flecture 4
DESCRIPTION
FLecture 4TRANSCRIPT
Shallow Foundations – Bearing Capacity
Introduction
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o Shallow foundations must satisfy various performance requirements and one of them is the bearing capacity (strength requirement)
o Since shallow foundations induce loads the near-surface soils, it induces both compressive and shear stresses in the soils
o The magnitudes of these stresses depend largely on the bearing pressure and the size of the footing
o If the bearing pressure is large enough, or the footing is small enough, the shear stresses may exceed the shear strength of the soil or rock that will result in a bearing capacity failure.
Bearing Capacity Failures
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o general shear failure
o local shear failure
o punching shear failure
General Shear Failure
o Most common mode of failure
o occurs in soils that are relatively incompressible and reasonably strong, in rock, and in saturated, normally consolidated clays that are loaded rapidly enough that the undrained condition prevails
o the failure surface is well defined and failure occurs quite suddenly
o a clearly formed bulge appears on the ground surface adjacent to the foundation
o Ultimate failure occurs on one side only and is often accompanied by rotation of the foundation
Load displacement curve for general shear failure
general shear failure
Local Shear Failure
o is an intermediate case
o shear surfaces are well defined under the foundation, but vague near the ground surface
o a small bulge may occur, but considerable settlement, perhaps on the order of half the foundation width, is necessary before a clear surface forms near the ground
o sudden failure does not occur
o The footing just continues to sink into the ground Load displacement curve for local shear failure
local shear failure
Punching Shear Failure
o opposite extreme
o occurs in very loose sands, in a thin crust of strong soil underlain by very weak soil, or in weak clays loaded under, slow drained conditions
o the high compressibility of the soil causes large settlements and poorly defined vertical shear surfaces
o little or no bulging occurs at the ground surface and failure develops gradually
Load displacement curve for
Punching shear failure
punching shear failure
Vesić’s investigation
o Vesic investigated these three modes of failure by conducting load tests on model circular foundations in sand
o shallow foundations (D/B < 2) can fail in any of the three modes
o deep foundations (D/B > 4) are always governed by punching shear
o The results show a general relationship between the mode of failure, relative density, and the D/B ratio.
Modes of failure of model circular foundations in Chattahoochee Sand
Bearing Capacity Failures
o The following guidelines are helpful to determine which of the three modes of failure will govern Shallow foundations in rock and undrained clays are governed by the
general shear case
Shallow foundations in dense sands are governed by the general shear case. In this context, a dense sand is one with a relative density, Dr, greater than about 67%
Shallow foundations on loose to medium sands (30% < Dr < 67%) are probably governed by local shear
Shallow foundations on very loose sand (Dr < 30%) are probably governed by punching shear
Bearing Capacity Analyses in Soil General Shear Case
o To be able to analyze and design spread footings, we must understand the relationship between bearing capacity, load, footing dimensions and soil properties
o The relationships have been studied using different approaches such as: assessments of the performance of real foundation, including full-scale
load tests
load tests on model footings
limit equilibrium analyses
detailed stress analyses, such as FEM analyses
Simple Bearing Capacity Formula
Bearing capacity analysis along a circular failure surface
Consider a continuous footing
a. assume this footing experiences a bearing capacity failure
b. failure occurs along a circular shear surface
c. soil is an undrained clay (f = 0)with a shear strength su
d. neglect the shear strength between the ground surface and a depth D
e. soil in this zone is considered to be only surcharge load that produces a vertical total stress of szD = gD at a depth D.
Simple Bearing Capacity Formula
Bearing capacity analysis along a circular failure surface
Take moments about Point A
Define a new parameter, called a bearing capacity factor, Nc
The above equation is known as the bearing capacity formula where Nc = 2p = 6.28.
