TECHNICAL PAPER
JOURNAL OF THE SOUTH AFRICAN
INSTITUTION OF CIVIL ENGINEERING
Vol 56 No 1, April 2014, Pages 54–62, Paper 992
PROF ASRAT WORKU is an Associate Professor of
Civil Engineering at Addis Ababa University
(AAU), Ethiopia, and is currently also working as
Operations Manager for Geotechnics at Gibb
International in Nairobi. He completed his BSc
degree in Civil Engineering in 1983, and,
specialising in geotechnics, his MSc degree in
1989, both from AAU. IN 1996 he earned his
DrIng degree from Wuppertal University, Germany, with a dissertation on
seismic soil structure interaction. He has practised in both structural and
geotechnical engineering. His industry experience in geotechnics spans
various types of major projects in several African countries.
Contact details:
PO Box 30020
Nairobi 00100
Kenya
T: +254 20 225 1880/0577
M: +254 725 617420
F: +254 20 221 0694 / +254 20 224 4493
E: aworku@gibbinternational.com
E: asratie@gmail.com
Soil-structureinteraction provisions
A potential tool to consider for
economical seismic design of buildings?
A Worku
Contemporary seismic design codes have become more stringent with respect to the
requirements for design forces and deformations in building design. This paper demonstrates
that it could be worthwhile to consider the introduction of soil-structure-interaction provisions
into local design codes. This is partly to be able to offset the costs incurred by the high
magnitude of base shear demand in most buildings attributed to site amplifications due to soft
soil sites, as per the requirements of current codes, including the recent South African seismic
design code. This beneficial effect of site soils is as a result of lengthening of the fundamental
period and of the increased effective damping of the overall system due to soil-structure
interaction, which in most cases lead to reduced design spectral values. The paper shows that, if
pertinent provisions in some international codes are properly adapted, a substantial reduction
in the base shear force can be achieved, in many cases resulting in structural-work cost saving.
With this, the paper attempts to address the legitimate concern of design engineers regarding
the potential escalation of construction costs associated with the introduction of stringent
requirements of contemporary seismic design spectra, especially for soft soil sites.
INTRODUCTION
Keywords: soil-structure-interaction, fixed-base structure, flexible-base
structure, period lengthening, effective damping, base shear,
site amplification
54
The behaviour of site soils is one of the three
major factors that can significantly influence the intensity of ground shaking due to
an earthquake at any given site, the other
two factors being the earthquake source
mechanism and the geology of the seismicwave path. The influence of site geotechnical
conditions on ground-shaking intensity is
studied following one of two approaches.
The first is an empirical approach based
on comparison of ensembles of recorded
ground motions at nearby rock and soil sites
of known geotechnical characteristics whenever these are available. The results of such
studies are presented in the form of smoothened, statistically averaged site-dependent
design spectra. These spectra are factored
forms of the basic design spectrum for the
corresponding rock site. The amplification
factors are in general dependent on the
nature of the site and the seismicity of the
region. In the absence or scarcity of recorded
ground motions for a given seismic region, it
is common practice to adapt design spectra
from regions of similar geologic and tectonic
setup.
The second approach is appropriate for
site-specific studies which involve modelling of the site soil as any other dynamic
system subjected to the ground motion at
the rock level. The soil can be modelled as
a continuous or discrete mass system. The
end results could be ground motion time
histories, peak ground motions or response
spectra at the ground surface.
This effect of site soils to amplify the
rock-level ground motion is generally detrimental to the integrity of structures built on
them.
Another important influence of site soils
on structures is related to soil-structure
interaction (SSI), which is rendered unfairly
less attention, especially in routine building
design. When the ground motion, amplified
by the site soil in the manner described
above, strikes the foundation, two forms of
SSI take place. The first is attributed to the
difference in rigidity between the foundation unit and the soil, which causes, among
others, reflection and refraction of the
seismic waves back into the soil mass. As a
consequence, the motion of the foundation
and the free ground become different, with
the foundation motion usually being smaller.
This aspect of SSI is known as kinematic
SSI. Ideally, the foundation motion should be
used as input motion in the analysis of the
structure. However, studies have shown that
the difference between the two motions can
be regarded as negligible. For this reason,
the free-ground motion is used as the input
ground motion in practice (Fenves & Serino
1992; Stewart et al 1999; Stewart et al 2003).
