Foundation Analysis and Design
Content
This document downloaded from
vulcanhammer.net
since 1997,
your source for engineering information
for the deep foundation and marine
construction industries, and the historical
site for Vulcan Iron Works Inc.
Use subject to the “fine print” to the
right.
All of the information, data and computer software
("information") presented on this web site is for
general information only. While every effort will
be made to insure its accuracy, this information
should not be used or relied on for any specific
application without independent, competent
professional examination and verification of its
accuracy, suitability and applicability by a licensed
professional. Anyone making use of this
information does so at his or her own risk and
assumes any and all liability resulting from such
use. The entire risk as to quality or usability of the
information contained within is with the reader. In
no event will this web page or webmaster be held
liable, nor does this web page or its webmaster
provide insurance against liability, for any
damages including lost profits, lost savings or any
other incidental or consequential damages arising
from the use or inability to use the information
contained within.
This site is not an official site of Prentice-Hall, the
University of Tennessee at Chattanooga, Vulcan
Foundation Equipment or Vulcan Iron Works Inc.
(Tennessee Corporation). All references to
sources of equipment, parts, service or repairs do
not constitute an endorsement.
Don’t forget to visit our companion site http://www.vulcanhammer.org
ENCE 461
Foundation Analysis and
Design
Spread Footings: Structural Design, Flexural Design
Mat Foundations (Part I)
Reinforcing Steel for Flexural
Loads
Concrete's weakness in tension; thus, reinforcing
steel must be added when tension is anticipated,
which is virtually guaranteed with flexural
loading
Reinforcing steel in foundation almost inevitably
involves use of reinforcing bars (rebar); welded
wire fabric, needles, etc., are not generally used
Since flexural stresses are usually small, Grade 40 (Metric
Grade 300, fy = 40 ksi or 300 MPa) steel is usually adequate,
although unavailable for bars larger than #6, in which case
Grade 60 (Metric Grade 420, fy = 60 ksi or 420 MPa) steel
may have to be used
Principles for Flexural Design
Factored moment on the critical surface Muc
determines the necessary dimensions of the
member and the necessary size and location of
the reinforcing bars
This can be a complex process; however,
geotechnical considerations tend to simplify the
design process, as it dictates some of the options
Amount of the steel required flexure depends
upon the effective depth d
Principles for Flexural Design
Effective depth
Compression
Tension
Principles for Flexural Design
Nominal moment capacity of a flexural member
made of reinforced concrete with f'c < 30 MPa (4
a
ksi)
M n A s f y d
2
d f y
a
0.85 f' c
As
bd
Principles for Flexural Design
Setting Mu = Mn (where Mu = factored moment
at the section being analysed), As can be solved
to
f' c b
2.353 M uc
2
As
d d
f' c b
1.176 f y
Principles for Flexural Design
Variables for equation:
As = cross-sectional area of reinforcing steel
f'c = 28-day compressive strength of concrete
fy = yield strength of reinforcing steel
= steel ratio
b = width of flexural member
d = effective depth
= 0.9 for flexure in reinforced concrete
Muc = factored moment at the section being analysed
2.353 M uc
f' c b
2
d d
A s
1.176 f y
f' c b
Minimum and Maximum Steel
Minimum Steel Requirements (ACI 10.5.4 and
7.12.2
For Grade 40 (Metric Grade 300) Steel: As > 0.002 Ag
For Grade 60 (Metric Grade 420) Steel: As > 0.0018
Ag
Ag = Gross Cross-sectional area
= As/Ag
Maximum Steel Requirements (ACI 10.3) never
govern the design of spread footings, but may be
of concern in combined footings and mats
Minimum and Maximum
Spacing
Selection of reinforcing bar size and spacing
must satisfy the following minimum and
maximum spacing requirements
Clear space between bars must be at least equal to db,
25 mm (1"), or 4/3 times the nominal aggregate size
Centre-to-centre spacing of the reinforcement must not
exceed 3T or 500 mm (18"), whichever is less
Development Length
Development length Id is the length rebars must
extend through the concrete in order to develop
proper anchorage
Assumptions for calculations of minimum
development length
Clear spacing between the bars is at least 2db
Concrete cover is at least db
Development Length
Development length Id
(English Units)
Development Length
Id (SI Units)
Id 3 f y
Id 9 f y
d b 40 f' c c K tr d b 10 f' c c K tr
db
db
Atr f yt
Atr f yt
K tr
K tr
1500 s n
10 s n
For spread footings use Ktr = 0, which is conservative
Development Length
Variables for development length variables
Id = minimum required development length (in., mm)
db = nominal rebar diameter (in, mm)
fy = yield strength of reinforcing steel (psi, MPa)
fyt = yield strength of transverse reinforcing steel (psi,
MPa)
f'c = 28-day compressive strenght of concrete (psi,
MPa)
Development Length
Variables for development length variables
= reinforcement location factor
!
