compared to is values. of them lesser The e (3.17d) (3.17c) e (b) expression also gets replaced gets by also expression y x (3.18b) (3.18a) The Foundation Engineering Handbook Engineering Foundation The e R x and and e H and radius and B parallel to parallel CE e y ee on the footing. In such such In footing. on the R F R B , one must first and and H Effective area i once with once with H L L parallel parallel ′ . Thus, B - - ′

Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread zone that extends extends zone that index.where plasticity PI the is for 27 30forand and overconsolidated clays. Bowles (2002) following expression suggests the where following (Equation expression the 2.6d): using soils grained of fine- saturated strength undrained the to obtain used be Cone data penetration can Data CPT 3.2.7.1 3.2.7 moments wave due and to wind loading. by introduced variable are when eccentricities to preventnot eccentricities used be footing opposite moment However, to column. counteract moment the the can in technique this 3.9. and Figure equal shown in an as creates column the force in how It axial the seen is of by column avoided adistance be the can footings induced by on the offsetting M factors depth ( the However, it unmodified must the noted be that to avoid eccentricity. footings Designing FIGURE 3.9 To average an determine moment, unbalanced an carries for that to designed be acolumn are footings When , and an axial force, axial an , and N k Bearing Capacity Using Capacity Bearing N k is the cone factor that ranges between 15 cone 20 factor consolidated between the and ranges clay for that is normally d 2B ) in the bearing capacity equations. bearing the ) in below the footing and 1/2 and below footing the P , that are fixed in magnitude, the resulting eccentricity ( eccentricity theresulting magnitude,in fixed are , that M q c for a footing design, one would for influence afooting consider afooting In SituIn BL ′′ = N P 2 Test Data Test xe S k RR u =+    = 13 qP co co N − s 55 50 D − k . 1 B above the footing. above the    PI and and , R e (3.20)    , (3.21) − eccentricity e =Load x =Offset L e must be used when determining when determining must used be    (3.19) e = M/P x = 123 – e - ) , Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 where soils. on cohesionless footings (1977)Parry provided capacity allowablefollowing for expression of the spread bearing 3.2.7.2 124 Illustration of influence zones. of influence Illustration FIGURE 3.10 q following expressions: the location placedbe using same on the to capacity of afooting capacity bearing plate, of the expected bearing the predict one can the ultimate curve. Knowing the load–deflection of flattening eventual the from estimated 3.10, Figure capacity plate of the be bearing can ultimate scrutinizes the that it seen is 3.10Figure plot shows of atypical plate-load deposit. one on asand results test When 3.2.7.3 involving 3.22 Equations 3.23. and in zone suggested influence footing The 3.2. Section in presented methods equivalent of friction angle an 4.4 design. mat under Section footing vided in s is the settlement (inis mm). expression pro settlement and the this more is Amodified form of versatile For For Typically, when SPT provided, following correlation data to estimate are the use one can Clayey soils Clayey D N f Plate Load Test Data Standard Penetration Test Data

55 < B is the corrected SPT blow count corresponding to a 55% hammer efficiency and SPT blow efficiency corrected to a55% the is count corresponding hammer , Soft clay Sand ϕ 2B influenceofplate for the soil and determine the bearing capacity using the the capacity using bearing the determine and soil for the qN φ= n, al l 25 = Section 3.2.7.1 Section u(f) + 30 28 = 2B ofactualfooting    55 q u(p) N    q 25 55 (3.24) s    . 4 12 The Foundation Engineering Handbook Engineering Foundation The /    can also be used for computations used be also can (3.23) (3.22) - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Clay, stiff Clay, mediumstiff Clay, ordinary Clay, soft Clay, verysoft Soil Description (kPa) Codes Building Indicated from Capacities Bearing Presumptive TABLE 3.5 b a Note: Source: Bedrock Sedimentary layered rock (hard shale, Soft rock Hardpan, cementedsand, Sand–gravel, compact Gravel, looseandcompactcoarsesand Sand, looseandcoarse,orsand–gravel Sand, looseandfine Inorganic silt,compact Sand, compactandsilty Sand, compactandclean Clay, hard Footings Spread on the classification of the predominant soil type that site.at type soil the predominant of classification Tableson the 3.4a, 3.4b, 3.4c, 3.5, 3.6a, sites based building capacities for codes of suggest acertain some cities bearing building The 3.2.8 cases. both groundwater of the in table effects would the and soil similar be 3.10)of (Figure type zones ence same the to confined are plate the footing of the both and sandstone, siltstone) gravel mixture, orcompactandfine Bowles (2002)interpretation. Building Officials andCode AdministratorsInternational, Inc. It must be noted that the above expressions can be applied if it is known that the influ It the that applied above be must it the noted be that known can if is expressions Sandy soils Sandy where

Values convertedfrom poundspersquare foottokilopascalsandrounded. Soildescriptionsvarywidely between codes. The following represents author’s interpretations. between codes.Thefollowingrepresents author’s footing. of to that the equal area an having of acircle diameter the as determined be can Presumptive Load-Bearing Capacity Load-Bearing Presumptive Bowles, J.E., B p is the plate and the is diameter Foundation AnalysisandDesign Chicago, qq u 9600 1995 600 300 125 100 240 300 210 175 125 (f) 75 25 B = , McGraw-Hill,New York, 2002.With permission. f is the equivalent foundation the is diameter, which u( p)    B B Natl. BoardofFire p f Underwriters,    (3.25) 140–400 9600 1976 950 100 100 — — } } } } } BOCA, 1993 6000 6000 340 240 240 240 140 140 140 100 a Building Code, Uniform 1991 9600 1400 300 300 300 210 200 100 100 b 125 - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Source: Soft normallyconsolidatedalluvialclays(e.g., Firm normallyconsolidatedclays(atdepth), Stiff-fissured clays(e.g.,stiff “blue”andbrown Very stiff boulderclay, verystiff “blue”London Hard boulderclays,hard-fissured clays(e.g., Description of 1mbelow Level) Ground Depth (at aMinimum Strength Shear Undrained on Based Clayey in Soils for Capacities Foundations Bearing Presumptive TABLE 3.6 126 investigation followed design. by detailed subsequent for needed site the is foundation idea size potential of the where apreliminary situations capacity advocated in factors are sites. bearing Hence, of primarily these use the the location with water of the depth, load associated table, inclination, settlements the and However, types. soil it values do mustthe these noted be notfoundation that shape, reflect 3.6b, 3.7 and present acomprehensive capacities of for presumptive various list bearing marine, riverandestuarineclays) “brown” LondonClay fluvio-glacial andlakeclays,upperweathered London Clay),stiff weathered boulderclay Clay deeper LondonandGaultClays) Minimum Depth of 0.75 Depth mbelow Level) Ground Minimum SPT Data on (at Based a Soils Granular in for Capacities Foundations Bearing Presumptive TABLE 3.6 Note: Source: Loose sandsandgravels Medium-dense sandsandgravels Dense sandsandgravels Very densesandsandgravels Description ofSoil

Tomlinson, M.J.,Boorman, R., Brunthill, Harlow, England,1995.With permission.

The watertableisassumednottobeabovethebaseoffoundation.Presumed bearingvaluesfor pad foundationsupto3mwideare approximately twicetheabovevalues. b Tomlinson, M.J.,Boorman,R., Technical, Brunthill, Harlow, England,1995.With permission. a Foundation DesignandConstruction N valueinSPT Foundation DesignandConstruction 10–30 30–50 5–10 >50 Undrained Shear Strength (kN/m 150–300 75–150 20–40 40–75 >300 150–500 500–800 50–150 The Foundation Engineering Handbook Engineering Foundation The Presumed BearingValue (kN/m 1 m 800 2 ) Foundation ofWidth , LongmanScientificand Technical, 100–200 200–400 400–800 Presumed BearingValue (kN/m 50–100 1 m 800 100–400 400–600 50–100 , LongmanScientificand for FoundationofWidth 2 m 600 150–250 300–500 75–100 25–50 2 m 600 100–300 300–500 30–100 2 4 m ) for 500 Negligible 150–250 75–125 50–75 4 m 400 2 ) Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread Very marlylimestones: Igneous, oolitic,andmarly Pure limestonesand Rock Group mm) 50 Not Exceeding (Settlement Surface Rock on for Capacities Foundations Bearing Presumptive TABLE 3.7 sandstones; cemented poorly cemented with flatcleavage/foliation) (including slatesandschists metamorphic rocks carbonate mudstones; sandstones; indurated limestones; well-cemented sandstones oflowporosity dolomites, carbonate cleavage/foliation and schistswithsteep mudstones andshales;slates Moderately strong Moderately strong Moderately strong Moderately weak Moderately weak Moderately weak Strength Grade Very weak Very weak Very weak Strong Strong Strong Weak Weak Weak Discontinuity Spacing 600 to>1000 600 to>1000 600 to>1000 600 to>1000 600 to>1000 600 to>1000 600 to>1000 600 to>1000 200 to>1000 60 to>1000 200–600 200–600 200–600 200–600 200–600 200–600 200–600 200–600 200–600 60–200 60–200 60–200 60–200 60–200 60–200 60–200 60–200 60–200 (mm) >600 >600 <200 >200 All All Bearing Value (kN/m Presumed Allowable 10,000 to>12,500 10,000 to>12,500 8000 to>100,000 2000 to>3000 4000 to>6000 3000 to>5000 2000 to>4000 750 to>1000 7500–10,000 5000–10,000 5000–10,000 1000–3000 3000–5000 3000–7500 2500–5000 1500–3000 1500–4000 4000–8000 See note See note See note See note See note 750–1000 750–2000 750–2000 500–1500 >10,000 >12,500 250–750 250–500 500–750 250–750 >1000 >5000 ( continued a a b a b b b b b a b b a a b a 127 2 ) ) Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 128 uted load of depth where by follows: Boussinesq’s as estimated theory elastic approximately be extensive can mass, load founded soil on an on arelatively footing small by subsurface aconcentrated the load, induced vertical in the as stress such vertical The Methods Analytical 3.3.1.1 3.3.1 these techniques. provided be applicationreviewed. of the to anumber examples illustrate Then, will of be will due to footings increase to evaluate stress used ground commonly niques the of anumber tech section, 1.5. this Section in in Therefore detail in have discussed been foundations forMethodologies computation building used under settlement of ground Analysis Settlement 3.3 b a Note: Source: Rock Group 50 mm) Not Exceeding (Settlement Surface Rock on for Capacities Foundations Bearing Presumptive TABLE 3.7(C Uncemented mudstonesand shales mented asnecessarybyfieldorlaboratoryteststodeterminetheirstrength andcompressibility. Bearing pressures fortheseweakorcloselyjointedrocks shouldbeassessedaftervisualinspection,supple the jointsopenbearingpressure mustnotexceedhalftheunconfinedcompression strength ofthe rock. Bearing pressures mustnotexceedtheunconfinedcompression strength ofthe rock ifthejointsare tight.Where Equation 3.26 can be used to derive the magnitude of vertical stress imposed at any imposed stress toEquation of derive 3.26 vertical used magnitude be the can