Terzaghi’s Bearing Capacity Formulas
Assumptions of Terzaghi’s bearing capacity formulas
o the D B
o no sliding occurs between the foundation and the soil
o the soil beneath the foundation is a homogeneous semi-infinite mass
o s = c’ + s’tanf’
o the general shear mode of failure governs
o no consolidation of the soil occurs
o the foundation is very rigid in comparison to the soil
o the soil between the ground surface and a depth D has no shear strength, and serves only as a surcharge load
o applied load is compressive and applied vertically to the centroid of the foundation and no applied moment loads are present
Geometry of failure surface for Terzaghi’s bearing capacity formulas
Terzaghi’s Bearing Capacity Formulas
Assumptions of Terzaghi’s bearing capacity formulas
o three zones were considered
wedge zone – remains intact and moves downward with the foundation
radial shear zone – extends from each size of the wedge and the shape of the shear planes are logarithmic spirals
linear shear zone – the soil shears along planar surfaces
Geometry of failure surface for Terzaghi’s bearing capacity formulas
Terzaghi’s Bearing Capacity Formulas
o for square foundations
o for continuous foundations
o for circular foundations
qult = ultimate bearing capacity
c’ = effective cohesion for soil beneath foundation
f’ = friction angle for soil beneath foundation
szD‘ = vertical effective stress at depth D below the ground surface
g’ = effective unit weight of the soil if groundwater table is very deep
D = depth of foundation below ground surface
B = width (or diameter) of foundation
Nc , Nq , Ng = Terzaghi’s bearing capacity factors = f(f’)
Bearing Capacity Factors
f’
(deg)
TERZAGHI (Eq. 6.4 to 6.6)
VESIĆ (Eq. 6.13)
Nc Nq Ng Nc Nq Ng
0 5.7 1.0 0.0 5.1 1.0 0.0
1 6.0 1.1 0.1 5.4 1.1 0.1
2 6.3 1.2 0.1 5.6 1.2 0.2
3 6.6 1.3 0.2 5.9 1.3 0.2
4 7.0 1.5 0.3 6.2 1.4 0.3
5 7.3 1.6 0.4 6.5 1.6 0.4
6 7.7 1.8 0.5 6.8 1.7 0.6
7 8.2 2.0 0.6 7.2 1.9 0.7
8 8.6 2.2 0.7 7.5 2.1 0.9
9 9.1 2.4 0.9 7.9 2.3 1.0
10 9.6 2.7 1.0 8.3 2.5 1.2
Terzaghi bearing capacity factors
Example Problem
A square footing is to be constructed as shown. The groundwater table is at a depth of 50 ft. below the ground surface. Compute the ultimate bearing capacity and the column load required to produce a bearing capacity failure.
f’
(deg)
Terzaghi (Eq. 6.4 to 6.6)
Vesic (Eq. 6.13)
Nc Nq Ng Nc Nq Ng
30 37.2 22.5 20.1 30.1 18.4 22.4
31 40.4 25.3 23.7 32.7 20.6 26.0
32 44.0 28.5 28.0 35.5 23.2 30.2
Example Problem
The proposed continuous footing shown will support the exterior wall of a new industrial building. The underlying soil is an undrained clay, and the groundwater table is below the bottom of the footing. Compute the ultimate bearing capacity, and compute the wall load required to cause a bearing capacity failure.
f’
(deg)
TERZAGHI (Eq. 6.4 to 6.6)
VESIĆ (Eq. 6.13)
Nc Nq Ng Nc Nq Ng
0 5.7 1.0 0.0 5.1 1.0 0.0
1 6.0 1.1 0.1 5.4 1.1 0.1
2 6.3 1.2 0.1 5.6 1.2 0.2 0.2 m
0.2 m
Vesić’s Bearing Capacity Formulas
o Skempton (1951)
o Meyerhof (1953)
o Brinch Hansen (1961)
o DeBeer and Ladanyi (1961)
o Meyerhof (1963)
o Brinch Hansen (1970)
o Vesić (1973,1975)
developed formulas based on theoretical and experimental findings
excellent alternative to Terzaghi
produces more accurate bearing values
applies to a much broader range of loading and geometry conditions
Vesić’s Bearing Capacity Formulas
o Vesić retained Terzaghi’s basic format and added the following additional factors:
sc, sq, sg = shape factors
dc, dq, dg = depth factors
ic, iq, ig = load inclination factors
bc, bq, bg = base inclination factors
gc, gq, gg = ground inclination factors
o so that the bearing capacity formula is re-written as
Notation for Vesic’s load inclination, base inclination, and ground inclination factors. All angles are expressed in degrees
Vesić’s Shape Factors
For continuous footings, B/L 0, so sc, sq, sg = 1.
