Lecture
2
INTERNATIONAL UNIVERSITY FOR SCIENCE & TECHNOLOGY
�م وا����������� ا������ ا��و��� ا����� �
Dr. Abdulmannan Orabi
Civil Engineering and Environmental Department
303421: Foundation Engineering
Bearing Capacity of Foundation
References
ACI 318M-14 Building Code Requirements for Structural Concrete ( ACI 318M -14) and Commentary, American Concrete Institute, ISBN 978-0-87031-283-0.
Bowles , J.,E.,(1996) “Foundation Analysis and Design” -5th ed. McGraw-Hill, ISBN 0-07-912247-7.
Das, B., M. (2012), “ Principles of Foundation Engineering ” Eighth Edition, CENGAGE Learning, ISBN-13: 978-1-305-08155-0.
Syrian Arab Code for Construction 2012
Dr. Abdulmannan Orabi IUST 2
Bearing Capacity of Foundation
The soil must be capable of carrying the loads from any engineered structure placed upon it without a shear failure and with the resulting settlements being tolerable for that structure.
This lecture will be concerned with evaluation of the limiting shear resistance, or ultimate bearing capacity of the soil under a foundation load.����
3Dr. Abdulmannan Orabi IUST
It is necessary to investigate both base shear resistance and settlements for any structure.
Bearing Capacity of Foundation
In many cases settlement criteria will control the allowable bearing capacity; however, there are also a number of cases where base shear (in which a base punches into the ground - usually with a simultaneous rotation) dictates the recommended bearing capacity.
Dr. Abdulmannan Orabi IUST 4
Bearing Capacity of Foundation
Structures such as liquid storage tanks and mats are often founded on soft soils, which are usually more susceptible to base shear failure than to settlement. Base shear control, to avoid a combination base punching with rotation into the soil, is often of more concern than settlement for these foundation types.
Dr. Abdulmannan Orabi IUST 5
Allowable Bearing Capacity
The recommendation for the allowable bearing capacity to be used for design is based on the minimum of either :
����
1. Limiting the settlement to a tolerable amount
2. The ultimate bearing capacity, which considers soil strength, as computed in the following sections
Dr. Abdulmannan Orabi IUST 6
Allowable Bearing Capacity
The allowable bearing capacity based on shear control is obtained by reducing (or dividing) the ultimate bearing capacity (based on soil strength) by a safety factor SF that is deemed adequate to avoid a base shear failure to obtain
����
����
���� =��
�.�(2-1)
The safety factor is based on the type of soil (cohesive or cohesionless), reliability of the soil parameters, structural information (importance, use, etc.), and consultant caution.
Dr. Abdulmannan Orabi IUST 7
Dr. Abdulmannan Orabi IUST
Allowable Bearing Capacity
Most building codes provide an allowable settlement limit for a foundation, whichmay be well below the settlement derived corresponding to given by equations( 2-1). Thus, the bearing capacity corresponding to the allowable settlement must also be taken into consideration.
����
8
BEARING-CAPACITY EQUATIONS
Terzaghi’s Bearing Capacity Theory
Terzaghi (1943) was the first to present a comprehensive theory for the evaluation of the ultimate bearing capacity of rough shallow foundations. According to this theory, a foundation is shallow if its depth, Df (Figure slid 11), is less than or equal to its width. Later investigators, however, have suggested that foundations with Df equal to 3 to 4 times their width may be defined as shallow foundations.
Dr. Abdulmannan Orabi IUST 9
BEARING-CAPACITY EQUATIONS
Terzaghi’s Bearing Capacity Theory
The effect of soil above the bottom of the foundation may also be assumed to be replaced by an equivalent surcharge,
(where is the unit weight of soil).
� = �∗ ��
�
Terzaghi suggested that for a continuous, or strip, foundation (i.e., one whose width-to-length ratio approaches zero), the failure surface in soil at ultimate load may be assumed to be similar to that shown in Figure on Slide 11.