Presumed bearingvaluesforsquare foundationsupto3mwideare approximately twicetheabovevalues, or equaltotheabovevaluesifsettlementsare tobelimited25mm. Stress Distribution in Subsurface Soils due to Foundation Loading to Foundation due Soils in Subsurface Distribution Stress z r Tomlinson, M.J.,Boorman,R., Brunthill, Harlow, England,1995.With permission. vertically below the center of a circular foundation (of below center vertically of the a circular radius and and ontinued q z are indicated in Figure 3.11. Figure in indicated are as (Figure 3.12) (Figure as ) Moderately strong Moderately weak Strength Grade Very weak Strong Weak Foundation DesignandConstruction ∆ σ z = 3 2 Pz π () rz Discontinuity Spacing 22 + 3 200–600 200–600 200–600 200–600 60–200 60–200 60–200 60–200 52 The Foundation Engineering Handbook Engineering Foundation The (mm) / All (3.26) , LongmanScientificand Technical, R Bearing Value (kN/m Presumed Allowable ) carrying adistrib ) carrying 1300–1000 1000–2000 1250–2500 See note See note 400–1000 250–5000 150–250 125–400 b b 2 ) - - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread ( footing loaded rectangular by auniformly 3.13. Figure shown in 3.13, on Figure Based caused increment stress the that seen be it can 2:1 the is distribution distribution used A commonly approximateusing distributions. due increments toAt footings it more times, is convenientstress subsurface the to estimate 3.3.1.2 (1966). by presented Harr expressions (e.g., loaded footings of uniformly on analytical based strip) estimated be can rectangular, ( horizontal the in increments Stress footing. rectangular due adistributed to increase (b) Stress footing. circular due adistributed to (a) increase Stress FIGURE 3.12 load. due aconcentrated to increase Stress FIGURE 3.11 10-year settlement of the 1.5 × 1.5 m footing carrying a 200-kN load as shown in Figure Figure in shown load as a 200-kN carrying 1.5 of 1.5 the × 10-year mfooting settlement the and settlement consolidation to compute ultimate the it necessary is that Assume 3.4 Example (a) Approximate Stress Distribution Method R ∆σ q z ∆ z σ ∆ z σ =− x z and and q =    1 q    y P () [( ) and vertical ( vertical ) and Bz 1 B ++ + × BL Rz L () r ) at of adepth / 1 Lz )] 23 / ∆σ    2 (3.28)    z (b) z ) directions due shapes to other ) directions (3.27) z z is a b 129 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 130 Illustration for Example 3.4. Example for Illustration FIGURE 3.14 increment. stress vertical of subsurface estimation Approximate FIGURE 3.13 1.0 to be 10 × determined 3.15, by Figure ( represented of consolidation coefficient are the and clay layer) of the area mid-plane (from sample the of arepresentative characteristics tion consolida laboratory 3.14. Table the provided in that are assume 3.8. properties Also Soil From Figure 3.15, Figure From Overburden pressure at the footing depth =16.5 depth ×1.0 footing at =16.5 the pressure kPaOverburden z Contact pressure ( pressure Contact Preconsolidation pressure ( pressure Preconsolidation –8 m 2 /s based on the methodology presented in Section 1.5. Section in /s presented methodology the on based q ) =200/(1.5) B P ∆σ z –1.0 m –2.0 m –6.0 m –3.5 m p The Foundation Engineering Handbook Engineering Foundation The c 2 ) = 60 kPa) =60 = 88.89 kPa 2 q =P/A 1 clay Soft soil Sandy C 3.75 v ) of the clay was clay was ) of the m - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread Laboratory consolidation curve. consolidation Laboratory FIGURE 3.15 where where as: increase stress 3.28), the (Equation 2:1 attenuation estimate stress can one 3.13). (Figure attenuation appropriate stress an by commonplace using the Using mined deter be can footing the level clay layer mid-plane at of the the within increase stress Δ Thus, footing. the within minimum would the be settlement corner the whereas maximum, the be would settlement center the were aflexible to behavefooting, 0.001. as footing the If and footing where Δ that see example, can one OQ. this as In indicated distance needed) =the is,that figure, the of bottom at the indicated 3.16 scale Figure to the from footing bythe mapping obtained 3.16) (reproduced Figure be of clay. in depth can mid-plane chart the This by considering Newmark’s the influence using obtained be clay layer the can in average increase The stress On the other hand, if the footing were to behave as a rigid footing, then the average the then footing, were arigid to behave as footing the if hand, other the On at adepth increase stress The N B and and and and I is the influence factor of the diagram. For the chart shown in Figure 3.16,in Figure shown chart the For diagram. the factor of influence the is 1.045 1.085 L Soft clay Wet sand Dry sand Layer 3.3 Example for Properties Soil TABLE 3.8 I 0.93 are the number of elements of the Newmark’s chart covered by the scaled covered scaled by the Newmark’s of the of elements chart number the are are the footing dimensions. footing the are 0.9 1.0 1.1 e d 10 c (the depth from the footing to the location where the stress increase is is increase stress where the location to the footing the (the from depth Δ p center p conner Unit Weight (kN/m =(4 ×19) ×88.89 ×0.001 =6.75 kPa ∆ =(58) ×88.89 ×0.001 =5.2 kPa d pq c can be found using Equation 1.19 Equation using found be can = 18.0 17.5 16.5 C    r () Bd p = ++ NqI cc BL p 3 c () ) =60kPa Ld , (1.19) SPT( 10 14 15    N ′ 100 ) 120 E C (MPa) 2.56 10.7 11.5 c p (kPa) d c

= 3.75 m. I = - 131 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 132 Engineering, 3.3. Holtz, (From Example R.D., in Kovacz, W.D., of Newmark’sUse chart FIGURE 3.16 would one have obtained, footing, of the corner and center for the were one to have if It estimates averaged that noted must be above the stress Δ Thus, Prentice Hall, Englewood Cliffs, NJ, 1981.) Cliffs, Englewood Hall, Prentice p average ∆ p av =88.89[1.5/(1.5 +3.75)] erag e == 2 1 ( 6.75 + 5.2) 5.975 The Foundation Engineering Handbook Engineering Foundation The 2 =7.256 kPa kP a An Introduction to Geotechnical to Geotechnical Introduction An AB Depth ofpoint I =0.001 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 The following expression can be used to estimate the ultimate consolidation settlement consolidation settlement ultimate the to estimate used be following can expression The Footing (Flexible) of Center the beneath Settlement Ultimate Footings Spread age of consolidation at degree time Using Terzaghi’s (1-D) of one dimensional theory consolidation (Terzaghi, 1943), aver the s t average of consolidation, degree ( at time any intermediate footing of the settlement The Settlement 10-Year of Estimation tlement since average the to estimate consolidation set used be ultimate followingcan expression The (Rigid) Footing of Settlement Ultimate Average since consolidation settlement ultimate the to estimate used be following can expression The Footing (Flexible) of Corner the beneath Settlement Ultimate since in combination with any estimates. one above of with combination the settlement in ultimate layer is found from Equation 1.4b Equation from layer found as is 3.13. Figure clay offrom the mid-plane at the average overburden The pressure effective obtained 3.15 estimate approximate Figure to the in opposed as estimates ages of the Since Since σ σ Because the estimates are significantly different, the author suggests using the aver using the suggests author different, significantly are estimates the Because ′ ′ vo vo +∆ +∆ σ s ′ vo av pp pp s ce co ce er σ < nt ag nt rn ′ vo er p er er e c +∆ = = , one can assume that the overall clay layer is in an overconsolidated state. overconsolidated an clay layer overall in the is that assume can , one > <       σ pp s 11 11 c c ′ vo ce av + + (Figures 1.20c (Figures 3.15) and (Figures 1.20b (Figures 3.15): and s nt 25 25 er s =+ av . er . ce ag . . er 16 nt 06 06 = ec ag er .( > 52 er          = 11 = s       ce 0 0 +    ).    25 nt .l .l (Figures 1.20c (Figures 3.15) and 1 064 064 1 . U er 17 + H . + H 06 ave = e e 51 0 t 0 , of the clay layer corresponding to the particular time time clay, of particular layer the to the corresponding og og       (. ,       1    U    0 51    + C H 54 54 ave C ). .l 60 60 064 t r +− e = 0 lo . . can be determined from Table from determined be can 1.8 the knowing lo 8 8    80 gl U + + gl    og C ave σ (. 0 0 σ p 12 p r ′ vo .l .l c ′ vo 382 382 s lo 54 c ult 59 + g + ). .. (3.29) 54 85 C σ C og og + c . ′ vo c 8 σ 82 og og 554 5 (. + ′ vo 2 448 ∆ t    σ .. .. 75 ) can be estimated by using the the by using estimated be ) can 85 σ p = ′ vo 60 ′ + + ). vo 60    p 30 = p + 67 c + .m c 54 ∆ 975 ∆ 6 p 5 p 88k             = m Pa = 81 . 56 . 9 4 mm mm - 133 - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 3.14, example, that, for see this one can where expression: Time Factor ( 134 Immediate settlement computation. settlement Immediate FIGURE 3.17 Then, From Table 1.8, Figure 3.17.Figure Table in given layers are average 3.9. The three values for CPT the in 3.14, factor shown is plot Figure influence in shown foundation strain the for which of the settlement differential total to compute ultimate the it necessary is that Assume 3.5 Example H dr is the longest path accessible to draining pore clay water longest layer. the the path is in accessible to draining Figure In T ) corresponding to the time time to the ) corresponding Clay Wet sand Dry sand Soil Type Example 3.5 in Used Properties Soil TABLE 3.9 T –4.0 m = 10 0.6–1.75 m − 8 () q 10 c 2.675 2.875 (MPa) ×× 5 365 H U dr T 25 ave = 2.5 m. =2.5 . = = 0.77 = 2 24 t H Ct 11.5 ( . 10 ( v dr 2 T ×× can be determined using the following the using determined be can E , 60 E (1.16) s s = 2 =4 The Foundation Engineering Handbook Engineering Foundation The q 60 E –6.0 m –3.5 m –2.0 m –1.0 m q c from Table 1.7) s c 10.7 MPa from Table 1.7) = 0 . 504 clay Soft soil Sandy Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4

Spread Footings Spread plate of the following expressions: settlement the and on the mated based level, esti be contact foundation of stress the the can settlement in magnitude same the For would constructed. be footing the that depth location the at and same performed the a plate from load determined test be can of footing a shallow settlement immediate The 3.3.2 Sandy soils Sandy Clayey soils Clayey settlement to be 1.0 to be above Hence, the acceptable. in. value is settlement 6.0 be (3.06 will +2.94) corner at the that whereas or or 0.55 0.24 in, mm mm in. aboveFor the data, Solution is also satisfied. criterion settlement 0.013. is concrete in differential Hence, cracks the prevent structural or (14.00 –6.00)/(1.06)/1000 =0.007. s are moduli tic center Settlement Computation Based on Plate on Test Load Based Data Computation Settlement Differential Settlement Check Total Check Settlement 14.06 be will (= footing 8.19 of center the at the settlement + 5.87) total the Therefore, 1.12,From Equation According to most building codes, the maximum allowable differential settlement to settlement allowable differential maximum codes, the building to most According as settlement 1.13, by Equation applying Then, immediate the obtains one Settlement Immediate = {1 –0.5[16.5/(88.89 –16.5)]}[1.0][88.89 –16.5][0.41(1.0)/(11.5 ×10 + 0.033/(2.57 ×10 Overburden pressure at footing depth ( depth at footing pressure Overburden A 1 =0.5(0.1 ×0.75) +0.5(0.75 ×0.6) +0.5(0.25)(0.533 +0.6) =0.41 m Contact pressure ( pressure Contact ( . Areas of the strain-influence diagram covered by different elas covereddifferent by diagram strain-influence of the . Areas s s center corner . Most building codes stipulate the maximum allowable total allowable total maximum the stipulate codes . Most building 3 )] =5.9 mm – A can be deduced as 0.5(5.87) deduced be as can =2.95 mm. 2 s . The differential settlement is equal to equal is settlement differential . The =0.5(1.5)(0.533 +0.133) =0.5 m A corner 3 = 0.5(0.5)(0.133) = 0.033 m 0.033 0.5(0.5)(0.133) = = )/distance to corner center from ss fp = Δσ ss fp ) =200/(1.5) =    BB fp 2 + B B B p f f

