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Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 20
CHAPTER TWO
BEARING CAPACITY OF SHALLOW FOUNDATIONS
Table of Contents
2.0 Introduction ........................................................................................ 21
2.1 Bearing Failure Modes .......................................................................... 21
2.2 Ultimate Bearing Capacity Equations ...................................................... 22
2.2.1 Terzaghi’s Bearing Capacity equation ................................................... 22
2.2.2 Meyerhof’s Bearing Capacity equation .................................................. 24
2.2.3 Hansen’s Bearing Capacity Equation .................................................... 25
2.2.4 A comparative summary of the three bearing capacity equations ............ 28
2.2.5 Allowable bearing capacity and factor of safety ..................................... 30
2.2.6 Eccentric Loads ................................................................................. 31
2.3 Field Tests ........................................................................................... 32
2.3.1 Plate Loading Test ............................................................................. 32
2.3.2 Standard Penetration Test (SPT) ......................................................... 33
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 21
2.0 Introduction
A foundation, often constructed from concrete, steel or wood, is a structure
designed to transfer loads from a superstructure to the soil underneath the
superstructure. In general, foundations are categorized into two groups, namely,
shallow and deep foundations. Shallow foundations are comprised of footings, while
deep foundations include piles that are used when the soil near the ground surface has
no enough strength to stand the applied loading. The ultimate bearing capacity, qu,
(in kPa) is the load that causes the shear failure of the soil underneath and adjacent
to the footing. In this chapter, we will discuss equations used to estimate the ultimate
bearing capacity of soils. When you complete this chapter you should be able to:
� Calculate the bearing capacity of soils.
2.1 Bearing Failure Modes
Figure 2.1: Modes of bearing failures (a) General shear (b) Local shear and (c)
Punching shear.
Relative density of the soil and size of the foundation are among the major factors that
affect the mode of bearing failure likely to occur. The modes of bearing failure are
generally separated into three categories: The general shear failure (Fig. 1.1 a) is
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 22
usually associated with soils of low compressibility such as dense sand and stiff
cohesive soils. In this case, if load is gradually applied to the foundation, settlement
will increase. At a certain point – when the applied load per unit area equals to the
ultimate load qu – a sudden failure in the soil supporting the foundation will take place.
The failure surface in the soil will extend to the ground surface and full shear
resistance of the soil is developed along the failure surface. Bulging of the soil near the
footing is usually apparent.
For the local shear failure (Fig. 1.1 b), which is common in sands and clays of
medium compaction, the failure surface will gradually extend outward from the
foundation but will not reach the ground surface as shown by the solid segment in Fig.
1.1 b. The shear resistance is fully developed over only part of the failure surface
(solid segment of the line). There is a certain degree of bulging of the soil.
In the case of punching shear failure, a condition common in loose and very
compressible soils, considerable vertical settlement may take place with the failure
surfaces restricted to vertical planes immediately adjacent to the sides of the
foundation; the ground surface may be dragged down. After the first yield has
occurred the load-settlement curve will be steep slightly, but remain fairly flat.
2.2 Ultimate Bearing Capacity Equations
2.2.1 Terzaghi’s Bearing Capacity equation
Many of the present day principles regarding bearing capacity equations appear
to have had their origin on a failure mechanism proposed by Prandtl in the early 1920s
(refer literature for Prandtl’s failure mechanism). Prandtl developed a bearing capacity
Figure 2.2: Failure mechanism for Terzhagi’s bearing capacity solution.
equation assuming a smooth (frictionless) footing and ignoring the weight of the soil
in the failure zone. These assumptions are not true in practice and therefore Prandtl’s
equation is never used in practical design, but it was a beginning.
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 23
Terzhagi (1943) improved the Prandtl equation to include the roughness of the
footing and the weight of the failure zone. The failure mechanism in a c’, φ’ soil for
Terzhagi’s bearing capacity solution is shown in Fig. 2.2. Terzhagi’s ultimate bearing
capacity equations are given as follows:
Strip (or long) footing: γγγ NBDNNcq qcu 5.0' ++= (2.1)
Square footing: γγγ NBDNNcq qcu 4.0'3.1 ++= (2.2)
Circular footing: γγγ NBDNNcq qcu 3.0'3.1 ++= (2.3)
where Nc, Nq and Nγ are called the bearing capacity factors and are obtained as
follows:
)2/'45(cos2 2
'tan)'2/3(
φ
φφπ
+=
−eN q , )1('cot −= qc NN φ ,
−= 1
'cos'tan
221
φφ γ
γpK
N (2.4)
Figure 2.3: Terzhagi’s bearing capacity coefficients.
