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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 1
Bearing Capacity- shallow foundation systems
D. A. Cameron
Rock and Soil Mechanics 2006
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 2
TERMS
Thefoundation of a structure is the earth
upon which the structure is supported
structural loads are transferred to the soil
via the footing
Reinforced (?) concrete strip footings & pads
take column (point) loads & distributed
loading (line loads)
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 3
Foundation requirements
Capacity = ultimate
bearing capacity
qu Soil mass at point of
failing
SAFE bearing
capacity
qsafe FoS applied to qu
ALLOWABLE bearing
pressure
qa Allowablesettlementmay dictate
1. Safe bearing capacity orstrength2. Allowable bearing pressure
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 4
Presumptive qsafe values?
NoWTwithin depth, D = B, footing breadth
Dense gravel > 600 kPa
Loose gravel < 200 kPa
Stiff clays 150 - 300 kPa
Soft clays < 75 kPa
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 5
Modes of Failureof ShallowFoundations
General shear
Punching shear
Local shear
Fairly
incompressible soil
Soft, very
compressible soil
Intermediate soil
condition
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 6
Bearing Capacity Analysis
Solution is usually based on:
a) General bearing capacity failure
b) Strip footing - width, B, length L = ,founded at depth, D
c) Soil - homogeneous & isotropic,
unit weight,
d) Soil Strength - parameters c and
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SOIL
Strain
Stres
s
SURFACE FOOTING, B x L =
qaverage = Q/B per m length
Soil strength,cand
Force, Q
BRigid-plasticsoil behaviour
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Choice of Strengths
SAND Clean sand
c = 0 = c
, ,
not affected by
inundation
BUT lower qu!
CLAY Saturated, NC
undrained loading?
u = 0
usually more critical
drained loading?
c ,
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B
Q
The captured, driven soil wedge
Circular for u = 0
MECHANISM
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 10
Simple upper bound foru = 0 soils
Centre ofrotation?
B
Radius, B
qu
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 11
Solution
Take moments about the centre
Disturbing moment = restoring moment
quB(0.5B) = (cuB)B
qu = (2)cu
qu = 6.3 cu
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 12
Simple lower bound foru = 0 soils
B45 45
Active
wedge
Passive
wedge
Ha Hp
Ka = Kp = 1
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 13
Solution
Shear stress
Active statePassive state
Principal stress
= HaHp qu
cu
0
qu = 4cu
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 14
The Correct Answer
Lower bound estimate (safe)
qu = 4cu
The correct answer:
qu = 5.14cu
Upper bound estimate (unsafe)
qu = 6.3cu
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 15
The Accepted Failure Mechanism
1. Active wedge under footing
wedge at (45 + /2)
2. Rotational zone ofplasticity
(radial shear zone) log spiral
shear stresses on either side of the radialshear zone = fn(spiral)
3. Passive wedges to either side of footing wedges at (45 - /2)
resist soil rotation
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 16
45 - /2
Mechanismthe sameboth sides
Rotational or radial shear
zone soil plastic
Log spiral
A
P
45 + /2
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 17
Rotational or radial shear
zone soil is plastic
Log spiral
P
45 + /2
P
45 - /2
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 18
Log Spiraldictates change of shear in theradial shear zone
tan2
2
1 e
2
1
Normal forces offset by friction,
no moments about centre
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 19
Comment
Details of the geometry of thefailure mechanism are really only of
any practical purpose for
consideration of the influence ofadjacent or underground works, or
the influence of soil profile changes
N.B. = 0 soils have smallest
mechanism
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 20
The General Bearing
Capacity Equation
Ultimate bearing capacity for a strip footing,
subjected to vertical load
other situations handled by empiricalexpressions
qocuu N2
BNqNc1B
Qq
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 21
Considers 3 cases:
1. Surface footing, soil, with cohesion, but no friction
2. Surface footing, soil friction and soil
3. Burial (surcharge & extra shear resistance)
Contributions are simply summed
an approximation,
backed up by experience and experimentation
FORMULATIONLimit Equilibrium Analysis
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 22
Bearing Capacity Factors
Nc, Nq & N are the bearing
capacity factors corresponding to:
1. Cohesion case
2. Surcharge or burial case
3. Self weight of soil case
Nc, NqandNare all functions of
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 23
The Bearing Capacity Factors, Nc, Nq
0
20
40
60
80
100
120
140
160
0 10 20 30 40 50
Angle of Friction (kPa)
Bearing
CapacityFactor
Nc
Nq
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 24
0
5
10
15
20
25
30
0 5 10 15 20 25 30
Angle of Friction (kPa)
Bearing
Cap
acityFactor
Nc Nq
5.14
1.0
NC clay,saturated,short termloading
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 25
The Bearing Capacity Factor, N
0
50
100
150
200
250
0 10 20 30 40 50
Angle of Friction (kPa)
Bearing
CapacityFactor,
N
rough
smooth
Hansen
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 26
0
2
4
6
8
10
12
14
16
0 5 10 15 20 25 30
Angle of Friction (kPa)
B
earing
Capac
ityFactor,
N
rough
smooth
0.0
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 27
qocu N
2
BNqNcq
Cohesion term
Surface
surcharge term
Soil self-weight term
The Bearing Capacity Equation
- for a long strip footing
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 28
Variations in the Bearing Capacity
Factors between methods
1) Hansen [or Brinch-Hansen] analyses
generally accepted as most accurate
2) Terzaghi, the pioneer, misconstrued Nc
3) Meyerhofthe 2nd best of these 3
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 29
Effects of Soil Properties
Bearing capacity = fn(, , B, c and qo)
has the greatest influence
Both the 2nd and 3rd terms in the equation
depend on
if water is above the bottom of the footing,the surcharge weight is also affected
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 30
THE SURCHARGE TERM- most footings are buried
1
qo = 1D
D
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 31
Effect of Footing Size
Last term with N has B orfooting breadth
Wider footing, greater bearing capacity
BUT for = 0 soil, B has little effect
N = 0! for = 0
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 32
Factor of Safety, qu to qsafe?