"#
$
%
$
!
&$
!
"#
$
'
$
Development Length
Variables for development length variables
(
()
*&
*+$
(
*,
*$
-
' $
'$
!
% $
.
$
'
/ /
/
!
Development Length
Variables for development length variables
0
/
%
!
$
!
!
!
'
$%
!
.1#
%
%# 2
"
' /
/ !
%
$
.
3 45 6$ 7
8
$' 7,
9%
! :$;
Application to Spread Footings
a)
Footings in reality
bend in two
perpendicular
directions
b)
Footings are designed
as if they bend in only
one direction
Justification of One-Way Slab
Assumption
The full-scale load tests on which this analysis
method is based were interpreted this way
Foundations should be designed more
conservatively than the superstructure
Flexural stresses are low, so amount of steel
required is nominal and often governed by min
Additional construction cost due to this
simplified approach is minimal
Once footing is analysed one way, we place the
same steel area in the perpendicular direction
Steel Area
Usual procedure is to prepare a moment diagram
and select an appropriate amount of steel for each
portion of the member
For spread footings, we can simplify this by
identifying a critical section for bending and use
the moment determined there to design the steel
for the entire footing
Location of critical section for bending depends
upon the type of column being used
Critical Section for Bending
Moment at Critical Bending
Based on
Section
soil bearing
Factored
Assumes Pu acts
through centroid
of footing
bending moment
2
Pu l 2 M u l
M uc
2B
B
pressure with
assumed
eccentricity
of B/3
Muc = factored moment at critical section for bending
Pu = factored compressive load from column
Mu = factored moment load from column
l = cantilever distance
B = footing width
Design Cantilever Distances
Concrete Columns
l = (B – c/2)/2
Steel Columns
Notes
Masonry Columns
l = (B-c)/2
l = (2B – (c + cp))/4
Variables
B = footing width
c = column width
cp = base plate width
Treat timber columns in
same way as concrete
columns
If column has circular,
octagonal or other
similar shape, use a
square with equivalent
cross-sectional area
If column has a circular
or regular polygon crosssectional area, base the
analysis on an equivalent
square
Computation of Steel Area
Compute Muc
Find the steel area As and reinforcement ratio
2.353 M uc
f' c b
2
d d
A s
1.176 f y
f' c b
a
M n A s f y d
2
d f y
a
0.85 f' c
As
bd
: 7
.'
# <
= # "'
!
,
$
Computation of Development
Length
The development length is measured from the
critical section for bending to the end of the bars
(usually 70 mm (3") from the end of the footing,
even if loads don't require it)
Supplied development length
I d supplied l 70 mm 3 in
(Id)supplied = supplied development length
l = cantilever distance
This length must be greater than the required
development length. If not, best solution is to use
smaller rebars with shorter development lengths
Example of Flexure Design
Given
Find
Foundation design used in shear design example, T =
27"
Required steel area for flexure
Solution
Find Required Steel Area
Determine cantilever distance l
Bc 12621
l
52.5
2
2
Example of Flexure Design
Given
Find
Foundation design used in shear design example, T =
27"
Required steel area for flexure
Solution
Find Required Steel Area
Determine cantilever distance l
Bc 12621
l
52.5 inches
2
2
Example of Flexure Design
Find Required Steel Area
Determine factored bending moment at critical section
2
Pu l 2 M u l
M uc
2B
B
2
991,000 52.5
0
M uc
2 126
M uc 10,800,000 in-lbs
Example of Flexure Design
Find Required Steel Area
Determine steel area based on factored bending
moment at critical section
f' c b
2.353 M uc
2
A s
d d
f' c b
1.176 f y
4000 126
2.353 10,800,000
As
23
1.176 60,000
0.9 4000 126
2
A s 8.94 in
For flexure in
reinforced concrete
Example of Flexure Design
Check computed steel area against minimum
steel requirements
For Grade 60 steel, As > 0.018 Ag = 0.018 B T
Minimum steel area = (0.018)(27)(126) = 6.12 in2
Since As = 8.94 sq.in. > 6.12 sq.in., OK
Use #8 (1") bars: Area of each bar = 0.79 sq. in.
Number of bars = 8.94/0.79 = 11.3 so use 12 bars
Space between bars = 126/12 = 8.7"
Minimum spacing = 18" or 3T = (3)(27) = 81" so OK
either way
Example of Flexure Design
Check development length
(Id)supplied = l – 3 = 52.5 – 3 = 49.5"
Check required inequalities
3 45 6$ 7 > 46 ;
8
$' 7,>7,'
% $
'
$
9%
! :$;><+ ;.