q    ) =16.5 ×1.0 kPa =16.5 kPa 2 2 kPa =88.89 kPa , (3.31) 3 ) +0.5/(10.7 ×10 3 ) - (3.30) 135 - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 foundation designers, poten who from removal do the soils not of recommend organic Therefore, content deposit organic when significant. soil of the is case the the particularly is consolidation. This primary the compression than relatively is that magnitude in larger prolonged exhibit due soils settlement to secondary organic in Foundations constructed 3.3.3 136 e void secondary and ratio componentsmary ( ∊ ponent, component, compressibility aprimary constitutes extensive program: laboratory on an for based soils, testing Floridarelationships organic void ratios water and contents. (1998) al. et Gunaratne following specific the determined initial their and content organic soils of the organic between relationships covered linear where need. to presented address this is treatise foundation. following analytical The of compression the settlement total component secondary the predominates that expected the estimate to techniques sites,analytical must building alternatively specific tial use 0 , as illustrated in Equations 3.36a Equations 3.36b and in illustrated , as The The load increment) unit per 1-D (vertical ultimate The soils strain of organic compressibility where (Andersland Al-Khafaji, 1980; and Many researchers al., et Gunaratne 1998) have dis W W content organic (oc) ofThe asoil as defined is O S = = footing. of the to that equal area an having of a circle diameter the as determined be can where a Computation of Settlement in Organic Soils in Organic of Settlement Computation and and of weight sample of the on combustion) total weight of the solids in the soil sample soil weight the total in solids of the weight of organic matter in the soil sample (usually determined based on the loss loss on the based sample soil (usuallyweight the determined in matter of organic b e , as expressed below. expressed , as ∞ and and B b p is the plate diameter and and plate the diameter is parameters specific to any organic soil can be expressed in terms of the pri of in terms be expressed can anysoil to organic specific parameters w are the ultimate void ultimate the water ratio are the and content, respectively. oc e oc ∞ a b =∗ =0.46 +1.55(oc) ult =× B =− =− w f = W is the equivalent foundation diameter, which which diameter, foundation equivalent the is W 0 () O () S Δσ 1 1 .. 136 e + + 1 1 p [ 100 and and e e a + + ∂ ∂ ∂ ∂ 2 e e a σ σ ( b p s , and a secondary compressibility com compressibility asecondary , and %) e 031 ] s (3.35) The Foundation Engineering Handbook Engineering Foundation The (3.36b) (3.36a) , respectively) of the initial void, respectively) ratio, initial of the

(3.34) (3.33) (3.32) - - - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 (1998) that determined also Footings Spread (Equation 3.28) foundation design. in used commonly (Equationtion 3.27) or appropriate any other 2:1 attenuation the as such stress distribution duced at due depth footing to the overburden averageof [initial the stress, stress current where secondary) 1-D and mary settlement, of depth location,the where soils: organic 3.37b,and (1998) al. et Gunaratne Floridaforrelationships derived following specific the secondary and primary where Based on observed linear relationships such as that in Equation 3.33, in that as such al. et relationships Gunaratne linear on observed Based Hence, the total 1-D settlement can be determined as Hence, 1-D determined be total the can settlement alayer in of thickness, strain vertical The The a a F   ( and and IM z (, pp oc ) and ) and () σσ σ b , parameters specific for a deposit specific soil field parameters organic ). b =− ( z    ) are ) are () 27   9 z and and a , because of their strong stress dependency. stress strong of their , because (. and and

07 compressibilities, respectively. Finally, 3.37a Equations by using b a b a =   87 = b IM =− =− σ ss parameters in Equations 3.38a and 3.38b expressed in terms terms 3.38a Equations in 3.38b and in parameters expressed (. () (. 18 σ + 12 σσ sa () () 1 1 p+ , 6 z 42 79 + + σσ ]. ]. 1 1 s 360 .) σσ 97 =+ e e Δσ + 8 + () . Δ ∫ H 360 . 0 79          17 z s d [( d can be determined using the Bousinesq’s the using distribu determined be can ε 77 +− d d p+s   σ σ ul .) .) oc 17 are stress-dependent functions associated with with associated functions stress-dependent are t zb , as in Equation 3.39 in , as 9 IM IM F F )( = ss pp    Δ ((, () ( 22 () oc 22 97 o σ ∆ σ z + . cc, ∆ s + , can be expressed in terms of (pri its total terms in expressed be , can p+ + zz z (. + 2 σ σ (. )]( 01 s 05 oc oc ) ) (3.39) ∆ (. 62 23 d σ 24 01 40 d d d σ z σ . ) 13 . + σ 2 61 + d σσ+ ′ vo oc 31 oc σ , 06 +1 () (3.40) .) () σ .) σ 15 3 1       .) 2 33 ,

stress increment, stress (3.37a) (3.38a) (3.38b)

   would dependent be on Δσ (3.37b) z , pro , 137 - - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 due to a finite stress increment imposed imposed by a foundation.increment due stress to afinite layer soil organic of integrate an Equationnumerically settlement 3.40 total the to estimate 138 where expression: generalized following by the far. so chapter summarized be ASD adopted can this been in has which ASD, philosophy design the is above, successful the Of historically and more the popular of design are foundations in used commonly philosophies design two The 3.4 Because of the complex of the ofBecause nature FS = Q R 2. LRFD 1. n i = = Load and Resistance Factor Resistance Load and Criteria Design Allowable (ASD) design stress soils. organic of 0.107 compressions of 0.148ondary settlement 0.041 and produces atotal m, which m. kPa,50 respectively. where layer. 3.35 by organic Equation applying the Then throughout σ be would pressure final the charge, sur extensive 1m an due to within attenuation stress no significant is there Because Solution of kPa. 50 of kPa 50 placed due extensively surcharge to an overburden pressure layer soil (oc a1-m-thick =50%) organic in current the and expected 1-D settlement ultimate the to predict wishes one tests, consolidation laboratory on based that, Assume 3.6 Example Fox and Edil (1992)Fox Edil and the used sec and primary numerically, obtains one integration the Finally, performing the on factor of safety load effect nominal resistance nominal a ( σ ) and ) and b ( σ ) are obtained from Equation 3.38 using an oc and and oc an 3.38 Equation using from obtained ) are v + ∈= C ul a Δ α t and and / RQ σ C 100 z 50 ∫ n c = 50 + 50 =100 +50 =50 kPa /F concept to predict the secondary settlement of settlement secondary the to predict concept [( ab b S σσ functions (Equations 3.38a (Equations 3.38b), and functions one can )( ≥ + ∑ )]( i d The Foundation Engineering Handbook Engineering Foundation The (3.41) σ ) , σ values of 0.5 and values of 0.5 and - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 excessive or vibration. settlements as such limits service and failure or sliding capacity failure bearing as such limits strength States include design for Limit spread footing States. at primary The Limit various mance (3) FS. incorporated in is no of quantitative risk measure have levels of different uncertainty, (2) on judgment based experience, and only and FS is of loads types varying fact that the equation of without the part heeding resistance to the (1) are disadvantages ASD of methods the main The (FS) applied factor is of safety only Footings Spread Using 3.41 Equations 3.42 and assuming and 3.4.2 chapter. this in discussed be oftion LRFD. will procedures latter two The calibra the as known factors is of load resistance selection for and the procedure used The 1998): Equation 3.1b same the 3.2. Section in by described be state can evaluation limit of service LRFD-based The foundation types. of appropriate selection factors of the design to different suit the resistance facilitating of However, related probability failure. of to the risk of not limitation the has also LRFD loads provides and and a qualitative resistance measure both it in accounts for variability advantages that exploration, are of main LRFD of The failure. even consequences and the extent of soil control of of predictive construction, equations, quality reliability erties, of application of loads, of loads. combinations and where as summarized be state can limit evaluation of strength LRFD-based 3.4.1 the without reaching itsa foundation function serves that The goal of LRFD is to design, without being conservative as to be wasteful of resources, to design, without conservativeto is wasteful be as of goal being LRFD The perfor evaluation footing of the requires LRFD using of design spreadThe footings Three different methods are adopted to select the resistance and load and factors (FHWA, adopted resistance are the to select methods different Three prop of soil made be to factors incorporate can variability resistance hand, other the On of loads, location direction and magnitude in Load factors account uncertainties for the η γ ϕ 2. 3. 1. i = = = RD Philosophy LRFD Calibration by the theory of reliability theory Calibration by the ASDto Calibration by fitting extensiveCalibration by experience judgment requires Calibration by to ASD Fitting load modifier load factors factor resistance φ φη = RQ n γγ FS DD ≥ QQ () QQ ∑ η DL +

= 1.0 = + γ LL ii (3.42) (3.43) Limit States Limit . 139 - - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Hence, the resistance factor,Hence, resistance the γ where 140 different methods: (1) methods: different CPT, (2) (VST), test vane shear (3) and preconsolidation pressure ( strength shear undrained of the estimation the 3.18 deviation. related population Figure deviation standard dard to is the shows in that Using Teng data from (1992) al. et 3.18), (Figure how sample illustrated be the it stan can expressions: following the using sample finite of data unbiased an (on from obtained However, both where (COV)variation of property, agiven soil provided be of site can coefficient by soil variability the ofA quantitativeof the measure 3.4.3.1 next section. the in discussed procedure are calibration to the concepts relevant statistical The distributions. statistical their using design porated the in incor loads consideredloads the are random are and variables. Therefore, resistance the of the reliability, and theory foundation the the resistance using calibration LRFD the In 3.4.3 FS live load ratio 0.55. of 3is Similarly, the estimate one can D If one assumes adead load/live one assumes If load ratio (=

=1.25 and and and σ μ = = Calibration by Reliability Theory Q Q standard deviation of the entire population of deviation entire of the standard population of entire of the mean Variability of Soil Data D D denotes dead load and / Q L γ values well. as L μ =1.75, then and and σ can be estimated by their respective sample counterparts sample respective counterparts by their estimated be can ϕ , corresponding to an ASD safety factor adeal/ of and 2.5 ASD safety to an , corresponding φ= Q 12 L .( denotes live load. s 53 25 x 2 .( CO = .) x X 01 X 30 ∑ V( i = at site the .. , defined as follows: , defined = n + 1 ∑ + x () i n = n xx ) .( 1 10 n 75 − i = x Q − 1 S i ) σ µ 10 D u

.) / ) of clay at a particular site using three site three using ) of clay at aparticular , (3.44) Q 2 The Foundation Engineering Handbook Engineering Foundation The X

= L X ) of 3.0, FS =2.5, load and factors of ) of size ) of size 05 at site the . ϕ 5 values corresponding to other to other values corresponding n , obtained at the same site at same the , obtained x and and (3.45) (3.46) s - -

Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread case of a discrete random variable) that satisfies the statistics of that particular random random variable)that ofof particular adiscrete the case statistics satisfies that of acontinuous random (in case (in variable) the the function function or mass probability arandom variable describe completely appropriateone can an using density probability (1995)Phoon al. et Tables shown in are 3.10 3.11, and respectively. purposes, For analytical reliable deviation aless evaluation indicates standard method. evaluation of the is, method, that arelatively correlate reliability wellmethod with higher agiven evaluation from obtained deviation estimates standard the that realizes one also investigations. subsurface Intuitively, 3.18 Figure shown in planning that in used be can as such of aclayey findings site Alternatively, soil. research in contained information the ( strength shear “true” of the deviationaccurate undrained of estimate the standard shows laboratory method,also on VST the that, prediction based provides amuch more Moreover,estimation. the in used 3.18Figure technique specific the with varies make can deviation.lation However, standard deviation one that standard of the estimate best the popu the possibly as 7. be can interpreted deviation estimate standard corresponding The estimation 1992.) 107–116, 4, Teng, (From W.H. size. sample with al., et prediction strength of undrained variation Reliability FIGURE 3.18 () σ Source: Relative density—indirect Relative density—direct Dry density Bulk density Plasticity index Plastic limit Liquid limit Natural watercontent Property Tests Index in Variability Soil TABLE 3.10 The typical variability associated with soil index tests and strength tests as reported by reported as tests strength and tests index soil with associated variability typical The ′ p based on the laboratory 3.18 on the consolidation based Figure tests. shows case, the each that, in

