CES 451
Beam on Elastic Foundation
1
Introduction:
Rigid Footing:
• The design of shallow foundation , the designer used to
consider uniform contact pressure stress beneath the
footing.
• The footing thickness is proposed thick to achieve
uniform pressure assuming C1 = 5‐7.
• To carry economic design , new method of design is
proposed to reduce the footing thickness.
Dr. Mohamed Monier Morsy
CES 451
2
Non Rigid Methods:
•Nonrigid methods consider the deformation of the mat and
their influence of bearing pressure distribution.
•These methods produce more accurate values of mat
deformations and stresses.
• These methods are more difficult to implement than rigid
methods because of soil-structure interaction.
3
Coefficient of Subgrade Reaction:
•Nonrigid methods must take into account that both
the soil and the foundation have deformation
characteristics.
• These deformation characteristics can be either linear
or non‐linear (especially in the case of the soils).
• The deformation characteristics of the soil are
quantified in the coefficient of subgrade reaction or
subgrade modulus, which is similar to the modulus of
elasticity for unidirectional deformation
4
Coefficient of Subgrade Reaction:
Definition of Coefficient of Subgrade Reaction
q
ks 

ks = coefficient of subgrade reaction, units of
force/length3 (not the same as unit weight!)
q = bearing pressure.
 = settlement.
5
Dr. Mohamed Monier Morsy
CES 451
6
NATIONAL PIPE COMPANY
Dr. Mohamed Monier Morsy
CES 451
7
Dr. Mohamed Monier Morsy
CES 451
8
Coefficient of Subgrade Reaction:
Definition of Coefficient of Subgrade Reaction
q
ks 

ks = coefficient of subgrade reaction, units of
force/length3 (not the same as unit weight!)
q = bearing pressure.
 = settlement.
9
Winkler Assumption:
A
B
•The slab is divided into six sections.
•Each sections of the slab is undergoing settlement and not affected by the
adjacent sections.
•Settlement at points outside slab is zero (Points A and B)
10
Ohde Assumption:
A
B
H
Elastic Material
•Each sections of the slab is undergoing settlement and is affected by the
adjacent sections.
•Settlement at points outside slab is not equal zero (Points A and B)
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Elastic Analysis of Shallow Foundation:
N1
N2
L
The settlement and contact pressure profile
1
a
3
2
a
a
4
a
5
a
6
a
B
12
Elastic Analysis of Shallow Foundation:
1
2
3
4
ps1
ps4
ps2
2
6
ps6
ps5
ps3
1
5
3
4
psi
5
is stress under each section
6
Pi = psi * a * B
P1
P2
P3
P4
P5
P6
13
Elastic Analysis of Shallow Foundation:
N1
N2
1
2
3
4
5
6
P1
P2
P3
P4
P5
P6
Formulation of Equations from psi , I , and Mi
Moment around nodes 1,2,……..& 6:
M1 = 0.0
M2 = P1 * a = ps1 *(a*B)*a
M3 = P1 * 2 a + P2 * a ‐ N1 * a = ps1 *(a*B)*2a + ps2 * (a*B)*a – N1 * a
.
.
.
Six equations are formulated between Mi and psi .
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Elastic Analysis of Shallow Foundation:
6E I
M i 1  4M i  M i 1  2c   i 1  2 i   i 1 
a
Ec : Modulus of Elasticity of Concrete = 2 X 107 kN/m2
I = (Bt3/12) and t is the slab thickness.
Mi‐1
Mi
Mi+1
i
i+1
Δi
Δi+1
i‐1
Δi‐1
4 Moment Equation is applied only for points 2 to 5 (i.e. for points 1 and
6 is not applicable)
Point 2
Point 3
Point 4
Point 5
6E c I
 1  2 2   3 
a2
6E I
M 2  4M 3  M 4  2c   2  2 3   4 
a
6E I
M 3  4M 4  M 5  2c   3  2 4   5 
a
6E I
M 4  4M 5  M 6  2c   4  2 5   6 
a
M1  4M 2  M 3 
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Elastic Analysis of Shallow Foundation:
1
 