Vesić’s Depth Factors
o for relatively shallow foundations (D/B 1), use k = D/B.
o for deeper footings (D/B > 1), use k = tan-1(D/B)
Vesić’s Load Inclination Factors
o i factors are 1 if load acts perpendicular to the base of the footing
o i factors are 1 when f = 0
V = applied shear load
P = applied normal load
A = base area of footing
c’ = effective cohesion (use c = su for undrained analyses)
f’ = effective friction angle (use f = 0u for undrained analyses)
B = foundation width
L = foundation length
For loads inclined in the B direction:
For loads inclined in the L direction:
Vesić’s Base Inclination Factors
o if the base of the footing is level, which is the usual case, all b factors are equal to 1.
Vesić’s Ground Inclination Factors
𝑔𝑐 = 1−𝛽
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o if the ground surface is level (b= 0) the g factors are equal to 1.
𝑔𝑞 = 𝑔𝛾 = 1− 𝑡𝑎𝑛𝛽 2
Vesić’s Bearing Capacity Factors
𝑁𝑞 = 𝑒𝜋𝑡𝑎𝑛 𝜙 ′ tan 45+
𝜙 ′
2
𝑁𝑐 =𝑁𝑞 − 1
𝑡𝑎𝑛𝜙′
𝑁𝑐 = 5.14
For f’ > 0
For f’ = 0
𝑁𝛾 = 2 𝑁𝑞 + 1 𝑡𝑎𝑛𝜙′
o Vesic used the following formulas for computing the bearing capacity factors Nq and Nc:
Vesic recommended the following formula for Ng
Summary
o Bearing capacity failure occurs when the soil beneath the footing fails in shear
o There are three types of bearing capacity failures:
general shear failure
local shear failure
punching shear failure
o Most bearing capacity analyses for shallow foundations consider only the general shear case
Seatwork
A 1.2 m square, 0.4-m deep spread footing is underlain by a soil with the following properties:
g = 19.2 kN/m3
c’ = 5 kPa
f’ = 30o
The groundwater is at a great depth.
ID nos. ending in an ODD No.
Compute the ultimate bearing capacity using TERZAGHI’s method
ID nos. ending in an EVEN No.
Compute the ultimate bearing capacity using VESIĆ’s method
Seatwork
A 5-ft wide, 8 ft. long, 2 ft. deep spread footing is underlain by a soil with the following properties:
g = 120 lb/ft3
c’ = 100 lb/ft2
f’ = 28o
The groundwater is at a great depth. using VESIĆ’s method, compute the column load required to cause a bearing capacity failure.
Seatwork
A 1.5-m wide, 2.5 m. long, 0.6 m. deep spread footing is underlain by a soil with the following properties:
g = 19 kN/m3
c’ = 4.8 kN/m2
f’ = 28o
The groundwater is at a great depth. using VESIĆ’s method, compute the column load required to cause a bearing capacity failure.
Groundwater Effects
o The presence of shallow groundwater affects shear strength in two ways:
Reduction of apparent cohesion
Increase in pore water pressure
o Both of these affect bearing capacity, and thus need to be considered
Groundwater Effects
Three groundwater cases for bearing analyses
CASE I CASE 2 CASE 3
NOTE: If a total stress analysis is being performed, do not apply groundwater correction because the groundwater effects are supposedly implicit within the values of CT and fT.