Phoon, K.etal.,Reliability-based designoffoundationsfortransmissionlinestructures, TR105000 Alto, CA,1995.With permission. Final report, report prepared byCornellUniversity fortheElectricPowerResearch Institute,Palo

can be improved by increasing the sample size up to an optimum size of about size optimum upsample to the improved be an size can by increasing Coefficient of variation 0.2 0.4 0.6 0 05 Fine grained Fine grained Fine grained Fine grained Fine grained Fine grained Soil Type Sand Sand Sample size Inherent SoilVariability 10 (COV) 0.61 0.19 0.07 0.09 0.29 0.16 0.18 0.18 15 VST method CPT method stress method Preconsolidation Measurement Variability (COV) 0.01 0.24 0.1 0.07 0.08 — — — Soils Found Soils ., 32, 141 S u ) - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 142 where E ables, follows: as equation considered of distribution the computed mathematical are the that using quantities corresponding variable the that with deviationgiven of variable, standard the and mean it most the to common only is match appropriate for a an sidered distribution random mathematical variable. selecting When would that closely “model” con of the function properties a mathematical statistical the would However, obviously is its histogram. be own instances many in what assumed is the randomof variable properties the statistical all satisfies that variable. distribution The Source: Tan Tan Undrained strength (preconsolidation pressure from Undrained strength (unconsolidatedundrained Undrained strength (unconfinedcompression Property Tests Strength in Variability Soil TABLE 3.11 consolidated undrainedtriaxialtesting) triaxial testing) testing) The following statistical relations can be derived between two different random vari different two derived be can between relations following statistical The E ( ϕ ϕ x (direct shear) (triaxialcompression) ) =

a E Phoon, K. et al., Reliability-based design of foundations for transmission line structures, TR 105000 Final 1995. With permission. report, report prepared byCornellUniversityfortheElectricPowerResearch Institute,Palo Alto, CA, and and expected value of expected indicates the expected value mean. or the expected the indicates σµ 22 b µ= =− −∞ ∫ ∞ Ex () di () xf st ri == bbu −∞ ∫ ∞ ti xf () on x xx () xx ds ab dm ou = di tt σ ec st 2 he ( ( on ri om a a + + bu or dm en b b ti ig ) = ) = on to in om σ E fa of ab Fine grained 2 ( en Soil Type ( a Sand, silt Clay, silt Clay, silt Clay, silt re a ou th ) + ) + to Clay Clay ao ea tt X fa E σ he ft ( The Foundation Engineering Handbook Engineering Foundation The 2 b re ( he ), b xi ce ) (3.49a) ao (3.49b) sc nt = Variability (COV) ft ro Inherent Soil he en id e al tr 0.23 0.20 0.32 0.22 0.33 oi — — lo d ca aal ti lo on ca (m ti me on Variability (COV) an Measurement ) (3.47) 0.14 0.08 0.19 — — — — (3.48) - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 probability distribution about respectively. mean, the distribution probability of moments the area of the fourth the and related third be to the populationfor can the sample data (Harr, 1977). andkurtosis It coefficients mustof notedthe skewness be that computeandkurtosis (flatness) could also coefficients of the skewness computed the from more accurately, one deviation estimates, standard the addition and mean in to the then would that random the of represent variation variable that distribution aprobability select However, distributions. lognormal of forced agiven to is case variable, the analyst the in if 3.47 (Equations deviation criteria standard 3.48) the and and the and normal the are only Footings Spread λ to above simplify expressions arelatively range, is, that COV( minor whenwithin the and ln( of where 3.51a of a continuous randomIf variable Distribution Lognormal 3.4.3.3 3.47Equations 3.48. and EquationIt shown be that 3.50 imposed by can automaticallythe conditions satisfies given byis Equation 3.49 a continuous randomIf variable 3.4.3.2 Two very commonly used distributions that merely satisfy the above-mentionedTwo the merely that satisfy mean distributions used very commonly Furthermore, it can be shown that when the random when shown be that it the variableFurthermore, can x , ln( , x ) can be expressed by those of by those expressed be ) can λ x and and Normal (Gaussian) Distribution ), is normally distributed and its probability density function is given by is Equation function its density probability and ), distributed normally is ζ are the mean and the standard deviation of ln( standard the and mean the are fx () fx () X =− X λ is normally distributed, its probability density function function its density probability distributed, normally is =− is lognormally distributed, then the natural logarithm logarithm natural the then distributed, lognormally is ξ= ξπ = σπ x 1 ln 2 as 1 2     ln ex (( (( = ln( = 1 ex 1 p + +     p CO CO     μ 2 1 µ ) V    V (3.51d) 2 1 2 ln()    2 X x X x σ ξ − )) )) − µ (3.51c)     (3.51b) λ    X 2    ) is relatively) is (<0.2), small the     2 (3.50)     x ), respectively. statistics The (3.51a) X exhibits a variation avariation exhibits 143 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 or the lognormal distribution is the convenience that such a distribution provides in the provides the convenience in the adistribution such is that distribution lognormal or the normal the as such distributions probability expressed of mathematically use A primary 3.4.3.4 ξ and 144 mean[ variates, then mal well. Therefore, it follows both that, if variates, is, then that normal distributed, mally Q effects, order involves ofIn that to adesign compute load reliability the randomly distributed variables of random terms in expressed foundation are of the soil provided strength shear by the resistance afoundation, as the such of and aload effect applied the If on asubstructure 3.4.3.5 as of its distribution probability terms in expressed be range of [ the values in foundation of Accordingly, the soil. parameters if relevant the geotechnical and of structures loads applieddom on earthen characteristics ran incorporate that the procedures design the in estimates of reliability assessment the enhance computations significantly also computation Similar of estimates. probability in terms of the combined random combined of variable the terms in Using Equation 3.51b, Equation Using on Equation 3.49,Based (Harr, of statistics 1977) theorem states both axis that, Central if The Q Estimation of Probabilities Reliability of Design , and soil resistance, resistance, soil , and Q and and R , respectively, then the reliability of the design can be expressed as expressed be can , respectively, design of the reliability the then g me ′ ( R an , Q a st [( , ) =ln b gR an ], then the probability of finding values ofof probability finding the ], then g ′ ′ da ( R ,) R Q , rd R Q g –ln , it is convenient to express the interaction between , it convenient between interaction is the to express ) would have following characteristics: the ]l ′ devi ( = R , PX Q Q n at () )] =mean(ln would be     R Re io and and <= () = COV( = n[ = 11 cf + gR P CO RQ ′ Q ( (, g R ∫ are lognormally distributed, that is, that lognor distributed, lognormally are a c ( ≥ R VC Q normally distributed X , g RQ () Q Q 22 R ) xx ) ( ]= R (3.51e) ) =( ) ) −mean(ln     (3.53) d X , The Foundation Engineering Handbook Engineering Foundation The Q − is a random variable that assumes arandom is variable assumes that ) would be normally distributed as as ) would distributed normally be ln σσ R (3.52) ln 22 –     () RQ Q () + ). + Q ln () ) OV (3.54a)

as well.as X less than than less     R and and Q c are nor are (< (3.54b) R b ) can ) can and and - - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Footings Spread σ Equation 3.50then to simplifies variate normal standard of the terms in St Similarly, 3.51c Equations using Therefore, the estimation of probability in Equation 3.52 of in probability Therefore, estimation would the as follows simplified be of form Equation 3.56b, differential the from Then, (Equation 3.50) distribution normal for expression the mathematical the one expresses If coefficient of the of reciprocal of variation the Then, an da rd devi at io nC [( gR ′ ,) Q β PX ]l () =+ == σ= σ µ <=   fz cP µ= n( () ln PX 1 ln ()   =− ln () 11   ln σπ        () <= + R z OV Z z 11 Q R (d , where     cf 1 CO + = − 2 Q < R z R CO 2 Q x ) =d () () c )l VC 1 1 −∞ ex   σ −µ ∫ − c σ = RQ () () + + 22 + V 1 1 p µ X CO CO R   () + + 2 x () ,       xx (3.56b) (3.56a) (3.58) n CO CO () + = (() V V d 1 2 1 + c Q R −∞ 2 2 + ∫ V V − σ z OV g µ CO Q R 2 CO 2 2 ′     ( fz    (3.55a) R ()    

V , V Q Q   2 Q d 2 ) can be expressed as expressed be ) can (3.55b)   x   (3.59) (3.55c) (3.57) 145 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 146 Re design, of the reliability the Then, tables, follows: as distribution normal standard the in tabulated function is conveniently(erf), the error of which in terms the above one defines integral If Since value arbitrary the setting Then, Re computed be 3.56a Equations 3.52 can design and of using the as reliability The 3.4.3.6 Substituting from Equations 3.57 Equations from 3.58, and Substituting β = Reliability Index μ / σ (Equation 3.55c), = P ( R ≥ Fz Q () −= ) = PX β () PX P () 1 1 ( <= −= R −= c cz 2 1 <= − Re Re = 0 in Equation 3.60, =0in π =d =− 0 Q er c −∞ c ∫ −∞ − σ ∫ ≥0) =1− − σ fd 2 1 µ µ () π −= 2 σπ 1 2 1 =1 2 β 1 π 2 1 π c −∞ ∫ π − σ 1 π µ 2 −∞ −∞ − − ∫ ∫ σ

ex µ β −∞ − ex ∫ – σ µ ex ex ex F p ex p 2 1 (– pd pd P    π    p( p − [(    β       − −∞ ), − − R    ∫ 2 1 2 − 1 β −

− 2 1 The Foundation Engineering Handbook Engineering Foundation The 2 1 ex zz zzz 2 1 2 1 2 zz zz 2 Q p    2 2    zz 2       d ) <0] =1−    2    −    σ d 2 1 d z 2 )   

z (3.61a) P ( X < 0) < (3.61b) (3.60) Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Equation 3.55c as index where reliability the Footings Spread Table 3.12 values by the recommended indicates FHWA (1998) for coefficient principles of the on the of statistics, of based variation Then λ follows: as expressed be factor procedure, bias can prediction mean the For representative strata SPT not blow site. being of in the borings counts, soil and obtaining equipment the in capacity) in ing losses foundation energy resistance, to underpredict (e.g., method of tendency aparticular the as of such error sources various Hansen’s bear where R tance, resistance measured The 3.4.3.7 n number of sources of error with individual factors affecting the strength of resistance of resistance strength the factors individual affecting of with of error number sources R λ n R Resistance Statistics Note: Source: Inherent spatialvariability Equipment/procedure usedinSPT Model error Correction Statistics Resistance TABLE 3.12 , as represents the bias factor for resistance. The bias factor includes the net effect of factor bias factor bias net effect includes The for the the resistance. represents

L = lengthofpile. Federal Highway Administration, Federal Highway BridgeSuperstructures R m CO β can be expressed in terms of the predicted (nominal) resis predicted of the terms in expressed be can β= is defined in terms of the load and resistance statistics in statistics the andresistance loadof in terms defined is VC Rn 2 =+ ln   ln () OV 11     , Washington, DC,1998.With permission. + R Q R = 1 2 CO m = λ CO () () Load andResistanceFactorDesign (LRFD)for 1 VC 1 1 λ λ R 2 + + R 2 … R VC CO CO () 1.0 1.0 1.3 2 n λ λ 22 Statistics forCorrectionFactors (3.62) i n ++ + (3.63a) V V Q R 2 2 O     V Q 2 OV   (3.55c) (3.63b) 0.15–0.45 (use0.3) λ (0.44/ R and COV and COV 0.5 λ L R i ) 0.5 is given by given is R . 147 - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4

148 By eliminating By eliminating ϕ From Equation 3.42 where Equation 3.55c,By one obtains rearranging 3.4.3.9 3.63b and 3.63a of forsources given bias from Equations deadin a then situation, loadssignificant design live load model specifies for StateAASHTO (American Association Highway Transportation and Officials) LRFD loads. dead live with and associated where load the factor bias includes uncertainties various Q load, write measured for the one can Similarly 3.4.3.8 Using Equation 3.62 Equation Using λ QQ Q Determination of Resistance Factors Load Statistics D = values Table for found in commonplace are materials 3.13. hand, other the On Factory made Component of Coefficients Loads forDead Factors Bridge and Bias Foundation Variation TABLE 3.13 Source: Live load Asphaltic wearingsurface Cast-in-place m .