2
3
4
ps1
ks
p4
4  s
ks
ps5
ps3
Spring Stiffness
1 
6
P ps  a  B



ps1
  ks aB
5
ps4
ps2
i 
ps i
ks
2 
ps 2
ks
p5
5  s
ks
ps6
3 
ps 3
ks
6 
ps 6
ks
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Elastic Analysis of Shallow Foundation:
a n
 i   p si  Ci
E s i 1
1
2
3
4
5
6
Es : Average Modulus of Elasticity of Soil.
C : Coefficient of contact stress distribution
at point (i)
ps1
ps5
ps3
Ci 
Co
1  k1 (i) k 2
Co, k1, k2 are given constant
ps2
ps6
ps4
‐Point 1:
a
C0 ps1  C1ps 2  C2 ps 3  C3ps 4  C4 ps 5  C5ps 6
Es
‐Point 2:
1 
2 
a
C1ps1  C0 ps 2  C1ps 3  C2 ps 4  C3ps 5  C4 ps 6
Es
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1
2
3
4
ps1
ps3
5
ps5
6
ps6
ps4
ps2
‐Point 3:
a
1  C 2 ps1  C1ps 2  C0 ps 3  C1ps 4  C 2 ps 5  C3ps 6 
Es
‐Point 4:
a
 2  C3ps1  C 2 ps 2  C1ps 3  C0 ps 4  C1ps 5  C 2 ps 6 
Es
‐Point 5:
a
 2  C 4 ps1  C3ps 2  C 2 ps 3  C1ps 4  C0 ps 5  C1ps 6 
Es
‐Point 6:
a
 2  C5 ps1  C 4 ps 2  C3ps 3  C2 ps 4  C1ps 5  C0 ps 6
Es
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N2
N1
1
2
3
4
5
6
O
4.4.1Based on the following equations :
1‐ Six equations (Mi & psi)
2‐ Four equations (I & Mi)
3‐ Six equations (psi & i)
ps1
ps5
ps3
ps2
ps6
ps4
4.4.2 The results of the equations in item (4.4.1) is four equations with six unknowns
ps1, ps2, ps3, ps4, ps5, & ps6
4.4.3. Additional two equations
 Fy  0.0
a  Bps1  ps 2  ps 3  ps 4  ps 5  ps 6  N1  N 2
 M o  0.0
a  Bps1  5.5a  ps 2  4.5a  ps 3  3.5a  ps 4  2.5a  ps 5 1.5a  ps 6  0.5a   0.0
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1‐ In case of symmetry:
1.1 M1 = M6 , M2 = M5 , M3 = M4
N
1
2
N
3
4
5
6
ps1 = ps6 , ps2 = ps5 , ps3 = ps4
1 = 2 , 2 = 5 , 3 = 4
1.2 The results of the equations in item (4.4.1) is two equations with three
unknowns ps1, ps2, ps3
1.3 One additional equation is determined as follows:
 Fy  0.0
a  B  2  ps1  ps 2  ps 3  2 N
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N
A
1
2
N
3
4
5
6
2‐ In case of calculating the settlement for point A outside the slab:
2.1 A new element is added to include point A.
2.2 The value of psA = 0.0.
2.3 in item (4.3) the value of A is determined:
A 
a
C0  0  C1ps1  C2 ps 2  C3ps 3  C4 ps 4  C5ps 5  C6 ps 6
Es
21
M
N
3
1
2
4
3‐ In case of calculating single load and moment.
5
6
3.1 A load of N/2 is at given point 4 and assume N/2 at the symmetric point 3.
N/2
N/2
1
2
3
4
5
6
X
3.2 Calculate the value of X so N/2 * X = M
3.3 Then the footing is symmetric as shown.
3.4 The M direction and the moment resulting from the N are opposite.
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Pseudo‐Coupled Method :
•An attempt to overcome both the lack of coupling in the
Winkler method and the difficulties of the coupling
springs.
• Does so by using springs that act independently (like
Winkler springs), but have different ks values depending
upon their location on the mat.
• Most commercial mat design software uses the Winkler
method; thus, pseudo‐coupled methods can be used with
these packages for more conservative and accurate results.
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Pseudo‐Coupled Method :
. Implementation:
• Divide the mat into two or more concentric zones.
• The innermost zone should be about half as wide and half as
long as the mat.
• Assign a ks value to each zone.
• These should progressively increase from the centre.
• The outermost zone ks should be about twice as large as the
inner most zone.
• Evaluate the shears, moments and deformations using the
Winkler method.
• Adjust mat thickness and reinforcement to satisfy strength
and serviceability requirements.
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Pseudo‐Coupled Method :
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