Journal of the South African Institution of Civil Engineering • Volume 56 Number 1 April 2014
The second, and more important, form
of SSI is manifested when the superstructure
starts to vibrate as a result of inertial forces
triggered by the excitation at the foundation
level. The inertial forces distributed over
the height of the structure cause a resultant
base shear and an overturning moment
at the foundation, which in turn cause
deformation of the soil. This deformation
initiates new waves propagating into the
soil mass. These waves carry away part of
the energy imparted on the structure by
the incoming earthquake waves and act as
a means of energy dissipation in addition to
the material/hysteretic damping inherent
in the system. This form of SSI is known
as inertial SSI. Its effect in most structures
is to increase total displacement due to the
additional soil deformation, and to decrease
the base shear demand due to the associated reduced structural inertia forces as a
result of the additional energy dissipation
into the soil (Fenves & Serino 1992; Worku
1996; Stewart et al 1999; Stewart et al 2003;
Tileylioglu et al 2011).
In the sense of the reduced base shear,
the consideration of SSI effect is beneficial
for most building structures. Unfortunately
this important effect is mostly ignored by
engineers, with the notion that the design
is on the safe side without the additional
computational effort needed to account for
SSI effects (Stewart et al 1999; Stewart et al
2003). This tendency lacks scientific rationality and is happening despite the fact that
provisions for this phenomenon have been
made available in some design codes since
the 1980s. The original versions of these
provisions have meanwhile been updated
through calibrations with actual records
from relatively recent strong earthquakes,
including the 1989 Loma-Prieta and the 1994
Northridge earthquakes (Fenves & Serino
1992; Stewart et al 2003; BSSC 2004; BSSC
2010). Results of such calibration works
and recent experimental verifications are
encouraging the use of the recent versions
of code-based SSI provisions (Stewart et al
1999; Stewart et al 2003; Tileylioglu 2011).
However, it is also worth pointing out that,
in certain seismic and soil environments, an
increase in the fundamental natural period
of a moderately flexible structure due to SSI
may have detrimental effects on the imposed
seismic demand (Mylonakis & Gazetas 2000;
Ziotopoulou & Gazetas 2010). In both cases
it is becoming more evident that neglecting
seismic SSI is not sustainable.
The recently revised South African
seismic code (SANS 10160-4) adapted the
site-dependent design spectra of EC8 (2004)
with some modifications. These spectra are
in general more demanding than those of
Psinωt
Psinωt
Psinωt
h0
R0, A0
w(0,t)
· (0,t)
mf w
K
z
dz
N
N(0,t)
mw
· dz
R, A
α
E, ρ
C
dz
(b)
α
(d)
N + N'dz
(a)
(c)
Figure 1 A circular rigid foundation on the surface of a half space subjected to a vertical harmonic
load and its simplified representations
the previous versions (SABS 0160 1989; EC8
1994). Some South African engineers have
expressed concern during the preparation
period leading to the issue of the recent
design code regarding the potential escalation of material and construction costs
associated with such stringent requirements
(Wium 2010).
With due account for this concern, this
paper attempts to demonstrate that a good
potential exists for some of the costs associated with site amplification to be partially
offset by the beneficial effects of inertial SSI
on many structures. This happens if engineers are allowed to exercise some degree of
freedom to employ SSI provisions available
in some international codes until these make
their way to the South African seismic code
in due course.
INERTIAL SOIL-STRUCTURE
INTERACTION AND
IMPEDANCE FUNCTIONS
In order to understand the influence of
inertial SSI on the response of building
structures subjected to seismic ground
motions, it is helpful to briefly introduce the
basic principles and concepts of dynamics
of foundations supported by flexible media
like soils. For this purpose, we consider the
vibration of the rigid circular foundation of
radius R0 resting on the surface of the soil
idealised as a homogenous elastic half space
shown in Figure 1 (a) and excited by the
vertical harmonic load. Let the half space
have an elastic modulus of E and a mass
density of ρ. For purposes of mathematical
expediency and better insight, let us further
represent the half space by the rudimentary
model of the truncated solid cone of crosssectional area of A0 at the ground level
which is the same as the contact area of
the foundation. The cone defines the angle
α with the horizontal and the height h0 up
to its apex above the ground (Worku 1996;
Wolf & Deeks 2004).
After formulating the equation of motion
of the conical soil beam based on the equilibrium of the differential soil element shown
in Figure 1(c), it can be shown, without
resorting to the details, that the differential
equation for the capping rigid circular foundation of Figure 1(b) becomes (Worku 1996):
mf ẅ0(t) +
EA0
EA0
ẇ (t) +
w (t) = Po sin ωt (1)
cL 0
h0 0
where mf is the mass of the foundation,
cL = √E/ρ is the velocity of the longitudinal
elastic wave travelling away from the foundation through the conical soil column,
and w 0 is the vertical displacement of the
foundation.