Dr. Abdulmannan Orabi IUST 10
The failure zone under the foundation can be separated into three parts (see Figure 4.6):1. The triangular zone ACD immediately under the foundation2. The radial shear zones ADF and CDE, with the curves DE and DF being arcs of a logarithmic spiral3. Two triangular Rankine passive zones AFH and CEG
BEARING-CAPACITY EQUATIONS
Terzaghi’s Bearing Capacity Theory
Dr. Abdulmannan Orabi IUST 11
BEARING-CAPACITY EQUATIONSTerzaghi’s Bearing Capacity Theory
The angles CAD and ACD are assumed to be equal to the soil friction angle .Note that, with the replacement of the soil above the bottom of the foundation by an equivalent surcharge q, the shear resistance of the soil along the failure surfaces GI and HJ was neglected.
∅�
Dr. Abdulmannan Orabi IUST 12
BEARING-CAPACITY EQUATIONS
Terzaghi’s Bearing Capacity Theory
The ultimate bearing capacity, , of the foundation now can be obtained by considering the equilibrium of the triangular wedge ACD shown in Figure below
����
Dr. Abdulmannan Orabi IUST 15
BEARING-CAPACITY EQUATIONS
Terzaghi’s Bearing Capacity Equation
���� = 0.5������ + ����� + �����
�� , �� , � !���"#�#�"$ %&�'�&$()*�&(+",
and depends on angle of shearing resistance (ø)(.�/. 4 − 1, ��,)
4ℎ#"#:
� = 4$!(ℎ+*(ℎ#*++($ %
� = 7 $(4#$%ℎ(+*(ℎ#,+$8/#8+4(ℎ#*+7 !�($+ 8#9#8
� = � ∗ ��
�� = 0, �� = 1� !�: = 1.5; + 1 = 5.71*+"∅ = 0Dr. Abdulmannan Orabi IUST 16
BEARING-CAPACITY EQUATIONS
Terzaghi’s Bearing Capacity Equation
���� = 0.5������ + ����� + �����
�� , �� , � !�� = �ℎ�'*�&(+",
Shape factors for the Terzaghi equations
Sγ Sq SC
Square 0.8 1 1.3
Circular 0.6 1 1.3
Rectangular 1-0.2 B/L 1 1+0.3 B/L
Dr. Abdulmannan Orabi IUST 17
BEARING-CAPACITY EQUATIONS
Terzaghi’s Bearing Capacity Theoryfor Local Shear Failure
Terzaghi suggested the following relationships for local shear failure in soil:
where
��� , ��
� , � !����"#=+!$*$#!/#�"$ %*�&(+",
���� = 0.5������� + ���
��� + �������
�� =2
3� tan∅� = tan(
2
3∅)
Dr. Abdulmannan Orabi IUST 18
BEARING-CAPACITY EQUATIONS
Meyerhof ’s Bearing Capacity Equation
In 1951, Meyerhof published a bearing capacity theory that could be applied to rough, shallow, and deep foundations. The failure surface at ultimate load under a continuous shallow foundation assumed by Meyerhof is shown in Figure below.
Dr. Abdulmannan Orabi IUST 19
BEARING-CAPACITY EQUATIONS
Meyerhof ’s Bearing Capacity Equation
In this figure abc is the elastic triangular, bcd is the radial shear zone with cd being an arc of a log spiral, and bde is a mixed shear zone in which the shear varies between the limits of radial and plane shears depending on the depth and roughness of the foundation.
The plane be is called an equivalent free surface.