Example Problem
A 30-m by 50-m foundation is to be built as shown in the figure. Compute the ultimate bearing capacity.
f’
(deg)
Terzaghi (Eq. 6.4 to 6.6)
Vesic (Eq. 6.13)
Nc Nq Ng Nc Nq Ng
30 37.2 22.5 20.1 30.1 18.4 22.4
31 40.4 25.3 23.7 32.7 20.6 26.0
32 44.0 28.5 28.0 35.5 23.2 30.2
Solution
f’
(deg)
Vesic (Eq. 6.13)
Nc Nq Ng
30 30.1 18.4 22.4
31 32.7 20.6 26.0
32 35.5 23.2 30.2
Allowable Bearing Capacity
To obtain the allowable bearing capacity, qa, the ultimate bearing capacity is divided by a factor of safety
where
qa = allowable bearing capacity
qult = ultimate bearing capacity
F = factor of safety
The foundation is then designed so that the bearing pressure, q, does not exceed the allowable bearing capacity, qa.
𝑞𝑎 =𝑞𝑢𝑙𝑡
𝐹
Allowable Bearing Capacity
o Soil Type
o Site characterization data
o Soil variability
o Importance of the structure and the consequences of a failure
o The likelihood of the design load ever actually occurring
Allowable Bearing Capacity
o Soil Type
o Site characterization data
o Soil variability
o Importance of the structure and the consequences of a failure
o The likelihood of the design load ever actually occurring
o Design F - Extreme Values
Typical Range
Sand Clay
Extensive Minimal
Uniform Erratic
Low High
Low High
2.0 4.0
2.5 3.5
Factors affecting the design factor of safety, and typical values of F.
Allowable Bearing Capacity
The true factor of safety is probably much greater than the design factor of safety, due to the following:
o The shear strength data are normally interpreted conservatively, so the design values of c and f implicitly contain another factor of safety.
o The service loads are probably less than the design loads
o Settlement, not bearing capacity, often controls the final design, so the footing will likely be larger than that required to satisfy bearing capacity criteria.
o Spread footings are commonly built somewhat larger than the plan dimensions.
Example Problem
A column has the following design vertical loads: PD = 300 k, PL = 140 k, PW = 160 k will be supported on a spread footing located 3 ft. below the ground surface. The underlying soil has an undrained shear strength of 2000 lb/ft2 and a unit weight of 109 lb/ft3. The groundwater table is at a depth of 4 ft. Determine the minimum required footing width to maintain a factor of safety of 3 against a bearing capacity failure (use Terzaghi’s method).
f’
(deg)
TERZAGHI (Eq. 6.4 to 6.6)
VESIĆ (Eq. 6.13)
Nc Nq Ng Nc Nq Ng
0 5.7 1.0 0.0 5.1 1.0 0.0
1 6.0 1.1 0.1 5.4 1.1 0.1
2 6.3 1.2 0.1 5.6 1.2 0.2
Design Loads
ASD design load combinations [ANSI/ASCE 2.4.1]
o D
o D + L + F + H + T + (Lr or S or R)
o D + L + (Lr or S or R) + (W or E)
o D + (W or E)
Alternate method of evaluating wind and seismic loads
o 0.75[D + L + (Lr or S or R) + (W or E)]
o 0.75[D + (W or E)]
Bearing Capacity on Layered Soils
Many soil profiles are not uniform. To compute the bearing capacity of foundation on soils where c, f and g vary with depth, we can use three methods:
o Evaluate the bearing capacity using the lowest values of c’, f’ and g in the zone between the bottom of the foundation and a depth B below the bottom.
this is the zone where bearing capacity failures occur
this method is conservative
however many design problems are controlled by settlement, so a conservative bearing capacity analysis may be the simplest and easiest solution
Bearing Capacity on Layered Soils
o use weighted average values of c’, f’ and g based on the relative thickness of each stratum in the zone between the bottom of the footing and a depth B below the bottom
this method could be both conservative and unconservative
provides acceptable results as long as the differences in the strength parameters are not too great
Bearing Capacity on Layered Soils
o consider a series of trial failure surfaces beneath the footing and evaluate the stresses on each surface using methods employed in slope stability analyses.