$%
!
Example of Flexure Design
Check development length
Id 3 f y
d b 40 f' c c K tr
db
I d 3 60000 1 1 1 1
2.5
d b 40 4000
For 1" bars, I = 28"
Id
Since (I )
= 49.5,
28
development length is OK
db
d
d applied
Example of Flexure Design
Final Design
Mat Foundations
Reasons to Consider Raft or
Mat Foundations
Structural loads are so high and soil conditions
are so poor that spread footings would be
exceptionally large.
If spread footings are more that 1/3 of building
footprint area, a mat or a deep foundation would
probably be more economical
Soil is very erratic or prone to differential
settlements, or soils that are expansive or subject
to differential heaves
Structural loads are erratic; flexural strength of
the mat will absorb these irregularities
Reasons to Consider Raft or
Mat Foundations
Lateral loads are not uniformly distributed
through the structure and thus may cause
differential horizontal movements in spread
footings or pile caps.
Uplift loads are larger than spread footings can
accommodate.
Bottom of the structure is located below the
groundwater table, so waterproofing is an
important concern. Mats are monolithic, so
easier to waterproof. Weight of mat also resists
hydrostatic uplift forces.
Pile Supported Mats
Downdrag Problems
Transfer of Column Loads to
Mats
Methods of Designing Mat
Foundations
Rigid Methods
Considers the mat far more rigid than the surrounding
soils, so flexure of the mat does not affect distribution
of bearing pressure
Nonrigid Metods
Considers the flexibility of the mat relative to the soil
Rigid Methods
Magnitude and distribution of bearing pressure
only depend on the applied loads and the weight
of the mat
Distribution is either uniform or varies linearly across
the mat (same as spread footings)
Application of Rigid Methods
Although rigid methods are appropriate for
spread footings, it is not really applicable for mat
foundations
Portions of a mat beneath column and bearing
walls will settle more than portions with less
load, which increases bearing pressure beneath
heavily-loaded zones
This is especially true with foundations on stiff
soils and rock, but is true with all foundations
Simplified bearing pressure is in reality not
correct in any case; more important with large
mat
Distribution with Column Loads
Questions?
vulcanhammer.net
since 1997,
your source for engineering information
for the deep foundation and marine
construction industries, and the historical
site for Vulcan Iron Works Inc.
Use subject to the “fine print” to the
right.
All of the information, data and computer software
("information") presented on this web site is for
general information only. While every effort will
be made to insure its accuracy, this information
should not be used or relied on for any specific
application without independent, competent
professional examination and verification of its
accuracy, suitability and applicability by a licensed
professional. Anyone making use of this
information does so at his or her own risk and
assumes any and all liability resulting from such
use. The entire risk as to quality or usability of the
information contained within is with the reader. In
no event will this web page or webmaster be held
liable, nor does this web page or its webmaster
provide insurance against liability, for any
damages including lost profits, lost savings or any
other incidental or consequential damages arising
from the use or inability to use the information
contained within.
This site is not an official site of Prentice-Hall, the
University of Tennessee at Chattanooga, Vulcan
Foundation Equipment or Vulcan Iron Works Inc.
(Tennessee Corporation). All references to
sources of equipment, parts, service or repairs do
not constitute an endorsement.