Dr. Abdulmannan Orabi IUST 21
BEARING-CAPACITY EQUATIONS
Meyerhof ’s Bearing Capacity Equation
Vertical load:
���� = 0.5������!� + �����!� + �����!�
���� = 0.5����$�!� + ���$�!� + ���$�!�
Inclined load:
Dr. Abdulmannan Orabi IUST 22
BEARING-CAPACITY EQUATIONS
Meyerhof ’s Bearing Capacity Equation
tan 2( ) tan (45 )2
qN e π φ φ= +
( 1) cotC qN N φ= −
( 1) tan(1.4 )qN Nγ φ= −
where�� , �� , � !�� �"#�#�"$ %&�'�&$()*�&(+",
�� = 0, �� = 1� !�: = ; + 2 = 5.14*+"∅ = 0
Dr. Abdulmannan Orabi IUST 23
BEARING-CAPACITY EQUATIONS
Meyerhof ’s Bearing Capacity Equation
where
�� , �� , � !�� = �ℎ�'#*�&(+",
1 0.1 .......... 10q p
BS S K for
Lγ φ= = + × ≥
1 0 .2c p
BS K
L= + ×
CD = (� E(45 +∅
2)
�� = �� = 1*+"∅ = 0
24Dr. Abdulmannan Orabi IUST
!� , !� , � !!� = �#'(ℎ*�&(+",
BEARING-CAPACITY EQUATIONS
Meyerhof ’s Bearing Capacity Equation
where
1 0 .2 fC p
Dd K
B= + ×
1 0 .1 .... 1 0fq p
Dd d K f o r
Bγ φ= = + × ≥
!� =!� = 1*+"∅ = 0
Dr. Abdulmannan Orabi IUST 25
$� , $� , � !$� = F &8$ �($+ *�&(+",
BEARING-CAPACITY EQUATIONS
Meyerhof ’s Bearing Capacity Equation
where
$� = $� = 1 −G
90
E
� )∅
$� = 1 −G
∅
E
∅ > 0
$� = 0*+"G ≠ 0� !∅ = 0
G<∅
V
H
R
Dr. Abdulmannan Orabi IUST 26
BEARING-CAPACITY EQUATIONS
Hansen’s Bearing Capacity Equation ( General Equation )
Df
V
η
+β
Dr. Abdulmannan Orabi IUST 27
BEARING-CAPACITY EQUATIONS
Hansen’s Bearing Capacity Equation ( General Equation )
'0.5ult q q q q q q C C C C C Cq B N S d i g b qN S d i g b CN S d i g bγ γ γ γ γ γγ= + +
tan 2( ) tan (45 )2
qN e π φ φ= +
( 1) cotC qN N φ= −
( 1) tan(1.4 )qN Nγ φ= −
where�� , �� , � !�� �"#�#�"$ %&�'�&$()*�&(+",
�� = 0, �� = 1� !�: = ; + 2 = 5.14*+"∅ = 0
Dr. Abdulmannan Orabi IUST 28
BEARING-CAPACITY EQUATIONS
Hansen’s Bearing Capacity Equation ( General Equation )
'0.5ult q q q q q q C C C C C Cq B N S d i g b qN S d i g b CN S d i g bγ γ γ γ γ γγ= + +
where
�� , �� , � !�� = �ℎ�'#*�&(+",
'
'1
q
C
C
N BS
N L= + ×
'
'1 sinq
BS
Lφ= +
'
'1 0.4
BS
Lγ = − �� ≥ 0.6
Dr. Abdulmannan Orabi IUST 29
!� , !� , � !!� = �#'(ℎ*�&(+",
where
BEARING-CAPACITY EQUATIONS
Hansen’s Bearing Capacity Equation ( General Equation )
'0.5ult q q q q q q C C C C C Cq B N S d i g b qN S d i g b CN S d i g bγ γ γ γ γ γγ= + +
1 0 .4Cd K= +
21 2 ta n (1 s in )qd Kφ φ= + −
1
1
tan ( ) 1
f f
f f
D DK fo r
B B
D DK fo r
B B
−
= ≤
= f
!� = 1*+"�88∅
30
BEARING-CAPACITY EQUATIONS
Hansen’s Bearing Capacity Equation ( General Equation )
'0.5ult q q q q q q C C C C C Cq B N S d i g b qN S d i g b CN S d i g bγ γ γ γ γ γγ= + +
$� , $� , � !$� = F &8$ �($+ *�&(+",
where
1
1
q
C q
