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1
ENCE 461Foundation Anal ysis and
Design
Bearing Capacity of Shallow Foundations
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Bearin g Capacit y Failure
a) General Shear FailureMost common type of shear
failure; occurs in strong soils and rocks
b) Local Shear FailureIntermediate between
general and punching shear failure
c) Punching Shear FailureOccurs in very loose sands
and weak clays
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General and Punchin g Shear Failure
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Soil Conditions and Bearin g Capacit y Failure
(after Vesi� (1973)
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Load Displacement Curves
a) General Shear Failure
b) Local Shear Failure
c) Punching Shear Failure
Ductile Failure Brittle Failure
(after Vesi� (1973)
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Comments on Shear Failure
� Usually only necessary to analyse general shear failure
� Local and punching shear failure can usually be anticipated by settlement analysis
� Failure in shallow foundations is generally settlement failure; bearing capacity failure must be analysed, but in practical terms is usually secondary to settlement analysis
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Development of Bearin g Capacit y Theor y
� Application of limit equilibrium methods first done by Prandtl on the punching of thick masses of metal
� Prandtl's methods adapted by Terzaghi to bearing capacity failure of shallow foundations
� Vesi� and others improved on Terzaghi's original theory and added other factors for a more complete analysis
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Sample Limit Equilibrium Method
� M a�qult Bb�B2�suBb�B��zD Bb�
B2�0
Assume: No soil strength due to internal friction, shear strength above foundation base neglected
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Sample Limit Equilibrium Method
qult�2�su��zD
Nc�2��6.28qult�Nc su��zD
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Assumptions for Terza ghi's Method
� Depth of foundation is less than or equal to its width
� No sliding occurs between foundation and soil (rough foundation)
� Soil beneath foundation is homogeneous semi-infinite mass
� Mohr-Coulomb model for soil
� General shear failure mode is the governing mode (but not the only mode)
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Assumptions for Terza ghi's Method
� No soil consolidation occurs
� Foundation is very rigid relative to the soil
� Soil above bottom of foundation has no shear strength; is only a surcharge load against the overturning load
� Applied load is compressive and applied vertically to the centroid of the foundation
� No applied moments present
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Failure Geometr y for Terza ghi's Method
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Log-Spiral Geometr y
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Application of Lo g-Spiral Geometr y
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Bearin g Capacit y Factors
� Nc = factor of soil cohesion
� Nq = factor of overburden pressure
� N = factor for unit weight of soil
� A, B, C = constants depending upon geometry, soil properties, etc.
� qult
= ultimate bearing capacity of the soil
qult�A Nc�B Nq�C N
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Terzaghi Equations and Factors
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Terzaghi Bearin g Factors
� Equations for Nc, N
q
shown at left
� Values for N
originally derived from Cullman's graphical method; can be approximated by equations
� All strictly a function of
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Tabulated Terza ghi Bearin g Factors
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Application to Square and Circular Foundations
qult�1.3c' Nc�� ' zD Nq�0.4 'BN
�Square�qult�1.3c' Nc�� ' zD Nq�0.3 'DN
�Circular �
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Example of Terza ghi's Method
� Given
� Square Foundation as Shown
� Grounwater table is 50' below surface
� Ignore slab-on-grade flooring
� Find
� Ultimate bearing capacity and column load to produce same
Use Terzaghi's Method
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Example of Terza ghi's Method
� Solve for qult
� Obtain Bearing Capacity Factors
qult�1.3c' Nc�� ' zD Nq�0.4 'BN
�Square�
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Example of Terza ghi's Methodqult�1.3c' Nc�� ' zD Nq�0.4 'BN
qult��1.3��150��37.16���121��2��22.46���0.4��121��3.25��19.7�qult�7246�5435�3099
qult�15,780psf
� Compute weight of foundation
� Wf = (3.25)2(2)(150) = 3169 lbs.