R n Federal Highway Administration, 1998, for HighwayBridgeSuperstructures RQ from Equations 3.66b Equations from 3.66c, relation and the using and =+ ex pl λ CO {} Q β L = 1.15 COV and T VC Qn 2 n D CO R   R =+ () n 11 VC m = = QQ 22 RR OV λ CO γ =+ Q == λ Q D Q 1 2 = n D VC OV = , Washington, DC,1998.With permission. mn (1.15 = Q RQ λ 22 CO 1 D γ Q λ () 1.00 1.05 1.03 λ + D L DL λ 2 Q Q = 0.18 forlive two the loads. are there If … VC D + Load and Resistance Factor Design (LRFD) Load andResistanceFactorDesign(LRFD) λ λ R 2 D 22 Q λ Q CO + L n ++ i L ) (3.65a) OV Q The Foundation Engineering Handbook Engineering Foundation The (3.66b) γ V L L , Q Q (3.64) 2   L (3.66d) (3.66c) OV () () 1 1 (3.65b) + + (0.18 =COV CO CO COV 0.10 0.08 0.25 V V Q R 2 2 Q D

Q L ) (3.66a) Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 where derived be factor can as resistance the Footings Spread prediction methods in common use. common in methods prediction provided Table are in indices reliability 3.15.on selected and Q Table 3.16 of foundation factors for strength avariety suggested resistance the outlines Finally, factors suggested by FHWA resistance the (1998) based for CPT SPT results and β T is the target reliability index evaluated index Table from reliability target the is 3.14. CPT SPT Test Measurements of Strength Soil Coefficients for Factors and Bias Variation TABLE 3.15 Source: Earth pressure coefficient (K) Wall friction( Cohesion Angle offriction(

Source: 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 Reliability Index Distribution Lognormal for Index Reliability and of Failure Probability between Relationship TABLE 3.14 φ Federal Highway Administration, Federal for HighwayBridge Superstructures R =

δ Q ) Federal Highway Administration, Federal DC, 1998.With permission. Design (LRFD)forHighwayBridgeSuperstructures m ϕ ex ) λγ p R {} [] β TD DD QQ ln m +   = () γ 11 LL ++ λ Q CO D Q , Washington, DC,1998.With permission. VC () D Load and Resistance Factor Design (LRFD) 1 QQ + 22 ++ λ Probability ofFailure CO 1.0 1.0 1.0 1.0 1.0 1.3 λ Q () R 1 L Load andResistanceFactor Q OV VC + L QQ 2.46 ×10 2.12 ×10 1.82 ×10 1.56 ×10 1.34 ×10 1.15 ×10 0.99 ×10 0.85 ×10 CO 22 (3.67a) DL L V () R 2 OV + −8 −7 −6 −5 −4 −3 −2 −1 , Washington, CO V R 0.6–0.8 COV 2 0.15 0.2 0.4 0.1 0.4   R (3.67) 149 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 150 For relatively values of small Similarly, numerator the ofas Equation simplified 3.55c be can as 3.68 written be can TaylorUsing the for expansion, relatively values series of COV small of denominator as Equationfollows:The simplified 3.55c be can 3.4.3.10 Determination of the Simplified Resistance Factor Source: Plate loadtest Semiempirical procedure Rock Rational methodusingshearstrength ( Semiempirical procedure usingCPTdata Clay Rational methodusingshearstrength ( Semiempirical procedure usingCPT data Semiempirical procedure usingSPT data Sand Method/Soil/Condition Foundations State for Shallow Limit Strength Factors for Geotechnical Resistance TABLE 3.16 Field vanesheartests Laboratory test(UUtriaxial) CPT data CPT data SPT data ln ln     

Q R   Federal Highway Administration, Federal for HighwayBridgeSuperstructures () 11 + () () ln 1 1 ln CO + + Q R   CO CO () VC ++ 11 RQ 22 ln + COV COV V V () CO Q R 2 2 () + 11      VC (e.g., <0.3), above the simplified to be can expression =+ RQ 22 OV CO ln () Q R VC + QR 22   =+ ln S N OV −+ u ) from ln , Washington, DC,1998.With permission. ) from ln Load and Resistance Factor Design (LRFD) Load andResistanceFactorDesign(LRFD) (() () 1 11   + () ≈+ CO CO CO The Foundation Engineering Handbook Engineering Foundation The V VC OV VC Q RQ 2 22 RQ 22 ++ −+ ln ln ≈ Resistance Factor OV ( ln () 1 Q R 0.45 0.35 0.55 0.45 0.55 0.5 0.6 0.6 0.6 0.5 OV CO

(e.g., <0.3), Equation V ) R 2 (3.68) Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 By defining the By defining Hence, for expression the Footings Spread 3.5 are listed in Table in 3.5 listed are 3.17. where load and factors follows: as resistance the to express method imate of Equation comparison 3.70ctics, then Equation with 3.42 a convenient yields approx and load and independent factors of are other’s each resistance the that one statis assumes if (COV of other each statistics depend on the From Equation 3.70c, factors, load resistance the that and it seen is ϕ Equation 3.42, Recalling tively, obtain one can and in Equations load3.62 andresistance 3.64,respec the nominal of definitions Using the Separately combining Equation 3.69, in terms rearranging and one obtains SPT tests. SPT on based designed to be is that for factor a bridge footing asuitable resistance Estimate 3.7 Example β T is the target reliability. Resistance factors corresponding to a target reliability of reliability to atarget reliability. factors target corresponding the Resistance is α factor as follows as factor R and and β can be simplified simplified to be can CO Q terms, one obtains terms, VC λλ RQ R 22 Re += Re β= n Q R − − φλ OV γλ αβ αβ = = = CO CO CO e R αβ V V R R R n R n VC () e and COV and = e CO ln = α = − RQ αβ 22 αβ () Qe VC γ + T CO Q R RQ T Q CO nn CO + Qe αβ n V OV (3.42) OV V VC Q CO R RQ (3.70e) αβ (3.70d) V + Q Q (3.70a) CO ) as well.) as However, for convenience, (3.69) (3.70b) V OV Q (3.70c) γ and and ϕ , respectively, 151 - - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 152 Source: CPT SPT Method Estimation Calibration Reliability-Based Using Sand on for Capacity Spread Footings Evaluation of Bearing Factors for Semiempirical Resistance TABLE 3.17 λ From Table 3.13 3.65, Equation and Solution and assuming that that assuming and From Table 3.15, for SPT,

Also, since it is recommended that that it recommended is since Also, Federal Highway Administration, Federal Superstructures Source: Culverts Vertical earthpressure Horizontal earthpressure Wearing surfacesand utilities Downdrag Components andattachments Type ofLoad Loads FactorsLoad for Permanent TABLE 3.18 Safety, FS Factor of Earth structures Flexible metalbox Flexible buriedstructure Rigid frames Rigid buriedstructure Retaining structure Overall stability At rest Active 2.5 4.0

Federal Highway Administration, Federal permission. (LRFD) forHighway BridgeSuperstructures , Washington, DC,1998.With permission. γ Reliability L Average Index, CO = 1.75 and 3.2 4.2 λ V R Q =1.3, COV β D =+ Q γ D Reliability D (. Index, = 1.03(1.05)(1.00) = 1.08 = 1.03(1.05)(1.00) = 00 Target = 1.25 (Table 3.18). Load andResistanceFactorDesign(LRFD)forHighwayBridge 3.5 3.5 80 R λ

22 = 0.7. = Q β L T = 1.15, COV (. 10 Span (m) )( 50 10 50 10 Load andResistanceFactorDesign += Maximum 1.50 1.50 1.95 1.35 1.30 1.35 1.35 1.35 1.25 The Foundation Engineering Handbook Engineering Foundation The .) 1.5 1.5 1.8 25 , Washington, DC,1998.With Q L = 0.18 (FHWA, 1998; Table 3.13) with ASD 2 Fitting Load Factor 0.60 0.60 0.37 0.37 0 . 289 Resistance Factor Minimum Reliability N/A 0.75 0.90 0.90 0.90 0.90 1.00 0.65 0.45 0.90 0.9 0.9 Based 0.57 0.52 0.53 0.49 Selected 0.55 0.55 0.45 0.45 Φ Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread Illustration for Example 3.7. Example for Illustration FIGURE 3.19 to carry a column load of 400 kN. The subsoil can be considered as ahomogenous as considered be can subsoil The load kN. of 400 acolumn to carry 3.19, Figure in shown column For the use 3.8 Example dead load/live the and of failure load ratio (Tableprobability 3.19). of 3.67 the Equation Using Table terms and in 3.14, expressed be can factor resistance the Applying Equation 3.67, Equation Applying 3.0 2.0 1.0 Factor Variation of Resistance TABLE 3.19 Q Q φ D L R =    10 .. 81 Q Q D L + 13 .. φ 15    R 12    = ex 51    Q 10 Q pl 0.516 0.529 0.554 0.085 β .. =2 D {} L 11 β 81 Φ .. + T Q Q R 31 with with B P =400kN D L    . 75 n[ + 25 ((    LRFD concepts LRFD 10 Q 15 ++ Probability ofFailure Q Q (( β 0.0099 D 0.402 0.412 0.432 10 =2.5 / D L    Q ++ ex .) + 289 L and Required Reliability 17 pp{ ( . 1 .) 289 05 + 2 5 .}    ((. β 07 ((. 22 to design a suitable footing asuitable footing to design T 01 D 0.00115 )) 0.313 0.321 0.336 β 2 (. =3 81 01 ) 22 )(( 8 ) + (. 07 0.000134 β 0.2250 0.244 0.262 =3.5 )) ]

153 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 154 loadings, etc., have to satisfy special design criteria in addition to the regular bearing bearing additionloadings, regular in etc., to the criteria design have special to satisfy wave due loads that as such machines, to operating to dynamic Foundations subjected 3.5 Design of Footings to Withstand Vibrations Withstand to of Footings Design the vicinity): the not in table groundwater is the (assume that properties following the clay with silty q 1.0. is soil load for factor recompacted recommended level foundation =1.25 at the ×4400/( stress Factored contact level base ( footing at the stress effective vertical The parameters capacity Table bearing the 3.5 indicates c ϕ′ factors resistance Assume method (Example 3.1), (Example more conservative. slightly to be method seen is state design limit the with width above the footing compares one When B 37/ 6 37/ 6 expression Hansen’s From By applying Table from 3.18. dead load obtained load for factor the The is the It that noted must be capacity: bearing ultimate for the written be can expressions following the Then, 3.5), (Equation expression capacity Hansen’sUsing bearing Internal friction ( friction Internal ( weight Unit Unit ( cohesion qB ul t =+ =+ () 182 φη 12 γ RQ ) =17 kN/m n c .. () ) =20 kPa s 65 = 81 c d ϕ =1.359 N c (. =1.4 ) =15° c =8 ∑ 57 359 γ 47 ϕ )( ii R 14 B of 0.6 0.6 and (Table 3.16) for tan 3 .) ult ult with no load modifier ( no load modifier with = B

