principles of foundation engineering - virginia · pdf fileprinciples of foundation...
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Principles of Foundation Engineering
Braja M. Das
Chapter 3Shallow Foundations:
Ultimate Bearing Capacity
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Unit Weights
w – unit weight of waterd – dry unit weight (no moisture, just air)m – moist unit weight (has moisture & air)s – saturated unit weight (all moisture, no air)’ – bouyant unit weight = s - w
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Virtually every structure is supported by soil or rock. Those that aren't either fly, float, or fall over".
Richard Handy
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Transcona Grain Elevator Failure
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Intuition
What factors affectthe bearing capacityand settlement of afooting?
Df
D1
B
Sand
Clay
= 32c = 0 psf
= 32c = 200 psf
300 kips
50 kips
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General ConceptsShallow foundations must satisfy 2 criteria:• Adequate safety against shear failure of soils• Do not have excessive settlement
Ultimate Bearing Capacity• The load per unit area at which there is a shear failure of the
soils supporting the foundation
Failure Modes:• General Shear Failure – sudden failure of soil• Local Shear Failure – foundation movement by sudden jerks
requiring substantial movement for failure to reach groundsurface
• Punching Shear Failure – shear failure surface will not reachground surface
• Modes dependent on soil conditions
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BC Zones
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TEDSS
TC
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Nature of bearing capacity failure in soil
(a) general shear failure;(b) local shear failure;(c) punching shear failure (redrawn after Vesic, 1973)
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Variation of qu(1)/0.5B and qu/0.5Bfor circular and rectangular plates on
the surface of a sand
ExampleDry unit weight = 90 pcfFooting width = 4 feetd/w = 90/62.4 = 1.441/2B = 0.5*90*4 = 180 psfPlot Red Linequ/(1/2B) = 105qu = 105*180 = 18,900 psfqu(1) = 65*180 = 11,700 psf
qu(1) – circular footingqu – rectangular footing
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Modes of foundation failure in sand
General shear failureis most likely failureMode unless:
• Small width footing• Very loose soils• Deep embedment
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Settlement of circular and rectangular plates at ultimate load (D/B = 0) in sand
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This nomograph shows thatit takes a lot of settlementbefore full ultimate bearingcapacity is achieved.
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Settlement Usually Governs
Allowable settlements for structures• Strip foundation (masonry) < ¾ inch• Square footing < 1 inch
Example
A 150 kip column load is supported by a square footing with anallowable bearing pressure of 2 kips per square foot. Footing with is
Footing Width (B)150
8.7 feet2
For 1 inch settlement S/B = (1/12)/8.7 = 0.0096 or 0.96% about 1%
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Example Settlement
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Terzaghi Bearing Capacity
Assumptions• General shear failure• Continuous strip foundation (Width/Length approaches zero)• No contribution by shear J-H and I-G• Soil above base acts as surcharge load q = •Df
PassiveFailure
Soil Wedge = ’
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Effects of Cohesion
Typically B to 1.5 B
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Terzaghi Equation for BC
where:qu = ultimate bearing capacity = moist unit weight of soilB = footing widthC = effective cohesionDf = depth of embedmentN, Nc, Nq are bearing capacity factors depending on
qu B N
2C Nc Df Nqu
Equation for a continuous strip footing
Table 3.1Or Eq. 3.4 – 3.6
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Modification for Other Shapes
qu 0.4 B N 1.3C Nc Df Nq
qu 0.3 B N 1.3C Nc Df Nq
Square Footing
Circular Footing
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Local Shear Equations
qu
B N
223
C Nc Df Nq
qu 0.3 B N 0.867C Nc Df Nq
qu 0.4 B N 0.867 C Nc Df Nq Df
strip
square
circular
Where N’, N’c, and N’q are BC factors N, Nc, and Nqmodified by substituting ’ with
tan23
tan
-1 Table 3.2
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Factor of Safety
qallowable = qultimate / FS
Selected factor of safety is dependent on:• Amount of subsurface information available• How accurate the subsurface data• Amount of structural information available• Sustained load and live load development• Potential for future changes at the site (flooding, erosion)• Experience of the engineer in those soil conditions
The greater the confidence in the available information the lower the safety factor can be.• Minimum safety factor of 2 for good accurate data• Safety factor of 4 for very low level of confidence
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Modification of bearing capacity equations for water table
•Df = •D1 + (sat – w)•D2
Case I
Case II
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= 110 pcfsat = 120 pcf= 33C = 200 psf
Df = 4 ftD1 = 2 ft
Df
D1
Diameter = 8 ft
A Circular Foundation
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SolutionFirst, we have a circular foundation, so we use qu = 0.3BN+1.3CNc+DfNq
Determine Df = q = D1+(sat-w)D2 = (110)(2)+(120-62.4)(4-2) = 335.2 psf
Next, determine for first part of equation =’=sat-w = 120-62.4 = 57.6 pcf
Obtain N, Nc and Nq for = 33. From Table 3.1 (page 139) N=31.94, Nc=48.09, Nq=32.23
qu = 0.3(57.6)(8)(31.94) + 1.3(200)(48.09) + 335.2(32.23) = 27722.3 psf
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Structural Information
Structural information is critical to your analysisof foundations.
