Ground Response Analysis - Geotechnical Earthquake engineering

Soil Dynamics
5. Ground Response Analysis
Cristian Soriano Camelo1
1Federal University of Rio de Janeiro
Geotechnical Engineering
August 31st, 2017
Cristian Soriano Camelo (UFRJ) Soil Dynamics August 31st, 2017 1 / 40

Outline
1 Introduction
Ground Response Analysis
2 One-Dimensional Ground Response Analysis
Linear approach, Non-linear approach

Outline
1 Introduction

Introduction
Ground response analyses:
- Prediction of surface motions, development of design response
spectra, evaluation of dynamic stress and strains, to determine
earthquake induced forces.
The inﬂuence of local soil conditions on the nature of earthquake
damage has been recognized for many years, e.g., 1985 Mexico City EQ
and 1989 Loma Prieta EQ.

Introduction
Eﬀects of soil on ground motion( Gazetas, 2015 )
Diﬀerent motions: amplitude and frequency (periods).

Introduction
Mexico, 1985( Gazetas, 2015 )
8.1 Earthquake Magnitude at the source, diﬀerent damage.

Introduction
Lake Zone: very soft soils. SCT is the aﬀected area.

Introduction
Peak accelerations, velocities (diﬀerent soils, essentially the same
distance from the source, diﬀerent responses).

Introduction
Mexico, 1985 - Records of acceleration ( Gazetas, 2015 )
Epicentral region and SCT, similar amplitude, but, diﬀerent frequencies
of oscillation - RESPONSE SPECTRA.

Introduction
Diﬀerent periods of response vs Natural periods of structures, which
one is more detrimental?.

Introduction
Diﬀerent depths of soil, given similar soils, what happens with the 40 m
of soil ?.

Introduction
Agreement with simplest theory?.

Outline
1 Introduction

One-Dimensional Ground Response Analysis
Assumption:all the boundaries are horizontal and the response of a
soil deposit is predominantly caused by SH-waves propagating
vertically from the underlying bedrock.

One-Dimensional Ground Response Analysis
Ground response nomenclature
Motion at the surface of a soil deposit - free surface motion; Motion at the base of
the soil deposit - bedrock motion; Motion at a location where bedrock is exposed -
outcropping motion.

One-Dimensional Ground Response Analysis - Linear
Approach
Transfer Functions With Linear Systems
A transfer function is a derived equation that allows the evaluation
from the input motion (at the base) to the output motion (in the mass).
Main assumption: vs, G, ε (dynamic properties) DO NOT CHANGE
WITH γ

Approach
Transfer Functions - TF
TF=|F(ω)| - Function of frequency of the loading.
Bedrock Site Response Ground motion

Approach
Case I: Homogeneous Undamped Soil on Rigid Rock:
The displacement in this case can be expressed as:
u(z, t) = 2A eikz+e−ikz
2 eiωt = 2Acos(kz)eiωt
Where k = ω/vs - The wave number.

Case I: Homogeneous Undamped Soil on Rigid Rock
Transfer Function
F(ω) = Umax(0,t)
Umax(H,t) = 2Acos(0)eiωt
2Acos(kH)eiωt
|F(ω)| = 1
cos(kH) = 1
cos(ωH/vs)

Case II: Homogeneous Damped Soil on Rigid Rock
Transfer Function
|F(ω)| = 1√
cos2(kH)+(ξkH)2
= 1√
cos2(ωH/vs)+(ξωH/vs)2

Case II: Homogeneous Damped Soil on Rigid Rock
The greatest ampliﬁcation factor occur approximately at the lowest
natural frequency, known as the fundamental frequency ω0 = πvs
2H .
The period of vibration corresponding to the natural frequency is
called characteristic site period, Ts = 2π
ω0
= 4H
vs

Case III: Homogeneous Damped Soil on Elastic Rock
In the rigid model, the waves are perfectly reflected by the rock. In this
case, there are reflected and transferred waves. In addition, the
presence of damping will imply complex stiffness.

Transfer Function
|F(ω)| = 1
cos(k∗
s H)+(iα∗
zsin(k∗
s H)
= 1
cos(ωH/v∗
s,s)+(iα∗
zsin(ωH/v∗
s,s)
Where,
k∗
= ω ρ
G∗ - Complex wave number
α∗
z =
ρsv∗
s,s
ρrv∗
s,r
- Complex impedance ratio
vs = G∗/ρ - Complex shear wave velocity
G∗
= G(1 + 2iξ) - Complex shear modulus

If it is assumed a damping ratio ξ = 0, the Transfer Function can be
expressed as:
|F(ω)| = 1√
cos2(ksH)+α2
zsin2ksH
Resonance cannot occur (denominator is always greater than
zero). The stiﬀer bedrock means that greater ampliﬁcation
may occur.