1.25 ×4400/( () =tan 2 17 ′ + 17 + =20(0.6) =12 kPa B 2 =182.65 +40.47 () –1 21 B d =1.6 m [0.6 tan [0.6 s =182.65 +57.47 q (. N q

= 1.294

= 1.26 q 26 =2 )( 1 π .) ϕ 294 B ] =9° 2 ) +17 The Foundation Engineering Handbook Engineering Foundation The B q = 1.55 m obtained from the ASD the from =1.55 mobtained + ) =(17)(depth) =17 B η

005 = 1.0) = .( B B 17 )( N ϕ d s B γ γ γ π and and =0.6 =1.0 =0.4 )( B 04 2 ) +(1.0)17 .) c (. , respectively. 06 B )( . 10 .) B . Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 ( of oscillation frequency of the terms amplitudes in limiting the levels of to different of severity. vibration corresponding magnitudes acceleration for agiven frequency. operating limiting the 3.20 Figure order the indicates of amplitude of vibration However, and related limiting to the are criteria design main the of design afoundation would the to that vibrations. consideredbe during subjected be Table criteria. capacity settlement and may that of 3.20 anumber additional criteria lists Footings Spread III. II. I. Functionalconsiderationsofinstallation Footings of Vibrating for Design of Criteria List TABLE 3.20 Source: H. F. E. D. C. B. A. F. E. C. B. A. E. D. C. B. A. D. G. Designconsiderationsforinstallationsinwhichtheequipmentproduces excitingforces For steady-state harmonic oscillations, the limiting accelerations can be deduced accelerations be can from limiting For the steady-state oscillations, harmonic Possible modesofvibration—couplingeffects Limiting dynamicconditions Settlement: static+repeated dynamicloads Bearing capacity:static+dynamicloads Static settlement Static bearingcapacity Cost ofreplacement Cost ofmaintenance Total operationalenvironment Causes offailure Modes offailure and thedesignobjectives Local isolationofindividualmachines Isolation offoundations Possible changesinambientvibrations Ambient vibrations Design considerationsforinstallationofsensitiveequipment 4. 3. 2. 1. Environmental demands 3. 2. Connections 1. Fatigue failures 3. Acceleration 2. Velocity 1. Initial costanditsrelation toitem A 2. 1. Limiting displacement,velocity, oraccelerationamplitudes

Resonance ofstructural components Sensitive equipmentnearby Psychological effect onpersons Physiological effect onpersons Supporting structure Machine components Vibration amplitudeatoperatingfrequency By newequipment By construction Richart, F.E. etal., permission. Vibrations ofSoilsandFoundations acceleration limit =displacement , Prentice Hall,EnglewoodCliffs, NJ,1970.With limit ω 2 (3.71) ω ) as 155 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 156 of vibrations: theory elementary derived the using modulus, density, mass Poisson’s and ratio foundation 3.21).of the soil, respectively (Figure respectively, where and given by are constants damping the and spring the foundation circular, the is If (Lysmer 1966) as written Richart, amplitude be force and simpleconstant can harmonic equation ofThe motion for foundation arigid of mass 3.5.1 modes ofof vibration. different for anumber sections ensuing provided the are in availablemulations analyses such from for Analytical principles by dynamics. of the soil determined be foundationvibrating can amplitudes (or maximum the hand, accelerations) other the On undergone by agiven F.E. al., et =25.4 (note: 1in mm). of Richart, vibration (From frequency aparticular for amplitude of displacement limits General FIGURE 3.20 Then, the following important parameters that relate vibratory be that to the parameters motion can following important the Then, 1.

Natural frequency of vibration frequency Natural Vertical Steady-State Vibrations Steady-State Vertical Vibrations of Soils and Foundations

log [Amplitude of vibration (in)] −4 −3 −2 −1 0 0123 B is the equivalent footing diameter, equivalent footing the is Frequency inlog(cyclesperminute) mz ++ cz zz kz f , Prentice Hall, Englewood Cliffs, NJ, 1970. by author.) Cliffs, Reproduced Englewood Hall, , Prentice n = == c PP z 2 1 = –2 k 08 z 1 . 00 3 = eP − 2 5 it ω ν 1 2 B 2 s 1 GB − =+ 2 πν s ν s G 1 [c 2 (3.73a) ss GB − ρ os The Foundation Engineering Handbook Engineering Foundation The s 4 () m (3.73b) s ωω m ti 1 subjected to a vertical steady-state to avertical subjected (3.74a) G si s , structural tolerance Lower limitof machine foundations machine an Upper limitof tolerance Upper limitofhuma tolerance Lower limitofhuma n( ρ s , and , and t )] (3.72) d ν s denote the shear denote shear the n n - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Footing subjected to vertical vibration. vertical to subjected Footing FIGURE 3.21 Footings Spread

where 2. 4. 3.

and the damping ratio, damping the and nondimensional the 3.22 Figure against in ted Amplitude of vibration ratio Damping ratiomass dimensionless where modified the frequency Resonant The amplitude of vibration can be expressed as follows: as amplitudeThe expressed of be vibration can = constant damping Critical ratioDamping = excitationFor force-type M is the amplification the factor, is D = (damping constant)/(critical constant) damping f D M r . = = πρ 10 A B z    2 B == 12 ( GB DD zs k − P k z =− s m    0 z (. == 21 ω MP ) ω 1/2 z () () s nn B Pk − z B o P    A z / 2 ν z    36 1 0 0 z 2 . 1 2 425 + ) B ρ , expressed in Equation 3.74f in , expressed plot and GB − z s m    fo B s ν

B frequency or frequency ratio, frequency or frequency 3 r s (3.74d) D z is given by given is (3.74c) B M ω z ω , > (3.74e)    03 2 . (3.74f) (3.74b) ω / ω n - 157 , Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 158 ( ( factor (a) Magnification FIGURE 3.22 ω / ω where 5. n ). The phase lag phase The (b (a)

–1.5 –0.5 Phase shift (rad) ϕ M ) 0.5 1.5 –1 Amplification factor can be determined from Equation 3.74g from determined be can 3.22b. or Figure 0 1 2 3 4 5 6 0 1 ) against frequency ratio ( ratio frequency ) against 0 0 ωπ 0.5 0.5 φ nn = == ta 2 n Fr Fr f − ω 1 e e /    quenc quenc ω ωω 2 n ). (b) Magnification factor ( factor ). (b) Magnification ωω 2 n 1 1 1 2 n − y ratio y ratio GB − s ν D 2 s The Foundation Engineering Handbook Engineering Foundation The    m 1 (3.74g)

1.5 1.5 D =0.6 D =0.5 D =0.4 D =0.3 D =0.2 D =0.1 D =0 D =0.6 D =0.5 D =0.4 D =0.3 D =0.2 D =0.1 D =0 M ) against frequency ratio ratio frequency ) against 2 2 (3.74a) Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread Illustration for Example 3.8. Example for Illustration FIGURE 3.23 soil is dense sand having the following properties; unit weight =17 unit kN/m properties; following the having sand dense is soil 20 force of sin 25 harmonic vertical equivalent an imparts machine 700 The kN. is foundation and weight machine of the total The 3.23). (Figure diameter 4min is amachine supporting foundation concrete circular A rigid 3.9 Example density.) mass and place mass computation of the in the in used weight are unit weight the (Note that and Poisson’sFor 0.33. soil, to be sandy assumed be ratio can Solution frequency, frequency. (3) and operating at resonant vibration at the of the the amplitude (1) frequency, (2) =55modulus MPa. resonant Determine the vibration of the amplitude Hence, the shear modulus and the mass ratio can be computed be as ratio can mass the and modulus shear Hence, the (2) (1)

Resonant frequency Resonant Natural frequency Natural f n f r = = B z 2 πρ 1 =− πν 10 B 21 () 1 2 GB GB − s (. s ν s s z s m − 11 ρ B s G m z B = 36 3 = 2 =− 21 ) π () 21 = (. E + 22 π s (. ν () 1 4 s 03 07 == 3 )( (. () )( 10 20 55 1 700 − 4 m ,, 000 71 // (. . 17 )( × 33 21 P =25sin20tkN 1000 000 × ,, .. 000 t 33 1000 kN on the footing. If the foundation foundation the If footing. the on kN )( )( /g 4 000 ) 20 / 700 17 98 )( 7 )( 08 )( () MP 1000 1000 . 08 .) 660 1 a − 6 //g / .) (. 36 /( 98 40 ) ). = 3 = = 66 93 . . 3 86 6 cp 3 and elastic elastic and cp s s 159 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 160 Footing subjected to rocking. subjected Footing FIGURE 3.24 B where moment about rocking the to motion asteady-state of foundationThe arigid subjected amplitude harmonic constant Oscillations Rocking 3.5.2 and and f D f h are the diameter and height and foundation, of 3.24). the diameter respectively (Figure the are (3) (4) Hence, Hence,

ance in Figure 3.20. Figure in ance toler of or of 0.011 0.27tude human mm upper limit below the would fall in Amplitude of vibration Amplitude Operating frequency Operating A Based on an operating frequency of 3.18 frequency or 191 operating cps above an on the ampli cpm, Based factor, 3.21, Figure In magnification the safe. considered is range frequency operating the Thus, z == P k 0 z MP f m / f o =6.63/3.18 >2. =2.08 0 1 2 y GB − axis, can be written as (Hall, 1967) (Hall, as written be can axis, X s ν s M = 0.425/ = =− Ic θθ m 25 θθ o = 20/(2 = / Im ++ (. θ f 10 n =+ = B

z 6.63/9.36 =0.71 =0.425/0.86 =0.491 33    π Y B kM Bh ) =3.18 cps 16 )( θθ 22 12 θ M .) = y / M 3 (. 22 = 1.2 =    ×× The Foundation Engineering Handbook Engineering Foundation The e (3.75b) it ω 07 (3.75a) X 10 000 ×= 40 ). 12 mm - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 and Footings Spread of vibrations: theory elementary relevant derived vibratoryfollowing parameters be the to the the motion can Then, using respectively; If the foundation is circular, the spring and the damping constants are given by are constants damping the and spring the foundation circular, the is If where 2. 4. 3. 1.

against the nondimensional Amplitude of vibration ratio Damping frequency Resonant of vibration frequency Natural The amplitude of vibration can be expressed as follows: as amplitudeThe expressed of be vibration can excitation Moment-type M B θ is the magnification factor, magnification the is , the inertia ratio, given by is inertia , the c

θ frequency frequency ff DD = rn B =− () == θ 11 k =− −+ f θ θ n 00 θ 12 ν = = = . 1 s () 31 5 1 2 () M () () k 1 B GB ω π Mk θ BB BB y / 4 s − θθ ν θθ ω y 01 M B () θ () s ν / 1 04 3 k . n I 1 θ θ s ; where . θ + , ρ + 5

θ B 5 I (3.77a) (3.76a) G , which is plotted in Figure 3.22a Figure plotted is in , which 5 ss (3.76c) 2 ρ