• The range of loads varies the sizes of foundations.• The sizes of foundations is a factor in bearing capacity.• The sizes of foundations effects the range of settlements.• Very high loads could require excessive footing sizes and
a change in foundation type.• A high variation in adjacent loads could lead to excessive
differential settlements.
Typically, the project structural engineer will provide loadingconditions for the project. Your experience in local soilconditions gives you a starting point for bearing capacityanalyses. You are never given footing sizes during design. Youmay have footing sizes after design to analyze changedconditions.
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Other Conditions
Not All Conditions Are Simple
Karl Terzaghi’s bearing capacity is fora special case of a strip foundation on flatground and embeded at nominal depths.Previous modifications are based onfoundation shape. There are more complexscenarios that require further changes.
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General Bearing Capacity Equation
Now considers footing shape, depth, load inclination, and change in bearing capacity factors.
• Bearing capacity factors now based on passive parameterswith = 45 + ’/2.
• Shape factors consider intermediate shapes (rectangles).• Depth factors consider the effects of embedment depth.• Inclination factors consider non-vertical loading conditions.
B and L
Df
qu12 B N Fs Fd Fi c Nc Fcs Fcd Fci Df Nq Fqs Fqd Fqi qi
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Modified Bearing Capacity Factors
Nq tan2 452
e tan ( )
Nc Nq 1 cot
N 2 Nq 1 tan
Studies indicate that 452
(passive)
Table 3.3
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Shape, Depth & Inclination Factors
Page 145
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Effect of CompressibilityTerzaghi’s classic equation for bearing capacitywas for a general shear failure. It was modifiedto account for local shear failure when theEngineer felt that failure mode was applicable.
The general bearing capacity equation can takefailure mode into account by additional factorsmultiplying the 3 portions of the formula assuggested by Vesic.
The rigidity index is used to take compressibilityinto account and modify the results. It answersthe question – “Which failure mode do I have?”
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Rigidity Index
Ir
Gs
c q tan
whereGs = shear modulus of soils
q’ = effective overburden pressure
Both for soils at a depth of Df+ B/2
Gs
Es
2 1 s
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Critical Rigidity Index
Ir cr( ) 0.5 e
3.3 0.45B
L
cot 45(
2
e
3.3 0.45B
L
cot 45( )
2
When does soil compressibility becomean issue? Use critical rigidity index.
If Ir > Ir(cr) then Fcc = Fqc = Fc = 1
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Table 3.6 for Ir on page 154
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Compressibility Factors
F = F = exp{(-4.4+0.6 )•tan ’+[ ]}c qc BL
(3.07sin ’)(log2I ) r
1+sin ’
F = 0.32 + 0.12 + 0.60logIcc r
BL
When ’ = 0
When ’ = 0, then Fc = Fqc = 1
When ’ > 0 Fcc Fqc
1 Fqc
Nq tan
If Ir < Ir(cr)
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Modified General Bearing Capacity Equation
qu12 B N Fs Fd F Fi c Df Nq Fqs Fqd F Fqi qi qcc Nc Fcs Fcd F Fci cc
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Variation of Fc = Fqc with Ir and ’
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No Footing Width? Assume qallowable
Calculate B Insert B into BC equation Calculate qult
Determine FS Reinterate as needed.