A Factor of Safety of 3?
oqoc
safe qSofF
NB5.01)(NqNcq
= [nett ultimate bearing capacity (FoS)] + surcharge
= [qu nett (FoS)] + qo
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 33
The General Bearing Capacity Eqn.
Considers soil rigidity, (r)
footing shape, (s)
depth of embedment, (d)
Inclined load, (i) base inclination, (b)
Ground inclination, (g)
qoqqqqqqcccccccu N2B
gbidsrNqgbidsrNcgbidsrq
Note: shape factors not used with inclination factors?
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 34
Effect of depth, D ( u = 0)
D/B Brinch -Hansen
0 5.14cu + qo
1 7.2cu + qo
8.4cu + qo
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 35
Effect of depth, D ( u = 0)
D = 0qu surface
D = B
D >> B
1.4qu surface
1.63qu surface
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 36
The Influence of a WT
If zw
within 1.5D, assume at underside
of footing and use in self weight termBuoyancy if too high?
zw1
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 37
Footing Shapeshape factors B=L
L/B 1 2 5sc 1.19 1.10 1.04
= 0 sq 1.00 1.00 1.00
s 0.6 0.8 0.92
sc 1.61 1.31 1.12
= 30 sq 1.58 1.29 1.12
s 0.6 0.8 0.92
sc 2.01 1.50 1.20= 45 sq 2.00 1.50 1.20
s 0.6 0.8 0.92
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 38
Breduced = B'
Plan
l
Q
e
Eccentricity of Loading
e2BBBeffective
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 39
EXAMPLE 8.7From Barnes, (changed slightly)
A long, reinforced concrete, retaining wall is to be founded at 1.5 m
depth below ground level in a clay with the water table at 1.5 m
below ground level. The width of the footing is 2.5 m and the base
is 1.5 m below ground level. The thickness of the footing is 0.5 m.
The top 1 m of excavation is to be backfilled.
A vertical line load of 90 kN/m is located 0.45 m off the centreline of
the footing.
If c = 4 kPa, = 23 and = 22 kN/m3; cu = 40 kPa and u = 0 ;
concrete = 25 kN/m3 and backfill = 21 kN/m3, then find the factor ofsafety against bearing capacity failure for both short term and long
term conditions
DO NOT ignore depth factors
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 40
Problem from Notes
A column carries 900 kN.
The foundation soil is dry sand, 18 kN/m3, = 40.A minimum factor of safety of 2.5 is required.
FIND the size of :
a) A square footing if it is placed at the ground surfaceb) A rectangular surface footing L/B = 2
c) A square footing if it is placed 1 m below the surface.
d) A square footing, 1 m below the surface - the water
table is expected to rise to the underside of thefooting. Below the water table, the unit weight is 21kN/m3
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 41
ANSWER (a)
e = 0, so, q = 900/B2 kPa
qu req = 2.5 x 900/B2 = 2250/B2
BUT for dry sand, AND a surface footingqu = 0.5(s)BN
2250/B2 = 0.5(s)18BN
250 = (s)B3N = 0.6 x 85.8 B3
SOLVE: B = 1.69 m
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 42
ANSWER (c)
q = 900/B2 kPa
qu req = 2.5 x 900/B2 - sqdqqo = 2250/B
2 - 18sqdq
BUT for dry sand, AND a square footing at 1 m,
qu = sqdqqoNq + 0.5(s)(d)BN
2250/B2 18(1.84)1.16= 1.84 (1.16)(18)64.2 +
0.5(0.6)(1)18B(85.8)
2250/B2
38.4 = 2466 + 463.3B
0 = 0.206B3 + 1.11B2 - 1
SOLVE: B = 0.88 m assumed B = 1 m for depth factors
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 43
ANSWER (b)
q = 900/B2 kPa
qu req = 2.5 x 900/B2 = 2250/B2 - surface footing
BUT for dry sand, and L/B = 2,
qu = 0.5(s)BN
2250/B2 = 0.5(0.8)18B(85.8)
3.642 = B3
SOLVE: B = 1.54 m
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 44
ANSWER (d)
q = 900/B2 kPa
qu req = 2.5 x 900/B2 - sqdqqo = 2250/B
2 - 18sqdq
BUT for wet sand, AND a square footing at 1 m,
qu = sqdqqoNq + 0.5(s)(d) BN
2250/B2 18(1.84)1.16= 1.84 (1.16)(18)64.2 +
0.5(0.6)(1)(11.2)B(85.8)
2250/B2
38.4 = 2466 + 288.2B
0 = 0.128B3 + 1.11B2 - 1
SOLVE: B = 0.90 m assumed B = 1 m for depth factors
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 45
Q
DB
a
Changed shearmechanism
Shallower,
longer?
