Amir M. Kaynia, PhD.Discipline Lead Vibrations & Earthquake Eng.Norwegian Geotech. Institute, Oslo, Norway

Indo-Norwegian Training Programme on Nonlinear Modelling and Seismic Response Evaluation of StructuresDecember 14-16, 2014 – Continuing Education Center, IIT Roorkee

Modelling of foundations - Continuum and Discrete Approaches

• Introduction and Objectives

• Notion of foundation spring

• Elastic and inelastic springs

• Calculations approaches

• Modelling of shallow foundations

• Modelling of deep foundations

• Codes and standards

• Eurocode 8, Part 5: Foundation, retaining structures and geotechnical aspects.

• ASCE 41-06: Seismic rehabilitation of existing buildings

• API (American Petroleum Institute)

• Kramer, S.L. (1996). Geotechnical Earthquake Engineering, Prentice Hall, USA

• Lysmer, J. et al. SASSI- A System for Analysis of Soil-Structure Interaction. Research Report GT 8102 . University of California, Berkeley.

• Gazetas, G. (1990). Foundation Vibrations. Chapter 15 in Foundation Engineering Handbook.

The objective is to give an overview of the methods for accounting for Soil-Structure Interaction (SSI) in nonlinear seismic analysis of structures. Key aspects include:• Except for very strong/stiff soil conditions, the response of the structure

is dependent on the soil/foundation flexibility.

• The soil/foundation flexibility often reduces the internal forces and increases the displacements

• During medium/strong seismic loading, the soil experiences nonlinear loading leading to larger displacements

• Nonlinear soil/foundation response can dramatically change the dynamic response of the structure

Three-Step Method

subject of this session

subject of structural sessions

Background• Stiffnesses have been derived primarily by analytical solutions.

• Advantage: Simple formula

• Disadvantage: Only for idealized conditions, like for homogeneous soil profiles, and for regular geometries such as circular and rectangular.

• Most often we use only horizontal & rocking stiffness (less common to use vertical & torsional stiffness)

• Several key issues related to foundation stiffness:

• Coupling between horizontal and rocking stiffness

• Added soil mass (for dynamic analyses)

• Foundation damping (for dynamic analyses)

• Stiff vs. flexible foundation assumption kvcv

khkθ

chcθ

θFIM

𝐾𝐾 = 𝐾𝐾𝐻𝐻𝐻𝐻 𝐾𝐾𝐻𝐻𝐻𝐻𝐾𝐾𝐻𝐻𝐻𝐻 𝐾𝐾𝐻𝐻𝐻𝐻

KMHKHH

1

KMMKHM

1

z

Es

d Es*

𝐾𝐾 = 𝐾𝐾𝐻𝐻𝐻𝐻 𝐾𝐾𝐻𝐻𝐻𝐻𝐾𝐾𝐻𝐻𝐻𝐻 𝐾𝐾𝐻𝐻𝐻𝐻

KHM = KHM is cross-coupling term

Cross-coupling term can be eliminated by using a rigid link with length L to foundation

Rigid link L = KHM / KHH

KHH ≅ 0.8 Es d (Ep / Es* )0.28

Parabolic soil modulus:

KMM ≅ 0.15 Es d3 (Ep / Es* )0.77

KMH = KHM ≅ - 0.24 Es d2 (Ep / Es* )0.53

=

-3.0E+11

-2.0E+11

-1.0E+11

0.0E+00

1.0E+11

2.0E+11

3.0E+11

4.0E+11

5.0E+11

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Horiz

onta

l Stif

fnes

s (N

/m)

Kxx - Real, computed

Kxx - Imag, computed

Kxx - Real, 1DOF equivalent

Kxx - Imag, 1DOF equivalent

-1.5E+15

-1.0E+15

-5.0E+14

0.0E+00

5.0E+14

1.0E+15

1.5E+15

0.0 2.0 4.0 6.0 8.0 10.0

Rock

ing

Stiff

ness

(Nm

/rad)

Krr - Real, computed

Krr - Imag, computed

Krr - Real, 1DOF equivalent

Krr - Imag, 1DOF equivalent

Added soil mass and foundation damping are derived from frequency domain analytical or FE solutions (e.g. computer program SASSI) which produce complex-valued stiffness matrices. By fitting a second degree parabola to the real part, one can compute the added mass, and by fitting a line to the imaginary part, one can compute the foundation damping.

• Real [K (ω)] = K0 – m ω2

• Imag [K (ω)] = C ω

See example =>

Simple expressions for added soil mass and damping are available a few idealized cases. For example, below are foundation parameters for circular foundations with radius r on uniform soil with shear modulus G, mass density ρ and Poisson’s ratio υ (G = ρ Vs

2).

Background• Soil behaves nonlinearly even at very small stresses/strains – at stresses

as low as 10% of yield stress nonlinearity starts.

• There are two ways of handling soil nonlinearity in nonlinear analyses:

• Equivalent linear method – through iterative method

• Nonlinear method

0

20

40

60

80

100

0 0.02 0.04 0.06 0.08

Last

(kN

)

Forskyvning (m)

0

20

40

60

80

100

0 0.02 0.04 0.06 0.08

Last

(kN

)

Forskyvning (m)

0

20

40

60

80

100

0 0.02 0.04 0.06 0.08

Last

(kN

)

Forskyvning (m)

0

20

40

60

80

100

0 0.02 0.04 0.06 0.08

Last

(kN

)

Forskyvning (m)

• Equivalent linear method – through iterative method

• Nonlinear method

0

20

40

60

80

100

0 0.02 0.04 0.06 0.08

Last

(kN

)

Forskyvning (m)

0

2000

4000

6000

8000

10000

12000

0 0.02 0.04 0.06 0.08

Stiv

het (

kN/m

)

Forskyvning (m)

By dividing forces by corresponding displacements, we get stiffness as function of displacement

Nonlinear force-displacement curve to be used directly in nonlinear structural analysis

Curve to be used by equivalent linear method in nonlinear structural analysis

1) Continuum models (finite element)

0

20

40

60

80

100

0 0.02 0.04 0.06 0.08

Last

(kN

)

Forskyvning (m)

Increasing horizontal load

Nonlinear force-displacement curve

Finite element model of soil

Contours of soil displacements

2) Discrete methods – use of p-y curves for piles

p-y curves give the relation between lateral pressure on the pile surface and corresponding displacement. The curves are established from geotechnical parameters for sand and clay.

Example:

P-y curves for soft clay (API)

Either include the piles and the p-y springs in the structural model (left figure), or compute the force-displacement relationship at the top of the piles and include them in the structural models without the piles (right figure)

p-y

t-z

Q-z

Attach p-y curves to all

nodesAttach t-z curves to all nodes

Attach Q-z curves to pile

tip nodes

3) Simplified methods – shallow foundations

Compute foundation bearing capacity Q, and stiffness, K, and construct a simple force-displacement curve shown below

K

F

δ

Q

δy

δy ≅ 1% Found. Dim.