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 24
Figure 2.3 shows the variation of the bearing capacity factors provided by Terzhagi.
Based on this figure, Aysen (2002) proposed the following equation to obtain the
value of Kpγ in the Nγ equation:
)2/'60(tan)8.3'4'8( 022 φφφγ ++−=pK (2.5)
Where 'φ in the first term is in radians. In the undrained conditions (cu and 0=uφ ):
1=qN , 71.5)1( 23 =+= πcN , 0=γN (2.6)
2.2.2 Meyerhof’s Bearing Capacity equation
Meyerhof (1951) developed a bearing capacity equation by extending Terzhagi’s
failure mechanism and taking into account the effects of footing shape, load
inclination and footing depth by adding the corresponding factors of s, d, and i. For a
rectangular footing of L by B (L > B) and inclined load:
γγγγγγ disNBdisDNdisNcq qqqqccccu 5.0' ++= (2.7)
For vertical load, ic = iq = iγ = 1
γγγγγ dsNBdsDNdsNcq qqqcccu 5.0' ++= (2.8)
Figure 2.4: Meyerhof’s bearing capacity coefficients.
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 25
The bearing capacity factors:
)2/'45(tan)'tanexp( 2 φφπ +=qN , )1('cot −= qc NN φ , )'4.1tan()1( φγ −= qNN (2.9)
In the undrained conditions (cu and 0=uφ ):
1=qN , 71.5)2( =+= πcN , 0=γN
The bearing capacity factors are graphically presented in Fig. 2.4. The shape,
inclination and depth factors are according to:
Shape Depth Inclination
Any 'φ L
BKs pc 2.01+=
B
DKd pc 2.01+=
2
0
0
901
−== α
qc ii
For 0'=φ sq = s γ= 1 dq = d γ= 1 i γ= 0
For 010'≥φ
L
BKss pq 1.01+== γ
B
DKdd pq 1.01+== γ
2
0
0
'1
−=
φα
γi
+=2
'45tan2 φ
pK , α =angle of resultant measured from vertical axis.
when triaxial 'φ is used for plane strain, adjust 'φ to obtain 'triaxialφφ
−=L
B1.01.1'
For the eccentric load, the length and width of the footing rectangle are modified to:
L’ = L – 2eL and B’ = B – 2eB (2.9)
where eL and eB represent the eccentricity along the appropriate directions.
2.2.3 Hansen’s Bearing Capacity Equation
Hansen (1961) extended Meyerhof’s solutions by considering the effects of
sloping ground surface and tilted base (Fig. 2.5) as well as modification of Nγ and other
factors. For a rectangular footing of L by B (L > B) and inclined ground surface, base
and load:
γγγγγγγγ gbidsNBgbidsDNgbidsNcq qqqqqqccccccu 5.0' ++= (2.10)
Equation 2.9 is sometimes referred to as the general bearing capacity equation. In the
special case of a horizontal ground surface,
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 26
γγγγγγγ bidsNBbidsDNbidsNcq qqqqqcccccu 5.0' ++= (2.11)
Figure 2.5: Identification of items in Hansen’s bearing capacity equation.
Figure 2.6 provides the relationships between Nc, Nq, and Nγ and the 'φ values, as
proposed by Hansen.
Figure 2.6: Hansen’s bearing capacity coefficients.
The bearing capacity factors Nc and Nq are identical with Meyerhof’s factors. Nγ is
defined by:
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 27
φγ tan)1(5.1 −= qNN (2.12)
Since failure can take place either along the long side or along the short side, Hansen
proposed two sets of shape, inclination and depth factors.