InclinedLoading
qo D
c , , Q/(BL) =qu/(cosa)
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 46
Inclined Load correction factors(Meyerhof: guide onlyuse Hansen)
FACTOR DEPTH OF
FOOTING
INCLINATION OF LOAD FROM
VERTICAL
D 0 10 20 30
N 0 1.0 0.5 0.2 0
N B 1.0 0.6 0.4 0.25
Nc 0 to B 1.0 0.8 0.6 0.4
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 47
DEFINITION of nfor inclined loading
Inclination of Load
correction factors, ic, iq, i
B
L Q
PLAN
n
Q
B
ELEVATION
a
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 48
Inclined Loading
Has a tremendous influence
Cannot use shape factors (strip solution)
The solution is always for the VERTICAL
component only of the force, Q
Q = [N2 + T2] and qu = N/A
MUST design against sliding arising from
the HORIZONTAL component of force
i.e. T = N(tan )
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 49
CLASS EXAMPLE
A footing 2 m wide by 4 m long is to be placed on adense layer of sand, overlain by poorly compacted fill
(fill = 16 kN/m3) to a depth of 2 m. The sand has the
properties c = 0 kPa, = 38 and = 20 kN/m3.
There is no water table near the footing.
Check the ultimate bearing capacity of the footing for
a vertical central load
a central load inclined at 10 to the vertical (in plan,the load is parallel to the short side or breadth, B, of
the footing)
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 50
Non-Homogeneous Soil
- soil profiles
Approximations can apply, SO LONG AS the
general mechanism of instability remains
NOT THE CASE for very soft clay, in
thin layer, over hard soil
Toothpaste tube?
NOR strong thin layer over weak layer
Punching shear
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 51
Non-Homogeneous Soil?
Stronger layer
Weaker Layer
21
Load spreadingtechnique
Load, QI l l
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 52
Footing, B x L
Depth to layer, DL
New footing area
(B+DL)(L+DL)
Pressure, qL = Q/area
Check qa2
Ensure lower layer not
overstressed by qL
- reduce qa as required
Ignore lower layerand check surface qa
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 53
Example
Returning to the previous question, had
the footing been a surface footing and
the poorly compacted fill had the
properties c = 5 kPa, = 25, compute
the ultimate bearing capacity under
vertical loading .
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 54
Alternative treatment ifu = 0 soils
Layer 1,
cu1
Layer 2 ,
cu2
Centre of
rotation?
B
Radius, B?
O h F
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 55
Other Factors
a) Adjacent footingssuitably spacedto reduce interaction (refer to failure
mechanisms)
b) Rate of loading appropriate
strength parameters?
c) Inclined slopes adjacent to footings
influence on slope stability?
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 56
SUMMARY
Bearing capacity is determined by limit
equilibrium methods
Complexity requires semi-empirical
solutions
Endnote: some new developments willimprove these methods
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 57
GROUND COVERED
1) types of failure
2) general bearing capacity analysis
3) correction factors for depth,shape, inclination
4) influence of eccentric load
5) dealing with soil profiles
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 58
BEARING CAPACITY ON ROCKS
Rock can be stronger than a concrete footing
BUT consideration of rock mass strength may
be necessary
closed joints frequently spaced, can treat as
Bell-Terzaghi bearing problem
ALSO defects and their orientation may impact
upon performance
R k M ??
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 59
Rock Mass??
Close, but open vertical joints
B
S
Unconfined compressivestrength of rock rules!
R k M ??
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Rock Mass??
Widely spaced joints
B
S
Tension splitting of rock slab?Bishnois theoretical solution
Refer: Sowers G F, Introductory Soil Mechanicsand Foundations
Oth C id ti
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DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT 61
Other Considerations
Rigid
Weak
Flexural failure
Punching failure
H
Weak
Rigid, but thinner
H
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KEY POINT
Mechanisms of bearing capacity
failure of rock masses are usually
quite different from soil