Numerical Methods in Soil Mechanics numerical elastic foundation

Numerical Methods in Soil Mechanics numerical elastic foundation The chapter reviews aspects of the soil–structure interface behavior at the element level and the numerical integration of the corresponding interface constitutive models. The design of structures subjected to soil–structure interaction and to contact with friction should be tackled using soil–structure interface constitutive equations. These laws differ from the laws for soils because of three main features: the size of the relative displacements and of relative rotations between grains, the high level of dilatancy and contraction under shearing, and the presence of an intense degradation effect resulting from localization in the pattern of a shear band. The elastoplastic interface constitutive equations are easy to use but do not modelize all these effects. The incrementally non-linear interface constitutive equations are versatile for modeling all these interdependent phenomena. In addition, applications to piles under tension loading are presented to illustrate the results of these procedures.

NUMERICAL REPRESENTATION OF ELASTIC FOUNDATION

2

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

4y

dx4 ' q

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

Stiffness Matrix with Springs

• Stiffness matrix, [k] was written

[k]'

AE

L3

6EI

L2 0 &12EI

L3

6EI

L2

L2

4EI

&6EI

L2

2EI L

&AE

AE

0 &12EI

L3

&6EI

L2 0 12EI

L3

&6EI

L2

Stiffness Matrix with Springs

• Recall differential equation for the deflection of a beam on elastic support

4y

dx4 ' &ky % q

6

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

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/L 3

– Hence term 2,2 was 12EI/L 3

• Unit displacement in DOF #2 yielded a force (moment)at DOF#3 of 6EI/L 2

– Hence term 2,3 was 6EI/L 2

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/L 3 + 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/L 2

– 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

Stiffness Matrix with Springs

• Stiffness matrix, [k] with foundation springs becomes

[k] '

AE

L3 %k t 6EI

L3

6EI

L2

L2

4EI

L2

2EI

L

&AE

AE

L %k a 0 0

0 &12EI

L3

&6EI

L2

L3%k t &6EI

L2

L2

2EI

&6EI

L2

4EI

L %kè

10

Stiffness Matrix with Springs

• Comments on units for spring stiffness

– Stiffness k in beam on elastic foundation

• Stiffness/unit length of beam (F/L2)

– Stiffness kaand kt, are “lumped” stiffness

• Stiffness terms with units (F/L)

• Depend upon spring spacing

– Stiffness k2 has units of moment per radian rotation