The shape factors are:
Bcc
qBc i
L
B
N
Ns ,, 1 ⋅+= , 'sin1 ,, φ⋅+= BqBq i
L
Bs , 6.04.01 ,, ≥−= BB i
L
Bs γγ (2.13)
Lcc
qLc i
B
L
N
Ns ,, 1 ⋅+= , 'sin1 ,, φ⋅+= LqLq i
B
Ls , 6.04.01 ,, ≥−= LL i
B
Ls γγ (2.14)
For cu, φu=0 soil: BcBc iL
Bs ,, 2.0= , LcLc i
B
Ls ,, 2.0= (2.15)
The inclination factors are:
1
1 ,,, −
−−=
q
iqiqic
N
iii ,
1
'cot
5.01,
α
φ
+−=
b
iiq
AcV
Hi ,
2
'cot
7.01,
α
γ φ
+−=
b
ii
AcV
Hi (2.16)
where the suffix i (in Eqn. 2.15) stands for B or L. 52 1 ≤≤ α . 52 2 ≤≤ α . A is the area
of the footing base and cb is the cohesion mobilized in the footing-soil contact area. For
the tilted base:
2
'cot
)4507.0(1
00
,
α
γ φη
+−−=
b
ii
AcV
Hi (2.17)
For cu, φu=0 soil: biic AcHi −−= 15.05.0, (2.18)
In the above equations, B and L may be replaced by their effective values (B’ and L’)
expressed by Eqn. (2.9).
The depth factors are expressed in two sets:
For D/B ≤ 1 & D/L ≤ 1:
BDd Bc ⋅+= 4.01, , B
Dd Bq ⋅−+= 2, )'sin1('tan21 φφ (2.19)
LDd Lc ⋅+= 4.01, , L
Dd Lq ⋅−+= 2, )'sin1('tan21 φφ (2.20)
For D/B > 1 & D/L > 1:
( )BDd Bc
1, tan4.01 −⋅+= , )(tan)'sin1('tan21 12
, BDd Bq
−⋅−+= φφ (2.21)
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 28
( )LDd Lc
1, tan4.01 −⋅+= , )(tan)'sin1('tan21 12
, LDd Lq
−⋅−+= φφ (2.22)
For both sets: 1=γd (2.23)
For cu, φu soil: BDd Bc ⋅= 4.0, , L
Dd Lc ⋅= 4.0, (2.24)
For the sloping ground and tilted base, the ground factors gi and base factors bi are
proposed by the following equations. The angles β and η are at the same plane, either
parallel to B or L.
0
0
1471 β−=cg , ( )5tan5.01 βγ −== ggq (2.25)
For cu, φu soil: 0
0
147β=cg (2.26)
0
0
1471 η−=cb ,
'tan2 φη−= ebq , 'tan7.2 φη
γ−= eb (2.27)
For cu, φu soil: 0
0
147η=cb (2.28)
2.2.4 A comparative summary of the three bearing capacity equations
Terzaghi’s equations were and are still widely used, perhaps because they are
somewhat simpler than Meyerhof’s and Hansen’s. Practitioners use Terzaghi’s
equations for a very cohesive soil and D/B < 1. However, Terzaghi’s equations have
the following major drawbacks:
� Shape, depth and inclination factors are not considered.
� Terzaghi’s equations are suitable for a concentrically loaded horizontal
footing but are not suitable for eccentrically (for example, columns with
moment or titled forces) loaded footings that are very common in practice.
� The equations are generally conservative than Meyerhof’s and Hansen’s.
Currently, Meyerhof’s and Hansen’s equations are more widely used than
Terzaghi’s. Both are viewed as somewhat less conservative and applicable to more
general conditions. Hansen’s is, however, used when the base is tilted or when the
footing is on a slope and for D/B > 1.
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 29
EXAMPLE 2.1
Given the data in Fig. E2.1, determine the ultimate bearing capacity qu using:
a)Terzaghi’s, b) Meyerhof’s and c) Hansen’s bearing capacity equations.
Figure E2.1: An isolated footing.
EAMPLE 2.2
Determine the ultimate bearing capacity of a square footing 1.5 m, at a depth of 1 m
in a soil c’ = 10 kPa, 'φ =280, cu = 105 kPa, uφ =0 andγ = 19 kN/m3. Use Terzaghi’s,
Meyerhof’s and Hansen’s bearing capacity equations.