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EccentricityAll previous shallow footing problems hadone thing in common:
A vertical load has been applied at the centerof the foundation.
Eccentricity occurs when loading conditionsshift the center of loading. This occurs when:
• A moment is applied• The load is shifted off center by design• Figure 3.14 p. 158
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Eccentric Failure
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Eccentricity Examples
Exxon
Signs create momentsby wind loading Retaining walls or
structural conditions
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Eccentrically loaded foundations
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Avoid This
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Change Mode of Failure
B
e
BNoneccentric Failure
Eccentric Failure
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Max & Min Pressures
qmaxQ
B L6 M
B2 L
Q = vertical load, M = moment, e = eccentricity
qmaxQ
B L1
6 eB
qminQ
B L1
6 eB
qminQ
B L6 M
B2 L
MMoments
Eccentricity
e= MQ
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Excess Eccentricity
qmax4 Q
3 L B 2 e( ) qmin = 0
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Effective WidthModify the footing width “B” for the effectsof eccentricity
B’ = effective width = B – 2eL’ = effective length = L
Use B’ and L’ substituting forB and L in general bearingcapacity equation
For Fcd, Fqd & Fyd do not replace B with B’
Qult = qu(B’)(L’)43
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Prakash and Saran
Vertically & Eccentrically Loaded Continuous (Strip) Footing
Qult B c Nc e( ) q Nq e( ) 0.5 B N e( )
Qult B L c Nc e( ) Fcs e( ) q Nq e( ) Fqs e( ) 0.5 B N e( ) Fs e( )
Vertically & Eccentrically Loaded Rectangular Footing
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Shape Factors
Fcs e( ) 1.2 0.025LBcs
Fqs e( ) 1qs
Fs e( ) 1.0 2eB
0.68
BL
0.4332
eB
BL
s
<=1
2
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Nc(e)
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Nq(e)
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N(e)
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Two Way Eccentricity
Eccentricity can be one way and two way.
Examples of two way eccentricity:• Sign foundation• Water tower with tank on top
Now we have moments that effect bothfooting dimensions B and L.
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Analysis of foundation with two-way eccentricity
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1 WAY VERSUS 2 WAY
eB
My
QulteL
Mx
Qult
One waymoment
Two waymoment
Q = q’ •A’ult u A’ = B’•L’where q’u is calculated from bearing capacity equation
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Two Way Eccentricity Cases
Depending on loading conditionstwo way eccentricity is analyzedone of five ways.
1. eL/L >= 1/6 and eB/B >= 1/62. eL/L < 1/2 and eB/B < 1/63. eL/L < 1/6 and eB/B < 1/24. eL/L < 1/6 and eB/B < 1/65. Circular footing – always 1 way
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Effective area for the case of eL/L >= 1/6 and eB/B >= 1/6
B1 B 1.53 eB
B
L1 L 1.53 eL
B
A’ = ½B •L1 1
L’ = larger of B1 or L1SO B’ = A’/L’
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A’ = ½(L1 + L2)BL’ = larger of L1 or L2B’ = A’/ L’
Effective area for the case of eL/L < 0.5and 0 < eB/B < 1/6
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A’ = ½(B1 + B2)LB’ = A’/ LL’ = L
Effective area for the case of eL/L < 1/6 and 0 < eB/B < 0.5
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Effective area for the case of eL/L < 1/6 and eB/B < 1/6
A’ = L2B+½(B + B2)(L-L2)B’ = A’/ LL’ = L
2 overlap
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Effective area for circular foundation
Eccentricity on acircular footing is alwaysone way.
Use Table 3.8 to obtainA’ and B’ and solve forL’ = A’/B’
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Homework
CE 430 3.1a & c 3.2 3.3 a & c 3.8
CE 530Same as CE 430 plus 3.9 3.13
From Chapter 3
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