� Compute design load
� 15780 = (P + 3169)/(3.25)2 – 0
� P = 163,507 lbs. = 163.5 kips
q�P�W f
A�uD
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Notes on Terza ghi's Method
� Since soil cohesion can be difficult to quantify, conservative values of c (cohesion) should be used
� Frictional strength is more reliable and does not need to be as conservative as cohesion
� Terzaghi's method is simple and and familiar to many geotechnical engineers; however, it does not take into account many factors, nor does it consider cases such as rectangular foundations
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Vesi�'s Method
� Similar in basic format to Terzaghi's Method, but takes into account a large number of factors
� Some variations in the way it is implemented
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Factors in Vesi � Method
� Bearing Capacity Factor (N)
� Shape Factor (s)
� Depth Factor (d)
� Inclination Factor (i)
� Ground Factor (g)
� Base Factor (b)
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Notations for Vesi �'s Method
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Vesi�'s Bearin g Capacit y Factors
2.0
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Tabulated Vesi � “N” Factors
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Shape Factor (s)
sc�1�BL
Nq
Nc
sq�1�BL
tan '
s�1�0.4BL
For continuous footings, s = 1
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Depth Factor (d)
� Values of k
� D/B < 1, k = D/B
� D/B > 1, k = arctan (D/B), result in radians
� Discontinuity when D = B
dc�1�0.4k
dq�1�2k tan ' �1�sin ' �2
d�1
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Load Inclination Factor (i)
i c�1�mV
Ac'Nc
0
i q��1�V
P�Ac'
tan '
�m
0
i��1�V
P�Ac'
tan '
�m�1
0
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Load Inclination Factor (i)
� Variables
� V = applied shear load
� P = applied normal load
� A = base area of footing
� c' = effective cohesion (use c = su for undrained
analyses)
� �� = effective friction angle (use � = 0 for undrained analyses)
� B = foundation width
� L = foundation length
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Load Inclination Factor (i)� Values of m
� Loads inclined in the B direction:
� Loads inclined in the L direction:
� i = 1 if either loads act perpendicular to footing or soil is purely cohesive (� = 0)
� Applies to loads that are not perpendicular to the base of the foundation; does not apply to eccentric loads
m�2�B� L1�B� L
m�2�L �B1�L �B
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Ground Inclination Factor ( g)
� For level ground surface, g = 1
gc�1��
147ºgq�g��1�tan��2
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Base Factor (b)
� For footings with angled foundation bases
� When footing is level, b = 1
bc�1��
147ºbq�b��1�tan��2
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Allowable Bearin g Capacit y
� Most foundations designed by ASD for geotechnical strength
� Foundation is then designed so that the allowable bearing pressure is not exceeded
qa�qult
F
Factors when considering selection of a factor of safety
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General Bearin g Capacit y Example
� Given
� Continuous Foundation
� Width of foundation = 1100 mm (B)
� Depth of Foundation = 1500 mm (D)
� Soil cohesion c = 15 kPa
� Soil internal friction angle � = 28º, � = 19 kN/m3
� Water table even with depth of foundation
� Find
� Design loading, FS = 3
� Use depth factor term but ignore foundation weight
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Bearin g Example
� For continuous footing, s = 1
� For perpendicular load, i = 1
� For level foundation, b =1
� For level ground, g = 1
� Need to compute factors N, d
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“N” Factors for Example
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Depth Factor (d)
� Values of k
� D/B > 1.5/1.1 = 1.363
� k = arctan (1.363) = 0.938 radians
dc�1�0.4k�1.375
dq�1�2k tan ' �1�sin ' �2�1.281d�1
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Solution of Bearin g Capacit y Equation
� Allowable wall loading per lineal metre
� (385 kPa) (1.1 m wide) (1 m long) = 423 kN/m
qult�c' Nc dc�� ' zD Nq dq�0.5 ' B N d
�15��25.8��1.375���19��1.5��14.72��1.281���0.5��19�9.81��1.1��16.72��1�
qult�532.13�537.4�84.51�1154kPaqa�1154�3�385kPa
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Example Usin g Square Foundation
� Given
� Square foundation, load of 1500 kN
� On Soil Surface
� Soil Conditions� Sand, No Cohesion, � = 20 kN/m3, � = 36º
� Find
� Acceptable foundation size for square foundation
� Neglect effect of foundation weight
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Bearin g Capacit y Equation for Sand
“q” factor zero if foundation is at the surface
Surcharge Overburden Pressure
=1 – 0.4 = 0.6 in this case
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Example Usin g Square Foundation
� For � = 36º, N� = 56.31
� s� = 0.6
� qult
= (0.5)(20)(B)(56.31)(0.6) = 337.86B kPa
� qa = q
ult/FS = 337.86/3 = 112.62B kPa
� qa = Q/A = 1500 kN/A = 1500 kN/B2
qult�12 ' B N s
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Example Usin g Square Foundation
� qa = q
ult/FS = 337.86/3 = 112.62B kPa
� qa = Q/A = 1500 kN/A = 1500 kN/B2
� 1500/B2 = 112.62B
� B3 = 1500/112.62 = 13.32
� B = 2.37 m
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Bearin g Capacit y in Cla y
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Questions?