Strategy It is a good policy to sketch a diagram illustrating the conditions given.
EAMPLE 2.3
A square footing 1.5 m is to be constructed in sand with c’ = 0, 'φ =400. The thickness
of the footing is 0.45 m and its top surface is level with the horizontal ground surface.
The footing is subjected to a central vertical force of 700 kN and a central horizontal
force (parallel to the sides) of 210 kN. Find the ultimate bearing capacity by a)
Meyerhof’s and b) Hansen’s equations. (Note that Terzaghi’s equations are not
applicable for inclined loads). The unit weight of the sand is 18 kN/m3.
2.2.1 Effects of Groundwater Table on Bearing Capacity
For all the bearing capacity equations, you will have to make some adjustments
for the groundwater condition. The term Dγ in the bearing capacity equations refers
to the vertical stress of the soil above the base of the footing. The last term Bγ refers
to the vertical stress of a soil mass of thickness B, below the base of the footing. You
need to check which one of the three groundwater situations is applicable to your
project.
Situation 1: Groundwater level at a depth B below the base of the footing. In this
case no modification of the bearing capacity equations is required.
Situation 2: Groundwater level within a depth B below the base of the footing. If
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 30
the groundwater level is at a depth z below the base, such that z < B, then the term
Bγ is )(' zBz −+γγ or )(' zBzsat −+γγ . The later equation is used if the soil above
the groundwater level is also saturated. The term Dγ remains unchanged.
Situation 3: Groundwater level within the embedment depth. If the groundwater
is at a depth z within the embedment such that z < D, then the term Dγ is
)(' zDz −+γγ or )(' zDzsat −+γγ . The latter equation is used if the soil above the
groundwater level is also saturated. The term Bγ becomes B'γ .
Figure E2.7: Groundwater within a) a depth B below base, b) embedment depth.
EAMPLE 2.4
Re-do example 2.3 assuming that the groundwater level is at the footing level (0.45 m
below the ground surface). The saturated unit weight is 21 kN/m3.
2.2.5 Allowable bearing capacity and factor of safety
The allowable bearing capacity, qa is calculated by dividing the ultimate bearing
capacity by a factor, called the factor of safety, FS. The FS is intended to compensate
for assumptions made in developing the bearing capacity equations, soil variability,
inaccurate soil data, and uncertainties of loads. The magnitude of FS applied to the
ultimate bearing capacity may be between 2 and 3. The allowable bearing capacity is:
FS
qq u
a = (2.29)
Alternatively, if the maximum applied foundation stress max)( aσ is known and the
dimension of the footing is also known then you can find a factor of safety by replacing
qa by max)( aσ in Eqn. (2.29):
max)( a
uqFS
σ= (2.30)
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 31
2.2.6 Eccentric Loads
Meyerhof (1963) proposed an approximate method for loads that are located
off-centered (or eccentric loads).
Figure A1
He proposed that for a rectangular footing of width B and length L, the base area
should be modified with the following dimensions:
B’ = B – 2eB and L’ =L - 2eL (1)
Where B’ and L’ are the modified width and length, eB and eL are the eccentricities in
the directions of the width and length, respectively. From your course in mechanics
you should recall that
P
Me y
B = and P
Me x
L = (2)
where P is the vertical load, and My and Mx are the moments about the y and x axes,
respectively, as shown in Fig. A1.
The maximum and minimum vertical stresses along the x axis are:
+=B
e
BL
P B61maxσ and
−=B
e
BL
P B61minσ (3)
and along the y axis are:
+=B
e
BL
P L61maxσ and
−=B
e
BL
P L61minσ (4)
Since the tensile strength of soils is approximately zero, minσ should always be
greater than zero. Therefore, eB & eL should always be less than B/6 & L/6, respectively.
The bearing capacity equations are modified for eccentric loads by replacing B with B’.
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 32
EXAMPLE 2.5
A footing 2 m square is located at a depth of 1 m below the ground surface in a deep
deposit of compacted sand, 'φ =300, c’=0, and satγ =18 kN/m3. The footing is
subjected to a vertical load of 500 kN and a moment about the Y-axis of 125 kN・m.