This equation is similar to the conventional equation of motion of the replacement
single-degree-of-freedom (SDOF) oscillator
shown in Figure 1(d) and given by:
mf ẅ0(t) + Cẇ0(t) + Kw0(t) = Po sin ωt
(2)
where K and C are the spring and dashpot
coefficients of the mechanical model respectively. Comparison of the two equations
results in the following expressions for the
parameters of the SDOF model of Figure 1(d)
in terms of the geometry of the foundation
and the elastic properties of the soil:
K=
EA0
EA0
;C=
h0
cL
(3)
This result obtained on the basis of a rudimentary idealisation of the soil-foundation
system as a truncated conical column capped
by the rigid foundation (Figure 1(a)) demonstrates the following fundamental facts:
■ The semi-infinite continuum can be
replaced by a simple SDOF mechanical
massless model supported by a spring
and a dashpot of coefficients, K and C,
respectively, arranged in parallel, and
these parameters can be expressed in
terms of the foundation geometry, the
elastic parameters of the continuum and
a pertinent wave velocity.
Journal of the South African Institution of Civil Engineering • Volume 56 Number 1 April 2014
55
not exceed 4:1. For other cases, suggested
modifications are available (Gazetas 1991).
The subsequent discussion will thus focus on
circular foundations. The same discussion
can easily be expanded to other shapes and
soil-foundation conditions.
The static spring coefficients in Equation
(7) for a circular foundation are given by the
expressions in Equation (8) for the horizontal
translation and rocking degrees of freedom
respectively that are important in seismic
design (Gazetas 1991; Worku 1996):
m
m
V
Ṽ
T̃, ζ̃
h
T, ζ
K̄h
K̄r
(a)
Ksh =
(b)
Figure 2 (a) An SDOF structural model with a flexible-base; (b) A replacement SDOF model
■ Unlike in conventional dynamic models
of structures, the damping term – the
second term in Equation (1) – is not an
assumed addition of viscous damping; it
is a mathematical outcome showing that
the damping is an intrinsic behaviour
of the system. This term represents an
additional equilibrant force due to energy
dissipation through waves propagating
away from the foundation as represented
by the wave velocity in the coefficient.
It is in addition to the material damping
of the continuum not considered in this
discussion.
The truncated-cone approach was first
devised and the above important outcomes
observed about eight decades ago (Reissner
1936; Ehlers 1942). Interestingly, this seemingly primitive approach is extensively used in
the recent book authored by Wolf and Deeks
(2004) in a systematic manner. The new simplified approach has the potential of enabling
engineers to easily solve a range of practical
problems in structural dynamics involving SSI
without reverting to complex finite-element
techniques to model the site soil.
In a more rigorous treatment of the soilfoundation system, the spring and dashpot
coefficients of Equation (3) are dependent
on the frequency of excitation among many
other factors (Luco & Westman 1971; Veletsos
& Wei 1971). These coefficients, commonly
termed as impedance functions, are now
available in the literature for a wide range of
conditions after several decades of intensive
research works. They have meanwhile been
well compiled, and have already made their
ways into design codes starting from around
1980 (Gazetas 1983, 1991; Pais & Kausel 1988;
Worku 1996; BSSC 2004; BSSC 2010).
Reverting to the mechanical model of
Figure 1(d), its equation of motion given by
Equation (2) for zero mass takes, for any
degree of freedom considered, the form:
Cẇ(t) + Kw(t) = Poeiwt
56
(4)
The subscript of the deformation is dropped
for brevity reasons, and the harmonic load is
represented in its complex form for purposes
of generality. The trial solution to this differential equation should also be complex. After
substituting a complex function for w(t) and
solving for the complex-valued impedance
function, which by definition is the ratio of
the load to the response, yields:
P(t)
= K̄ = K + iωC
w(t)
(5)
On the other hand, the complex-valued
impedance functions obtained from rigorous
mathematical treatments of the semi-infinite
continuum are often presented in the literature in the following form:
K̄ = Ks[α(ω) + ia0β(ω)]
(6)
where Ks is the static spring stiffness, a0 is a
dimensionless frequency parameter given by
a0 = ωR/Vs , Vs is the shear wave velocity of
the continuum, α(ω) and β(ω) are frequencydependent dynamic impedance coefficients
(also known as dynamic modifiers). By
equating Equations (5) and (6) one obtains
the following important relationships for the
real-valued, frequency-dependent parameters
of the massless spring-dashpot model in
Figure 1(d):
K = Ks α(ω); C = Ks
R
β(ω)
Vs
(7)
As indicated above, the impedance coefficients, α(ω) and β(ω), are available for various
foundation conditions, soil conditions and
vibration modes.
A circular foundation on the surface of
the homogenous viscoelastic half space is
the most basic and most important case.