the surface that produces the lowest value of qult is the critical failure surface
most precise but also requires the most effort to implement
appropriate only for critical projects on complex soil profiles
Example Problem
Using the second method, compute the factor of safety against a bearing capacity failure in the square footing shown.
f’
(deg)
TERZAGHI (Eq. 6.4 to 6.6)
VESIĆ (Eq. 6.13)
Nc Nq Ng Nc Nq Ng
33 48.1 32.2 33.3 38.6 26.1 35.2
34 52.6 36.5 39.6 42.2 29.4 41.1
35 57.8 41.4 47.3 46.1 33.3 48.0
Evaluations of bearing capacity failures on saturated clays (Bishop and Bjerrum, 1960)
Locality Clay Properties
Computed Factor of Safety F Moisture
content, w
Liquid limit, wL
Plastic limit, wP
Plasticity index, IP
Liquidity index, IL
Loading test, Marmorera 10 35 15 20 -0.25 0.92
Kensal Green 1.02
Silo, Transcona 50 110 30 80 0.25 1.09
Kippen 50 70 28 42 0.52 0.95
Screw pile, Lock Ryan 1.05
Screw pile, Newport 1.07
Oil tank, Fredrikstad 45 55 25 30 0.67 1.08
Oil tank A, Shellhaven 70 87 25 62 0.73 1.03
Oil tank B, Shellhaven 1.05
Silo, US 40 20 35 1.37 0.98
Loading test, Moss 9 16 8 1.39 1.10
Loading test, Hagalund 68 55 19 18 1.44 0.93
Loading test, Torp 27 24 0.96
Loading test, Rygge 45 37 0.95
Evaluations of bearing capacity failures on saturated clays (Bishop and Bjerrum, 1960)
Results of static load tests on full-sized spread footings (Adapted from Briaud and Gibbens, 1994)
Seatwork
A column carrying a vertical downward dead load and live load of 150 k and 120 k, respectively, is to be supported on a 3-ft deep square spread footing.
The soil beneath this footing is an undrained clay with su = 3000 lb/ft2 and g = 117 lb/ft3. The groundwater table is below the bottom of the footing.
Compute the width B required to obtain a factor of safety of 3 against a bearing capacity failure.
Seatwork
A 120-ft diameter cylindrical tank with an empty weight of 1,900,000 lb. (including the weight of the cylindrical mat foundation) is to be built. The bottom of the mat will be at a depth of 2 ft. below the ground surface.
This tank is to be filled with water. The underlying soil is an undrained clay with su = 1000 lb/ft2 and g = 118 lb/ft3, and the groundwater table is at a depth of 5 ft.
Using Terzaghi’s equations, compute the maximum allowable depth of the water in the tank that will maintain a factor of safety of 3.0 against a bearing capacity failure. Assume the weight of the water and tank is spread uniformly across the bottom of the tank.
Summary
o there are several formulas to compute the ultimate bearing capacity, qult. These are Terzaghi and Vesic’s formulas.
o Shallow GWTs reduce the effective stress in the near-surface soils and can therefore adversely affect bearing capacity. Adjustment factors are available to account for this effect.
o The allowable bearing capacity, qa, is the ultimate bearing capacity divided by a factor of safety. The bearing pressure, q, must not exceed qa.
Summary
o Bearing capacity analyses should be based on the worst-case soil conditions that are likely to occur during the life of the structure.
o Bearing capacity analyses on sands and gravels are normally based on the effective stress parameters, c’ and f’. However, those on saturated clays are normally based on the undrained strength, su.
Summary
o Bearing capacity computations may be performed for local and punching shear cases. These analyses use reduced values of c’ and f’.
o Bearing capacity analyses on layered soils are more complex because the values of c’ and f’ for each layer should be considered.
o Evaluations of foundation failures and static load tests indicate the bearing capacity analysis methods in this chapter are suitable for the practical design of shallow foundations.