q
ii i
N
−= −
−1
0.5(1 )
cot
iq
f a
Hi
V A C
α
φ= −
+
20.7
(1 )cot
i
f a
Hi
V A C
αγ φ= −
+2
(0.7 / 450)(1 )
cot
o
i
f a
Hi
V A C
αγ
ηφ
−= −
+
2 ≤ OP� !OE ≤ 5
Dr. Abdulmannan Orabi IUST 31
BEARING-CAPACITY EQUATIONS
Hansen’s Bearing Capacity Equation ( General Equation )
'0.5ult q q q q q q C C C C C Cq B N S d i g b qN S d i g b CN S d i g bγ γ γ γ γ γγ= + +
where
%� , %� , � !%� = Q"+7 !*�&(+",
%�� =
RS
147S%� = 1 −
RS
147S
%� = %� = 1 − 0.5(� R T
Dr. Abdulmannan Orabi IUST 32
BEARING-CAPACITY EQUATIONS
Hansen’s Bearing Capacity Equation ( General Equation )
'0.5ult q q q q q q C C C C C Cq B N S d i g b qN S d i g b CN S d i g bγ γ γ γ γ γγ= + +
where
/� , /� , � !/� = ��,#*�&(+",
/�� =
US
147S*+"∅ = 0
/� = 1 −US
147S
/� = exp(−2U(� ∅) /� = exp(−2.7U(� ∅)
U$ "�!$� ,Dr. Abdulmannan Orabi IUST 33
BEARING-CAPACITY EQUATIONS
Hansen’s Bearing Capacity Equation ( General Equation )
���� = 5.14�� 1 +��� + !�
� − $�� − %�
� − /�� + �*+"∅ = 0
Where :
��� = 0.2
��
Y�!�� = 0.4Z
$�� = 0.5 − 1 −
[\
]���
/�� =
US
147S*+"∅ = 0
%�� =
RS
147S
Dr. Abdulmannan Orabi IUST 34
The equation for ultimate bearing capacity by Terzaghihas been developed based on assumption that water table is located at a great depth .If the water table is located close to foundation ; the equation needs modification.The effective unit weight of the soil is used in the bearing-capacity equations for computing the ultimate capacity.
BEARING-CAPACITY EQUATIONS
Effect of Water Table on Bearing Capacity
Dr. Abdulmannan Orabi IUST 35
BEARING-CAPACITY EQUATIONS
Effect of Water Table on Bearing Capacity
The water table is seldom above the base of the footing, as this would, at the very least, cause construction problems. If it is, however, the q term requires adjusting so that the surcharge pressure is an effective value and an effective unit weight must be used in the 0.5 ϒB Nϒ term.
Water table above the base of footing
Water table below the base of footing
!^
�� ��
G.W.T.
G.W.T.
Dr. Abdulmannan Orabi IUST 36
BEARING-CAPACITY EQUATIONS
Effect of Water Table on Bearing Capacity
�� = 2[ − !^!^
[E� +
��
[E[ − !^
E
4ℎ#"#[ = 0.5� tan( 45 + ∅/2)
!^ = !#'(ℎ+*4�(#"(�/8#/#8+4/�,#+**++($ %
� = 7 $(4#$%ℎ(+*,+$8$ !#'(ℎ!^
�� = ,7/=#"%#!7 $(4#$%ℎ(/#8+44�(#"(�/8#
When the water table lies within the wedge zone, can compute the average effective weight of the soil in the wedge zone as
��
Dr. Abdulmannan Orabi IUST 37
When the water table is below the wedge zone [depth approximately ], the water table effects can be ignored for computing the bearing capacity.