Case IV: Layered, Damped Soil on Elastic Rock
Displacement compatibility: um(Zm = hm, t) = um+1(Zm+1 = 0, t)
Stress continuity: τm(Zm = hm, t) = τm+1(Zm+1 = 0, t)

Case IV: Layered, Damped Soil on Elastic Rock
- Solving recursion formulae are very complicated and nearly
impossible by hand, however various computer codes have been written
over the years to solve these systems.
- Diﬀerent codes can be used: SHAKE (Schnabel et al., 1972),
DEEPSOIL (elastic site response, non-linear site response), EERA
(Equivalent-linear Earthquake site Response Analyses).

SUMMARIZING - Linear Approach
- Elastic linear solutions to site response are ”easy” (simpliﬁed
equations).
- Elastic solutions are not real, but can still be useful e.g initial design
projects (they are quick and stable).
- Account damping in the soil and/or rock.
- Multi-layer systems, using recursion formulae, it can be solved the
displacements and/or stresses for any layer in the system.

SUMMARIZING - Linear approach
Advantages:
- Fast
- Direct Solution
- Good for very stiﬀ soil/rock and or very small ground motions.
Disadvantages
-Does not account for soil nonlinearity (soil dynamic properties do not
change with strain).

SUMMARIZING - Site response analysis

Site Response for Nonlinear Systems
Site response for linear systems computed directly by using Transfer
Functions. However, linear approach neglects the nonlinear
behavior of the soil. There are two methods to account for soil
non-linearity:
1) Equivalent Linear Approach: ”trick” the analysis by dealing
with linear systems.
2) Nonlinear approach.

Equivalent Linear Approach
In the equivalent linear approach, the analysis is performed dealing
with a linear system. The overall goal of this analysis is to:
Use a single modulus and damping value for each layer that
represents the average shear response during an earthquake.
To do this, it is used the modulus reduction and damping curves.

Equivalent Linear Approach: Reduction curves and damping
curves can be determined in a laboratory, where cyclic harmonic
loading is applied. However this loading is significantly different than
transient earthquake loading.
In this case it is used an effective shear strain to convert the
transient shear to a laboratory-equivalent shear.

Equivalent Linear Approach
γeff = Rγ(γmax), where, Rγ = Magnitude−1
10 Idriss and Seed (1992)
Rγ is usually a value between 50% and 70%.
Procedure:
1) Guess initial values of G and ξ (usually low-strain values)
2) Perform linear site response analysis for a layered system to obtain
shear strain time histories for every layer. Identify γmax for each layer.
3) Compute γeff for each layer using γeff = Rγ(γmax).

4) Using γeff for each layer, obtain new estimates of G and ξ values
from the modulus reduction and damping curves for each layer.
5) Repeat steps 2-4 until some tolerance in the computed shear strain
is met, or until the maximum number of iterations has been achieved.
The procedure is repeated for every single layer in the model.

Procedure(Hashash, 2016)

DECONVOLUTION
Because the equivalent linear method adopts the linear approach to site
response, it is possible to ”go both ways” with the transfer function.
When there are recorded grounds on motion.

DECONVOLUTION
Deconvolution starts with the surface ground motion and divides it by
the Transfer Function to obtain the bedrock ground motion.

DECONVOLUTION
Deconvolution must be used with extreme caution.
Ground surface recorded motions include things like 3-D effects, basin
effects, wave reflections, etc. Therefore, not all of the observed ground
motions are simply due to soil amplification. So, in reality, the
difference between the soft soil site and the stiff soil site probably is
not 1:1.
For example, what would happen if there is a high surface acceleration
at the same period where there is a low value on the transfer function ?.
Super-big bedrock motions that are not real.

DECONVOLUTION
Rather than use deconvolution, most professionals prefer to use ground
motions recorded on rock outcrops with similar shear wave velocities as
the bedrock at their site.

For Further Reading I
Steven L. Kramer.
Geotechnical Earthquake Engineering.
Prentice Hall, 1996.
CEEN 545 - Lecture 21 - Nonlinear Site Response. Link
DEEPSOIL, 1D equivalent linear and nonlinear site response analysis platform.
Link to DEEPSOIL

Ground Response Analysis - Geotechnical Earthquake engineering

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