The ground water table is 5 m below the ground surface. Use Meyerhof’s bearing
capacity equation and calculate the factor of safety. Assume the soil above the ground
water is also saturated.
2.3 Field Tests
Often, it is difficult to obtain undisturbed samples of especially coarse-grained
soils for laboratory testing and one has to use results from field tests to determine the
bearing capacity of shallow foundations. Some of the most common methods used for
field tests are briefly described below.
2.3.1 Plate Loading Test
Tests on full sized footings are desirable but expensive. The alternative is to
carry out plate loading tests. The plate loading test is carried out to estimate the
bearing capacity of single footings. The plates that are used in the field are usually
made of steel and are 25 mm thick and 150 mm to 762 mm in diameter. A circular
plate of 300 mm is commonly used in practice. Occasionally, square plates that are
300 mm×300 mm are also used.
To conduct a plate load test, a hole is excavated (Fig. 2.8) with a minimum
diameter 4BP (BP = diameter of the test plate) to a depth of D (D = depth of the
proposed foundation). The plate is placed at the center of the hole. Load is applied to
the plate in increments of 10% to 20% of the estimated ultimate load. Each load
increment is held until settlement ceases. The final settlement at the end of each
loading increment is recorded. The test should be conducted until the soil fails, or at
least until the plate has gone through 25 mm of settlement.
Figure 2.8: Plate Loading Test
For tests in clay,
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 33
)()( PuFu qq = (2.31)
where qu(F) & qu(P) are ultimate bearing capacity of foundation and plate, respectively.
Eqn. (2.31) implies that the bearing capacity in clays is independent of plate size.
For tests in sandy soil,
p
FPuFu
B
Bqq )()( = (2.32)
where BF and BP stand for width of foundation and plate, respectively.
There are several problems associated with the plate load test. The test is
reliable if the soil layer is thick and homogeneous, local conditions such as a pocket of
weak soil near the surface of plate can affect the test results but these may have no
significant effect on the real footing, the correlation between plate load results and
real footing is problematic, and performance of the test is generally difficult.
2.3.2 Standard Penetration Test (SPT)
The Standard Penetration Test (SPT) is used to determine the allowable bearing
capacity of cohesionless coarse-grained soils such as sands. The test procedure for
SPT has been introduced in Chapter 1. The N values obtained from SPT are usually
corrected for various effects such as overburden pressure and energy transfer. The
following are two of the most commonly used methods in practice for correcting the N
values.
2;8.95
'0
≤
= N
zN cc
σ (Liao and Whitman, 1985) (2.33)
kPa 24,2;1916
log77.0 '0'
010 >≤
= zN
zN cc σ
σ (Peck et al., 1974) (2.34)
where cN is a correction factor for overburden pressure, and '0zσ is the effective
overburden pressure in kPa. A further correction factor is imposed on N values if the
groundwater level is within a depth B below the base of the footing. The groundwater
correction factor is:
)(22
1
BD
zcW +
+= (2.35)
where z is the depth to the groundwater table, and D and B are the footing depth and
width. If the depth of the groundwater table is beyond B from the footing base cW = 1.
The corrected N value is:
NccN WN=cor
Addis Ababa University, Faculty of Technology, Department of Civil Engineering
Soil Mechanics II: Lecture Notes Instructor: Dr. Hadush Seged 34
Meyerhof (1956, 1974) proposed the following equations to determine the allowable
bearing capacity qa from SPT values.
dea kNSq cor25
12= B ≤ 1.22 m (2.36)
dea kB
BNSq
2305.0
25
8
+= cor B > 1.22 m (2.37)
where Se is the elastic settlement of the layer in mm and kd = 1 + 0.33D/B ≤ 1.33.
In practice, each value of N is a soil layer up to a depth B below the footing base is
corrected and an average value of Ncor is used in Eqn. (2.37).
Bowles (1996) modified Meyerhof’s equations by 50% increase in the allowable
bearing capacity. Bowles’s equations are:
dea kNSq cor25
20= B ≤ 1.22 m (2.36)
dea kB
BNSq
2305.0
25
5.12
+= cor B > 1.22 m (2.37)
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