Don’t forget to visit our companion site http://www.vulcanhammer.org
ENCE 461
Foundation Analysis and
Design
Spread Footings: Structural Design, Flexural Design
Mat Foundations (Part I)
Reinforcing Steel for Flexural
Loads
Concrete's weakness in tension; thus, reinforcing
steel must be added when tension is anticipated,
which is virtually guaranteed with flexural
loading
Reinforcing steel in foundation almost inevitably
involves use of reinforcing bars (rebar); welded
wire fabric, needles, etc., are not generally used
Since flexural stresses are usually small, Grade 40 (Metric
Grade 300, fy = 40 ksi or 300 MPa) steel is usually adequate,
although unavailable for bars larger than #6, in which case
Grade 60 (Metric Grade 420, fy = 60 ksi or 420 MPa) steel
may have to be used
Principles for Flexural Design
Factored moment on the critical surface Muc
determines the necessary dimensions of the
member and the necessary size and location of
the reinforcing bars
This can be a complex process; however,
geotechnical considerations tend to simplify the
design process, as it dictates some of the options
Amount of the steel required flexure depends
upon the effective depth d
Principles for Flexural Design
Effective depth
Compression
Tension
Principles for Flexural Design
Nominal moment capacity of a flexural member
made of reinforced concrete with f'c < 30 MPa (4
a
ksi)
M n A s f y d
2
d f y
a
0.85 f' c
As
bd
Principles for Flexural Design
Setting Mu = Mn (where Mu = factored moment
at the section being analysed), As can be solved
to
f' c b
2.353 M uc
2
As
d d
f' c b
1.176 f y
Principles for Flexural Design
Variables for equation:
As = cross-sectional area of reinforcing steel
f'c = 28-day compressive strength of concrete
fy = yield strength of reinforcing steel
= steel ratio
b = width of flexural member
d = effective depth
= 0.9 for flexure in reinforced concrete
Muc = factored moment at the section being analysed
2.353 M uc
f' c b
2
d d
A s
1.176 f y
f' c b
Minimum and Maximum Steel
Minimum Steel Requirements (ACI 10.5.4 and
7.12.2
For Grade 40 (Metric Grade 300) Steel: As > 0.002 Ag
For Grade 60 (Metric Grade 420) Steel: As > 0.0018
Ag
Ag = Gross Cross-sectional area
= As/Ag
Maximum Steel Requirements (ACI 10.3) never
govern the design of spread footings, but may be
of concern in combined footings and mats
Minimum and Maximum
Spacing
Selection of reinforcing bar size and spacing
must satisfy the following minimum and
maximum spacing requirements
Clear space between bars must be at least equal to db,
25 mm (1"), or 4/3 times the nominal aggregate size
Centre-to-centre spacing of the reinforcement must not
exceed 3T or 500 mm (18"), whichever is less
Development Length
Development length Id is the length rebars must
extend through the concrete in order to develop
proper anchorage
Assumptions for calculations of minimum
development length
Clear spacing between the bars is at least 2db
Concrete cover is at least db
Development Length
Development length Id
(English Units)
Development Length
Id (SI Units)
Id 3 f y
Id 9 f y
d b 40 f' c c K tr d b 10 f' c c K tr
db
db
Atr f yt
Atr f yt
K tr
K tr
1500 s n
10 s n
For spread footings use Ktr = 0, which is conservative
Development Length
Variables for development length variables
Id = minimum required development length (in., mm)
db = nominal rebar diameter (in, mm)
fy = yield strength of reinforcing steel (psi, MPa)
fyt = yield strength of transverse reinforcing steel (psi,
MPa)
f'c = 28-day compressive strenght of concrete (psi,
MPa)
Development Length
Variables for development length variables
= reinforcement location factor
!
"#
$
%
$
!
&$
!
"#
$
'
$
Development Length
Variables for development length variables
(
()
*&
*+$
(
*,
*$
-
' $
'$
!
% $
.
$
'
/ /
/
!
Development Length
Variables for development length variables
0
/
%
!
$
!
!
!
'
$%
!
.1#
%
%# 2
"
' /
/ !
%
$
.
3 45 6$ 7
8
$' 7,
9%
! :$;
Application to Spread Footings
a)
Footings in reality
bend in two
perpendicular
directions
b)
Footings are designed
as if they bend in only
one direction
Justification of One-Way Slab
Assumption
The full-scale load tests on which this analysis
method is based were interpreted this way
Foundations should be designed more
conservatively than the superstructure
Flexural stresses are low, so amount of steel
required is nominal and often governed by min
Additional construction cost due to this
simplified approach is minimal
Once footing is analysed one way, we place the
same steel area in the perpendicular direction
Steel Area
Usual procedure is to prepare a moment diagram
and select an appropriate amount of steel for each
portion of the member
For spread footings, we can simplify this by
identifying a critical section for bending and use
the moment determined there to design the steel
for the entire footing
Location of critical section for bending depends
upon the type of column being used
Critical Section for Bending
Moment at Critical Bending
Based on
Section
soil bearing
Factored
Assumes Pu acts
through centroid
of footing
bending moment
2
Pu l 2 M u l
M uc
2B
B
pressure with
assumed
eccentricity
of B/3
Muc = factored moment at critical section for bending
Pu = factored compressive load from column
Mu = factored moment load from column
l = cantilever distance
B = footing width
Design Cantilever Distances
Concrete Columns
l = (B – c/2)/2
Steel Columns
Notes
Masonry Columns
l = (B-c)/2
l = (2B – (c + cp))/4
Variables
B = footing width
c = column width
cp = base plate width
Treat timber columns in
same way as concrete
columns
If column has circular,
octagonal or other
similar shape, use a
square with equivalent
cross-sectional area
If column has a circular
or regular polygon crosssectional area, base the
analysis on an equivalent
square
Computation of Steel Area
Compute Muc
Find the steel area As and reinforcement ratio
2.353 M uc
f' c b
2
d d
A s
1.176 f y
f' c b
a
M n A s f y d
2
d f y
a
0.85 f' c
As
bd
: 7
.'