BEARING-CAPACITY EQUATIONS
Effect of Water Table on Bearing Capacity
[ = 0.5� tan( 45 + ∅/2)
Water table below the base of the footing
!^ > [
��
G.W.T.
Dr. Abdulmannan Orabi IUST 38
BEARING-CAPACITY EQUATIONS
Effect of Soil Compressibility
The change of failure mode is due to soil compressibility, to account for which Vesic (1973) proposed the followingmodification of bearing capacity equation :
���� = 0.5������!��� + �����!��� + �����!���
�� , �� � !�� �"#,+$8&+='"#,,$/$8$()*�&(+",
Where :
Dr. Abdulmannan Orabi IUST 39
BEARING-CAPACITY EQUATIONS
Effect of Soil Compressibility
The soil compressibility factors were derived by Vesic(1973) by analogy to the expansion of cavities.
According to that theory, in order to calculate ,the following steps should be taken:
��, ��� !��
Step 1. Calculate the rigidity index, Ir, of the soil at a depth approximately B/2 below the bottom of the foundation, or
F̀ =Qa
�� +��(� ∅�4ℎ#"#
Qa = ,ℎ#�"=+!787,+*,+$8
�� = #**#&($9#+9#"/7"!# '"#,,7"#�(�!#'(ℎ(�� +b
E)
Dr. Abdulmannan Orabi IUST 40
BEARING-CAPACITY EQUATIONS
Effect of Soil Compressibility
Step 2. The critical rigidity index, , can be expressed asF̀ (�`)
F̀ (�`) =1
2#c' 3.3 − 0.45
�
Y&+( 45 −
∅�
2
Step 3. If , thenF̀ ≥ F̀ (�`) �� =�� =�� = 1
However, if , thenF̀ < F̀ (�`)
�� = �� = #c' −4.4 + 0.6�
Y(� ∅� +
3.07,$ ∅� 8+%2F̀
1 + ,$ ∅�
�: = 0.32 + 0.12�
Y+ 0.68+%F̀ *+"∅ = 0
�: = �� − 1 − ��
��(� ∅�*+"∅ > 0
Dr. Abdulmannan Orabi IUST 41
BEARING-CAPACITY from SPT
Bearing Capacity from SPT
The SPT is widely used to obtain the bearing capacity of soils directly.
Considering the accumulation of field observations and the stated opinions of the authors and others, this author adjusted the Meyerhof equations for an approximate 50 percent increase in allowable bearing capacity to obtain the following:
2
55 0.3(1 0.33 )
0.08
fa
N DBq
B B
+ = +
� > 1.2= � ≤ 1.2=d + e. ffgh
i≤ d. ff
Dr. Abdulmannan Orabi IUST 42
BEARING-CAPACITY of Mat Foundation
The gross ultimate bearing capacity of a mat foundation can be determined by the same equation used for shallow foundations, or
���� = 0.5������$�!� + �����$�!� + �����$�!�
The term B in Eq. above is the smallest dimension of the mat.
The net ultimate capacity of a mat foundation is
����(jk�) = ���� − �
Dr. Abdulmannan Orabi IUST 43
BEARING-CAPACITY of Mat Foundation
The net allowable soil bearing capacity
����(jk�) =����(jk�)
�. l
For mats on clay, the factor of safety should not be less than 3 under dead load or maximum live load.
Under most working conditions, the factor of safety against bearing capacity failure of mats on sand is very large.
Dr. Abdulmannan Orabi IUST 44
Dr. Abdulmannan Orabi IUST
BEARING-CAPACITY of Mat Foundation
Bearing Capacity from SPT
The net allowable bearing capacity for mats constructed over granular soil deposits can be adequately determined from the standard penetration resistance numbers
����(jk�) =�TT
0.08ln
�k(==)
25
����(jk�) = 16.63�TT�k(==)
25
45