nonlinear modelling seismic response evaluation of structures_foundations _dec2014_kaynia
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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.
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