# <
= # "'
!
,
$
Computation of Development
Length
The development length is measured from the
critical section for bending to the end of the bars
(usually 70 mm (3") from the end of the footing,
even if loads don't require it)
Supplied development length
I d supplied l 70 mm 3 in
(Id)supplied = supplied development length
l = cantilever distance
This length must be greater than the required
development length. If not, best solution is to use
smaller rebars with shorter development lengths
Example of Flexure Design
Given
Find
Foundation design used in shear design example, T =
27"
Required steel area for flexure
Solution
Find Required Steel Area
Determine cantilever distance l
Bc 12621
l
52.5
2
2
Example of Flexure Design
Given
Find
Foundation design used in shear design example, T =
27"
Required steel area for flexure
Solution
Find Required Steel Area
Determine cantilever distance l
Bc 12621
l
52.5 inches
2
2
Example of Flexure Design
Find Required Steel Area
Determine factored bending moment at critical section
2
Pu l 2 M u l
M uc
2B
B
2
991,000 52.5
0
M uc
2 126
M uc 10,800,000 in-lbs
Example of Flexure Design
Find Required Steel Area
Determine steel area based on factored bending
moment at critical section
f' c b
2.353 M uc
2
A s
d d
f' c b
1.176 f y
4000 126
2.353 10,800,000
As
23
1.176 60,000
0.9 4000 126
2
A s 8.94 in
For flexure in
reinforced concrete
Example of Flexure Design
Check computed steel area against minimum
steel requirements
For Grade 60 steel, As > 0.018 Ag = 0.018 B T
Minimum steel area = (0.018)(27)(126) = 6.12 in2
Since As = 8.94 sq.in. > 6.12 sq.in., OK
Use #8 (1") bars: Area of each bar = 0.79 sq. in.
Number of bars = 8.94/0.79 = 11.3 so use 12 bars
Space between bars = 126/12 = 8.7"
Minimum spacing = 18" or 3T = (3)(27) = 81" so OK
either way
Example of Flexure Design
Check development length
(Id)supplied = l – 3 = 52.5 – 3 = 49.5"
Check required inequalities
3 45 6$ 7 > 46 ;
8
$' 7,>7,'
% $
'
$
9%
! :$;><+ ;.
$%
!
Example of Flexure Design
Check development length
Id 3 f y
d b 40 f' c c K tr
db
I d 3 60000 1 1 1 1
2.5
d b 40 4000
For 1" bars, I = 28"
Id
Since (I )
= 49.5,
28
development length is OK
db
d
d applied
Example of Flexure Design
Final Design
Mat Foundations
Reasons to Consider Raft or
Mat Foundations
Structural loads are so high and soil conditions
are so poor that spread footings would be
exceptionally large.
If spread footings are more that 1/3 of building
footprint area, a mat or a deep foundation would
probably be more economical
Soil is very erratic or prone to differential
settlements, or soils that are expansive or subject
to differential heaves
Structural loads are erratic; flexural strength of
the mat will absorb these irregularities
Reasons to Consider Raft or
Mat Foundations
Lateral loads are not uniformly distributed
through the structure and thus may cause
differential horizontal movements in spread
footings or pile caps.
Uplift loads are larger than spread footings can
accommodate.
Bottom of the structure is located below the
groundwater table, so waterproofing is an
important concern. Mats are monolithic, so
easier to waterproof. Weight of mat also resists
hydrostatic uplift forces.
Pile Supported Mats
Downdrag Problems
Transfer of Column Loads to
Mats
Methods of Designing Mat
Foundations
Rigid Methods
Considers the mat far more rigid than the surrounding
soils, so flexure of the mat does not affect distribution
of bearing pressure
Nonrigid Metods
Considers the flexibility of the mat relative to the soil
Rigid Methods
Magnitude and distribution of bearing pressure
only depend on the applied loads and the weight
of the mat
Distribution is either uniform or varies linearly across
the mat (same as spread footings)
Application of Rigid Methods
Although rigid methods are appropriate for
spread footings, it is not really applicable for mat
foundations
Portions of a mat beneath column and bearing
walls will settle more than portions with less
load, which increases bearing pressure beneath
heavily-loaded zones
This is especially true with foundations on stiff
soils and rock, but is true with all foundations
Simplified bearing pressure is in reality not
correct in any case; more important with large
mat
Distribution with Column Loads
Questions?