(3.77b) (3.76b) (3.77d) (3.77c) 161 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 footing about the footing equivalent an 162 Footing subjected to sliding oscillation. sliding to subjected Footing FIGURE 3.25 derived: respectively. and given by are constants damping the and spring the foundation circular, the is If by Equation 3.78 of mass footing of circular arigid oscillations sliding was developed analog (1967) by Hall A mass–spring–dashpot horizontal to simulate the Oscillations Sliding 3.5.3 Thus, (of height, footing same the above applied be The can to arectangular relations Then, the following important parameters with respect to the above to the be respect motion can with parameters following important the Then,

1. The phase lag phase The Natural frequency of vibration frequency Natural B e that is determined by equating the moment the of surface of the by of the equating area determined is that y axis ( axis ϕ can be determined from Equation 3.74g from determined be can 3.22b. or Figure I θ ) to that of the equivalent circular footing. equivalent of) to that circular the f c n z mx = = k ωπ 2 x 64 46 1 ++ 1 π nn = .( cx == π 78 xx 16 1 BB 16 2 − − e () 43 78 1 B () f = ν kx 78 1 ν − − P s s − 12 − ) ν 1 B ν ss ν = 2 ss ν s GB Pe k I L s GB θ 0 θ G The Foundation Engineering Handbook Engineering Foundation The m it ss (3.77a) ω ρ (3.79a) (Figure 3.25). This can be expressed expressed 3.25). be (Figure can This m 1 (3.78) (3.79b) (3.80a) h ) using using ) Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 amplificationin Equation factor 3.74f as be modified can new solutions, the with amplitudes of keeping the the vibrations. and In frequencies nance above of the of new motion equations all the have cases, reso the to solved be to determine madebe must to Equations sections the3.72, following in modifications 3.75a,and 3.78.In ( masses anced 3.5.1 Sections 3.5.3 in createdby through unbal are foundation described the vibrations If 3.5.4 M Footings Spread

1

where is plotted in Figure 3.26. Figure plotted is in 2. 4. 3.

Resonant frequency Resonant against the nondimensional frequency frequency nondimensional the against Amplitude of vibration ratio Damping ratiomass dimensionless where modified the Foundation Vibrations due to Rotating due Vibrations Foundation Masses The phase lag phase The Moment-type excitation Moment-type The amplitude of vibration can be expressed as follows: as amplitudeThe expressed of be vibration can M is the amplification the factor, is m 1 with an eccentricity of eccentricity an with ϕ can be determined from Equation 3.74g from determined be can 3.22b. or Figure M ωπ 1 nn = ==     2 12 B ff DD rn x − f =− =    == A e ω 41 ω ) rotating at an angular frequency of frequency at angular an ) rotating x 78 () nn = − x P 16       − 1 ω 0 2 k P / ω / ()     ν x 0 ν ω k 78 1 0 ω 2 s n s M x 0 . − + − , which is also plotted in Figure 3.22 Figure plotted in also is , which 288 n    B . B , where ρ 166 , 2 ν    x

m x B ss ν B 3 D s (3.80d) GB x (3.80b) is given by given is

ω ω m 1    2 (3.80a) (3.81) ω , then, the the , then, (3.80e) (3.80c) 163 - - Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 164 P 3.5.4.3 Equation modified by the amplitude3.77dThe expressed of be vibration can as force unbalanced M 3.5.4.2 Equation modified by the amplitude3.74eThe expressed of be vibration can as P 3.5.4.1 ( factor Magnification FIGURE 3.26 0 0 = = y = m m m 1 1

e e

1 ω ω ez Translational Oscillations Sliding Vibrations Rocking Oscillations 2 2 ω must be substituted in Equation must 3.78 in substituted be for Equation must 3.72 in substituted be for 2 must be substituted in Equations 3.75a Equations must in substituted be for M 1 ) against frequency ratio ( ratio frequency ) against

mz Frequency response 6 0 1 2 3 4 5 0 ++ cz zz kz Ickm mx == θθ 0.5 θθ PP ++ ++ cx xx θ= A 01 ω z em Fr / it = ω ω e me kx quenc θ n me I θω =+ ). 1 m 0 1 = z 1 = M M y ratio me et ωω 1 1 1 ez 1 2 P P (3.83b) (3.82b) ω [c The Foundation Engineering Handbook Engineering Foundation The 0 0 2 os 2 e e it it ω () ω M (3.84a) (3.83a) y 1.5 , where it si n( D =0.6 D =0.5 D =0.4 D =0.3 D =0.2 D =0.1 D =0 ω z = moment arm of the =moment of the arm )] (3.82a) 2 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Further details are also found in Das (1993). Das found in also are details Further Equation modified by as the amplitude3.80e The expressed of be vibration can Footings Spread Illustration for Example 3.10. Example for Illustration FIGURE 3.27 3.6 diinl Examples Additional soil can be obtained from Chapter 1(Tables Chapter from 1.5 1.6). and obtained be can soil be 1 to of consolidation coefficient 3.15. Figure from its Assume obtained be clay layer of the can properties Consolidation 3.27. Figure in shown as aload of kN 200 carries that a2mdiameter with footing for acircular components settlement following the Predict 3.10 Example (d) (b) (a) (c) Total ultimate settlement of the center of the footing of center the of the settlement Total ultimate method Schmertmann’s from settlement Elastic settlement differential ultimate Maximum 2:1 (use 5years the in distribu footing Average of the settlement consolidation tion method) tion Elev. –1.2 Elev. 0.0 m Unit weight=18kN/m Soft clay × 10 A x –8 = m me 3 2 m /s. Suitable elastic parameters of the sandy /s. sandy of the parameters Suitable elastic 1 200 kN M 1

GWT –elev.–2.0m Unit weight=16.5kN/m Dry NCsand,SPT=15 Elev. –5.0m Elev. –3.0m 3 - (3.84b) 165 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 166

For T region oc the it in Thus, is 3.15, ( From Figure pressure preconsolidation the Equation 3.28. clay layer, of the mid-plane at the apply stress vertical in To compute increase the (b) (a) Increase stress at the center of the soft clay (2-m-diameter footing), soft of the center at the stress Increase The average effective overburden pressure at the mid-plane of the soft clay, soft of the mid-plane at the average overburdenThe pressure effective At of clay mid-plane

T

= 0.394, 0.394, = Using Newmark’s chart, AB = 2.8 m (Figure 3.16) AB m(Figure =2.8 Newmark’sUsing chart, ∴

S Placing the center of footing at the center of center chart at the of center footing the Placing 5yrs =(5.5)(0.69) =3.795 mm N U ce nt avg er = 0.69 = =× 48 σσ T ∴= ′ vo == σ S += f 4 H oot Ct av =5years, ′ vo ∆4 == gr v dr 2 = =+ = 192 in (. 48    gr    16 11 1 11 ,. . 89 9k + ad + ×× H 52 .. 2 )( kP e ∆ ∆ ∆ iu . o ∆ 05 += σ 06 H σ σ    σ − )( a sm 11 ce 8    dr = = () < nt    == = C =2m, er    05 11 17 P 1 π 0 π lo c / .l /4 365 064 . .) 05 42 51 g 192 (. 2 () 59 () 200 σσ DZ Q 2 kP ×× ′ + vo C 28 og .( ×× + σ 95 1 +− . 24 28 u a + ′ vo 0 =1×10 () P 59 OQ 18 ∆ 48 kP 001 c ) 2 ) = 60 kPa.) =60 2 . 60 The Foundation Engineering Handbook Engineering Foundation The 95 . () ak 9    19 = < ×    03 () 60 P = . π (. –8 c / 55 6 200 m 42 . OQ 60 82 = )( () 2 mm 0 /s . ) 394 Pa 2 = ) 12 .k 2 Pa Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread

N E E (c) Initial effective overburden stress at the foundation level foundation ( at the overburden stress effective Initial

Determination of the immediate settlement immediate of the Determination is the limiting angular distortion needed for structural damage. for structural needed distortion angular limiting the is computed be as can footing of the distortion edge, angular the the under that Placing the edge of footing at the center of center chart at the edge of footing the Placing It is noted that the angular distortion is greater than 1/75 than greater is (= distortion 0.0133), angular the It that noted is which to equal is center the under settlement immediate the that it assumed If is stress increase at the foundation level foundation ( at the increase stress for saturated sand (Table sand for saturated 1.7) =250( E S ce for dry sand (Table sand for dry 1.7) =500( nt 30.5 mm/1 =1/32 m(= footing) of radius the er for (Table clays 1.7) = 300( = = = ∆ σσ 35       σ ′ vo 1 11 edge σσ .m + + H 3 += ′ vo 1 e ∆4 S o =× . += 06 edge    m ∆ 150 ce ∆4    S nt C    = = = = er    r edge 0 lo 35    47    .l 0 064 . 1 11 gl . . + + 001 H 334 9 89 σ edge P 2 .. e −= mm ′ vo c og o 89 . 06 += ×= ..    = 150 = + ..    79 12 () 48 += C    C π 60    r 95 / c . 0 26 200 9 lo 42 N .l og 30 064 + g N +6) = 10,000 kPa () Δσ 0 σσ +15) =15,000 kPa P 5 58 11 .l 2 c ) =200/ ′ 382 vo . og N mm σ + . P 4 + +15) =6750 kPa ′ c vo kP ∆ 95 < 48 ∆ σ og . a> P .. 48    99 c kP    60 π + q . P (2) 9 ) =1.2 a c + 660 2 5 12 =63.66 kPa    . 2 ×    16.5 =19.8 kPa 167 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 168 Illustration for Example 3.11. Example for Illustration FIGURE 3.28

q 3.3 Equation Meyerhoff’s capacity: Using bearing Chapter 2. in obtained be site can for the parameters 12. Suitable is soil SPT subsurface value for the a load of 150 kN/m, carries that 3.28. averagewall The Figure in corrected shown as 1.5 a suitable is m, design of embedment depth the that Assuming 3.11 Example γ (d) SC mo For SPT – c is =− =− = φφ From Table 2.5, t Total ultimate settlement Total ultimate Elev. –1.2m    47 == = 10 .m 12 18 30 Cq ° . () () 9 . ∆∆ A m 5 Elev. 0.0 σ kN N ssu 63 ´ =12 .. /m 66 me 19 ∑ − Z 3 0 . 8 19 Es lz =+ =+ 8 °    35 ce z    10 nt .. ult 34 + er = .l co cN 2 .. 74 medium stiff clay stiff medium ns og c = s o c d 01 01 lid c 00 . . + at    qN mm    io 15 ne 0 150 kN/m q s . , 192 q 000 d q + 0.5 + la st + ic 0 The Foundation Engineering Handbook Engineering Foundation The .. B se 108 γ N ttl 67 γ em + s 550 Me GWT –Elev.–2.0m γ 0 d en γ 416 dium stiffcl C γ γ t su sa == == φ b t + 15 T N =° = = .. i 10 From TableFrom 2.6, 18 0 91 04 strip . ks , . . 000 9 12 kN fM 8 8 ay kN footing for the for the footing , SP ks    /m 0 (. /m f 63 0718 T =12 3 66 3 − 19 Pa .) 8 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 N Footings Spread Illustration for Example 3.12. Example for Illustration FIGURE 3.29 From Table 3.1 (for ϕ the footing? of zone, safety the factor of what influence be outsidewould foundation the kN/m 16.5 weight of aunit 20° and of friction angle an with granular is site level. soil the If ground the 3.29 at of Figure 1.5 a location in shown m above column the on load acts horizontal A 5-kN Example 3.12 Therefore, From Table 3.2b Elev. 0. 0 3 B , determine a suitable footing size. If the groundwater table subsides to a depth table subsides groundwater to adepth the If size. asuitable footing , determine ∴= ≥1.0 m qB Elev. –2.2m ul t () fo Elev. 1.5m ϕ (. r 71

φ = 20°, = 0,= Meyerhoff) == 85 )( () A 04 P B 150 .) × ≤ γ = 14 1 K q 5 kN F ul d p d (. s s 16.5 kN/m 16.5 ≤ 10 t γ γ c c c =5.14, ++ =+ = =+ =    ta (. 10 10 10 10 369 n. . . 21 2    ) ++ .( .    21 21 73 N 5 3 + , q ). =1.0, 81 K B    1 03 φφ 2 B B p 15 .    =+ ) B =tan .    25    =→ 10 .    + 10 = N 50 kN (. 18 03 1 2 B γ 2 . (45 + = 0.0 = + B 91       )( 03 2 B + . .. 22 21 ϕ )( /2) =2.04 . 0 68    .) 01    (.

