Numerical Methods in Soil Mechanics numerical elastic foundation

CE 538/561
Dr. Eric Drumm and Dr. Richard Bennett

Numerical Representation
of Elastic Foundation

NUMERICAL
REPRESENTATION OF
ELASTIC FOUNDATION

Stiffness Matrix with Springs
• Recall differential equation for a beam
of EI
– for the deflection, y
– at any point, x,
– subjected to a load, q
EI

d 4y
'q
dx 4
2

1


CE 538/561
Dr. Eric Drumm and Dr. Richard Bennett

Numerical Representation
of Elastic Foundation

Stiffness Matrix with Springs
• From the previous discussion of frames
(bending elements)
– In local x! and y! coordinate systems,
– Degrees-of-Freedom (DOF’s) were
defined:
• DOF#1 = node 1, x! direction (axial)
• DOF#2 = node 1, y! direction (transverse)
• DOF#3 = node 1, rotation, etc.
3

Stiffness Matrix with Springs
• Stiffness matrix, [k] was written

[k] '

AE
L

0

0

& AE
L

0

12EI

L3

6EI
L2

0

0

6EI
L2

4EI
L

0

& AE
L

0

0

AE
L

0
0

&12EI &6EI
L3
L2
6EI
L2

2EI
L

0
0

0

0

&12EI
L3
&6EI
L2

6EI
L2

0

0

12EI
L3

&6EI

& 6EI
L2

L2

2EI
L

4EI
L
4

2


CE 538/561
Dr. Eric Drumm and Dr. Richard Bennett

Numerical Representation
of Elastic Foundation

Stiffness Matrix with Springs
• Recall differential equation for the
deflection of a beam on elastic support

d 4y
EI
' &ky % q

4
dx

5

Stiffness Matrix with Springs
• Same as equation for beam bending,
with additional “-ky” term as “load” term
– Can include spring stiffness in the stiffness
matrix in the same manner as discussed
before when [k] was developed
– Assume that springs act independently of
each other, and affect only the DOF to
which they are attached
6

3


CE 538/561
Dr. Eric Drumm and Dr. Richard Bennett

Numerical Representation
of Elastic Foundation

Stiffness Matrix with Springs
• Direct Stiffness Method
– Recall that the term in the stiffness matrix is
• coefficient which yields the force at that DOF on the
beam arising from a unit displacement at the same or

another DOF on the structure.

– Example
• Unit displacement in DOF #2 yielded a force at DOF#2 of
12EI/L3
– Hence term 2,2 was 12EI/L 3

• Unit displacement in DOF #2 yielded a force (moment)at
DOF#3 of 6EI/L2
– Hence term 2,3 was 6EI/L 2

7

Stiffness Matrix with Springs
• Direct Stiffness Method
– If a spring of stiffness k were located at DOF#2,
the force would increase by the amount k
• Unit displacement in DOF #2 then would yield a force at
DOF#2 of 12EI/L3 + k
– Hence term 2,2 becomes 12EI/L 3 + k

• Unit displacement in DOF #2 with a spring does not
affect the moment at DOF#3 of 6EI/L2
– Hence term 2,3 remains 6EI/L 2

• The spring does not result in additional forces elsewhere
in the beam (only affects the DOF where it is located)

– Thus can add spring forces to the diagonal terms
of stiffness matrix

8

4


CE 538/561
Dr. Eric Drumm and Dr. Richard Bennett

Numerical Representation
of Elastic Foundation

Stiffness Matrix with Springs
• Stiffness matrix, [k] with foundation
springs becomes
AE
%ka
L

0

0

0

12EI
%kt
L3

6EI
L2

0

6EI
L2

[k] '
&

AE
L
0
0

&

AE
L

0

0

0

&12EI
L3

6EI
L2

4EI
%kè
L

0

&6EI
L2

2EI
L

0

0

AE
%k
L a

0

0

&12EI
L3

&6EI
L2

0

12EI
%kt
L3

&6EI
L2

6EI

2EI
L

0

&6EI

4EI
%kè
L

L2

L2

9

Stiffness Matrix with Springs

• Comments on units for spring stiffness
– Stiffness k in beam on elastic foundation
• Stiffness/unit length of beam (F/L 2)

– Stiffness k a and kt, are “lumped” stiffness
• Stiffness terms with units (F/L)
• Depend upon spring spacing

– Stiffness k 2 has units of moment per radian
rotation
10

5