01 Granular soil GWT –Elev.–1.1 )( .) 00 + m 169 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 170 q expression: capacity bearing Meyerhoff’s Q B zone: influence With no water within B or q ult From Table 3.1, for From Table 3.2b q = ul t c N =+ =+ (. 172 25 c s c q d . 52 d SS 46 c i q =+ c )( + == 64 A P .) 54 qN γ B .    15 γ dd q s 1 D q 10 q ii ϕ d == += q ++ += q =20°, i 0 =− 59 q Mc . + 0.5 + .( γ =(16.5)(1.1) +(16.5 −9.8)(1.1) =25.52 kPa .. 314    B 12 IB 1 71 B 10 .)    04 += N 57 B (. 90 1 . q q γ′ 50 =6.4, × ul .. B B Moment =5×3.7 =18.5 204 1 22 N 87 t B tan tan    =(16.5)(2.2) =36.3 kPa =+ γ 22 s ++ )( 1 174 γ == d 04 .( 25 0 θ 204 N γ 0 .) =5/50; i 877 ., γ . . γ 877 33 52 D q BB =1.65 m =1.65 m = 2.9 = d ≤ A + += 54 18 ssu q ( BB F 005 ul 0 B θ γ . .. .) t . 12 × 15 5 = 5.71° = 314 me × ()    B + 2 B 2 1 ci 59 (. The Foundation Engineering Handbook Engineering Foundation The − 16 (s . rcul ≤ 57 in 7 174 20 59 . B ce ar −+ 1    D 33 or .) 82 = + ssq (. 11 05 5 .) ua . 25 91 441 B . )( 1 . m re 5 .) 204 f + oot 59 . in    7 1 g) B 0 . 314 B    (. 0 551 ) Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread (a) Illustration for Example 3.13. (b) Forcing function 3.13. Example for (a) function (b) Forcing Illustration FIGURE 3.30 (b) –15 –10 –5

Force (kN) 10 15 0 5 0

P by represented be can that force to aperiodic subjected 3.30a is Figure in shown foundation machine A concrete 3.13 Example to 3.35. 2.5 from increases factor safety the Therefore, ( t ) =5sin(45 q 0.05 ul t =+ = (. 3 36 220 0.1 36 kP )( t ) +10 sin(90 (a) a 0.15 .) 41 T ime (s)    q d 0.2 =+ 0 + 2.5sin(135t) 5 sin(45t)+10sin(90) . A P 314 B 4 m t ) + 2.5 sin(135) +2.5 0.25    γ D (. 1 += 204 0.3 Mc I )( 0 F .) 877 == 0.35 B 50 t 23 95 2 m 320 )kN P ++ ( . + t 5 (c) 36 ). (c) Resultant rocking response. ). (c) rocking Resultant ( 0 –1.5 –0.5 P(t) Response (rad) . . 3 51 0.5 1.5 33 –2 –1 3 m )( . 0 1 0 B × 10 5 185 )( BB 12 × –4 . 65 × 0.05 .) B 2 (. 29 = 95 )( 0.1 1 . .) 5 204 kP 1.5 m 0.15 a Ti    1 me (s) + 0 0.2 . 314 B    0.25 (. 05 1 ) 0.3 171 0.35 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 172 is the inertia ratio. inertia the is where is coefficient dashpot The soil, vibrating the and foundation of the to those respect with small negligibly is moment of inertia mass machine the that assumed be It will where 5 sin(45 force (kN) horizontal of the components to the corresponding ratios frequency The (d) (b) The natural frequency of the system is system of the frequency natural The is 3.76a 3.76c, Equations Using constant through spring static the h g W is foundation of the moment of inertia mass The 23kN/m to be weight of concrete unit the Assume (a) Plot (c) r 0 0 =9.81 m/s == = = 1.5 foundation m=height of the Express the rocking response in the form form the in response rocking the Express of components frequency three all with associated shifts phase the Estimate of rock amplitudes the force components, estimate horizontal the Neglecting ing vibration corresponding to all three frequency components of components frequency three to all corresponding vibration ing component of frequency least of the to show cycles at two scale least ω P 4 BLh ( i t , and , t BL ), 10 (90 sin 3 ). I ππ 0 () P 3 fo γ c ( conc. un t θ

) in components and as a as and components ) in θ da = 4 i , and plot , and the ti 2 =4(3)(1.5)(23,000) =414, 000 = acceleration due gravity to acceleration = on 43 () 11 3 () k 08 −+ θ =+ . 3 t ), sin(135 2.5 and µ = rG B W 0 4 == g () θ 31 00 () 8 = 18    Gr .m ω − 31 rh ρ B 43 4 () 22 0 θ n µ 36 8 == − resultant rocking = = µ 08 83 equi k I .(    () ρ 0 θ I = r 0 18 01 0 55 ( 31 .) 1 423 va × t (. = ) are − 43 98 resultant force le 62 46 31 − 01 002 . , . (. 83 nt .) 000 02 1 31 (. − , 8 (. ra × 01 178 10 ) N 02 84

   di ×× + response is the weight foundation of the the is 18 0 us 3 ), . 8 0 ) 4 1700 696 θω 4 = =× () 22 tA 83 62 86 + by selecting an appropriate time appropriate time an by selecting 1700 ) . The Foundation Engineering Handbook Engineering Foundation The =− (. . 3 . 15 18 178 . 31 54 ∑ . 3 4 ra =× I 1 3    0 0(machine) ) d/ 15 8 = = . ii Nm si 83 s 06 31 n( .9 , 178 96 /r ≈ 0. ≈ 0 t ad 7 kg φ ra Nm i m/ ) d/ , tabulate , tabulate P 2 ( s t P ra ). ( t d ). A i - ,

Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Spread Footings Spread Andersland, O.B.and Al-Khafaji, A.A.W.N., 1980,Organic materialandsoilcompressibility, AASHTO, 1996,Standard Specifications forHighwayBridges, American AssociationforState References J. Geotech. Eng. Div., ASCE Highway andTransportation Officials, Washington, DC. θ then is response rocking The 3.22b, Figure From 3.22a, Figure From ratio is damping The ( t ) =(2.25 ×10 −5 ) sin (45) sin 5.62 ×10 1.38 ×10 2.25 ×10 Mk θ Mk D Mk i Θ y (rad) θ 2 Θ y Θ y / 2 3 1 == / 3 / 1 t θ c −0.15) +(1.38 ×10 c −6 −4 −5 θ c , GT7,749. θ θ θ =⇒ =⇒ =⇒ 43 07 14 . . . BB θθ 01 () Θ . 1 Θ Θ ω ω ω ω ω ω 2 + 5 φ φ φ 1 3 n n n 2 = 3 1 2 3 1 ======43 =− =− = 14 07 .( 86 86 86 01 ω .( = .( 62 62 62 135 . 02 11 90 45 i −4 . . . 10 135 (rad) . . . . . 5000 54 90 45 54 54 2500 5 ) sin (90) sin 31 31 31 0 ,) 5 × ra 000 × × ra .( 696 = ra d d × 0 15 10 05 0 0 × d × . . . 8 8 8 01

2 2 . 2 ) 6 4 2 ) t 10 −1.15) +(5.62 ×10 5 =× =× + =× 22 56 13 . .) . 696 . 51 21 81 = 0 0 0 ϕ − 0 − –0.2 –1.15 5 6 − i . 0.15 (rad) 106 4 r r aad ra aad d −6 ) sin (1.35) sin t + 0.2) + 173 Downloaded By: 10.3.98.104 At: 20:35 01 Oct 2021; For: 9781439892787, chapter3, 10.1201/b15592-4 Hall, J.R.,Jr., 1967,Coupledrocking andslidingoscillationsofrigidcircular footings, Gunaratne, M.,Stinnette,P., Mullins,G.,Kuo,C.andEchelberger, W., 1998,Compressibility relations Fox, P.J. andEdil,T.B., 1992, Federal Highway Administration, 1998, Load and Resistance Factor Design(LRFD)forHighway Das, B.M.,1993, Bowles, J.E.,2002, 174 Vesic,1975, A.S., Vesic, A.S., 1973, Analysis ofultimateloadsshallowfoundations, Tomlinson, M.J. and Boorman, R., 1995, Terzaghi, K., 1943, Teng, W.H., Mesri,G.andHalim,I.,1992,Uncertaintyofmobilizedundrainedshearstrength, Richart, F.E., Hall,J.R.andWoods, R.D.,1970, Phoon, K., Kulhawy, F.H. and Grigoriu, M.D., 1995, Reliability-based design of foundations for trans Parry, R.H.,1977,Estimatingbearingcapacityofsandfrom SPTvalues, Meyerhoff, G.G.,1963,Somerecent research onthebearingcapacityoffoundations, Meyerhoff, G.G.,1951,Theultimatebearingcapacityoffoundations, Lysmer, J.andRichart,F.E., Jr., 1966,Dynamicresponse offootingstoverticalloading, Holtz, R.D.andKovacz,W.D., 1981, Harr, M.E.,1977, Harr, M.E.,1966, Hansen, J.B.,1970, A revised andextendedformulaforbearingcapacity, DanishGeotechnical Bridge Substructures Van Nostrand-Reinhold,New York. ASCE Technical, Brunthill, Harlow, England. Found. Englewood Cliffs, NJ. Electric PowerResearch Institute,Palo Alto, CA,340pp. mission linestructures, 103(GT9), 1014–1019. 1(1), 16–26. Found. Div., ASCE Englewood Cliffs, NJ. Institute, Copenhagen,BulletinNo.28. Aug. Symposium onWave Propagation andDynamicProperties ofEarthMaterials for Floridaorganic material, 118(GT8), 1256–1263. , 99(SM1),45–763. , 32(4),107–116. Principles ofSoilDynamics Foundation EngineeringHandbook Mechanics ofParticulateMedia Foundations ofTheoretical SoilMechanics Foundation AnalysisandDesign Theoretical SoilMechanics , 92(SMI),65–91. , NCHRP CourseNo.13068,HI98-032,July, Washington, DC. C α TR 105000FinalReport / C c concept applied to compression ofpeat, ASTM J.Test. Eval. An Introduction toGeotechnical Engineering Foundation Design and Construction , PWS-KentPublishers,Boston. , Wiley, New York. , McGraw-Hill,New York. , McGraw-Hill,New York. , 1sted.,editedbyWinterkorn, H.F. andFang,H.Y., Vibrations ofSoilsandFoundations , report prepared byCornellUniversityforthe , 26(1). , McGraw-Hill,New York. The Foundation Engineering Handbook Engineering Foundation The Geotechnique J. SoilMech.Found.Eng.Div., J. Geotech.Eng.Div. ASCE J. Geotech.Eng.Div., ASCE , Longman Scientific and , Albuquerque, NM, , Albuquerque, , 2(4),30–331. Proc. International , Prentice Hall, , Prentice Hall, Can. Geotech.J., J. SoilMech. Soils - , ,