Studies have shown that use of an equivalent
circular foundation gives satisfactory results
for foundations of other shapes, provided
that the aspect ratio of the encompassing rectangle of the foundation plan does
8GRh
8GRθ 3
; Ksθ =
3(1 – v)
2–v
(8)
Note that the radii in the two cases are
different for non-circular foundations and
are determined by equating the area A and
moment of inertia Iθ for rocking motion of
the actual foundation to those of the equivalent circular foundation. Thus,
Rh = √A/π ; Rθ = √4I θ /π
(9)
The corresponding dynamic modifiers for a
surface circular foundation were originally
provided by Vlelestos and his co-workers
(Veletsos & Wei 1971; Veletsos & Verbic
1973) and Luco and Westmann (1971),
independently of one another, as functions of
the frequence parameter, a0. For other cases,
appropriate impedance functions are available and should be used in order to determine
the dynamic spring and dashpot coefficients
as per Equation (7). Important factors to be
further accounted for when using impedance
functions include foundation embedment
depth, foundation depth, foundation flexibility, soil layering and increase in stiffness
of soil with depth. Relevant literature should
be consulted for this purpose (Gazetas 1991;
Pais & Kausel 1988; Worku 1996; Stewart et
al 1999).
FLEXIBLE-BASE MODEL
PARAMETERS
In the most general three-dimensional case,
a single mass oscillator fixed at its base
acquires six additional degrees of freedom
(DOF) when the base is released. The additional DOFs consist of a translational DOF
in each direction of the Cartesian coordinate
axes and a rotational DOF around each of
them.
For an excitation due to upward propagating seismic shear waves, inclusion of the
horizontal and rocking DOFs alone is sufficient in planar analysis. This condition is
depicted in Figure 2 for a superstructure
represented by an SDOF model, in which the
complex-valued springs are lumped at the
base in each of the horizontal and rotational
Journal of the South African Institution of Civil Engineering • Volume 56 Number 1 April 2014
T̃
k
kh2
+
= 1+
T
kθ
kh
30
System damping (%)
25
20
15
10
5
0
1.0
1.1
1.3
1.4
1.5
1.6
FD = 3%
10%
20%
5%
15%
FBSD = 5%
Figure 3 Variation of the system damping with T̃/T for different foundation damping
0.8
0.6
S̃ a
0.4
S̄ a
Sa
S̃ a
0.2
T
T̃
T
T̃
(10)
0
0
The fixed-base period is given by the wellknown relationship of T = 2π/√k/m, where
k is the stiffness of the structure and m
is its mass. According to Equation (10),
the flexible-base period T̃ is always larger
than the fixed-base period and increases
with decreasing stiffness of the foundation.
Measured period lengthening of more than
50% are reported by researchers (Stewart
et al 2003). Note that the period ratio is
dependent on frequency (or period) because
of the frequency-dependent foundation
stiffnesses. It is, however, sufficient to
establish the stiffnesses for the fundamental
frequency/period of the fixed-base system
(Stewart et al 2003; BSSC 2004).
The effective flexible-base damping ζ̃ is
contributed from both the structural viscous
damping ζ and the foundation damping ζ̃ 0
1.2
Period ratio
Sa (g)
DOFs. Accordingly, the system now has
three degrees of freedom. This representation is equivalent to a real-valued spring and
dashpot arranged in parallel for each DOF.
The height h refers to the height of the roof
in the case of a single-storey building and to
the centroid of the inertial forces associated
with the fundamental mode in the case of
a multi-storey building which is commonly
taken as 0.7h assuming a linear fundamental
mode of vibration (Stewart et al 1999; BSSC
2004).
In time-history analysis (THA), the
frequency dependence of the foundation
parameters and the nature of the system
damping renders flexible-base models more
difficult to analyse than fixed-base models.
Such systems are termed as non-classically
damped systems and can be solved using
specially tailored closed-form or iterative
analysis methods (Worku 1996, 2005, 2012).
In contrast to THA, in responsespectrum and pseudo-static analyses, SSI is
accounted for by dealing with an equivalent
SDOF system as shown in Figure 2(b) with
modified parameters to account for the
foundation flexibility. This was proposed by
Veletsos and Meek (1974), who drew a parallel between the two models and found that
the maximum displacement of the mass in
Figure 2(a) can be accurately predicted using
the replacement SDOF system in Figure 2(b)
with a modified natural period of T̃ and a
modified damping ratio of ζ̃ . These modified
parameters are called flexible-base parameters and have the convenience of enabling
the engineer to use the conventional codespecified seismic design spectra as usual.
Veletsos and Meek (1974) found out that
the flexible-base period may be determined
from:
0.3
0.5
1.5
1.0
2.0
T (s)
Flexible-base damping, ζ̃
Fixed-base damping, ζ
Figure 4 Schematic representation of influence of SSI on design spectra
(adapted from Stewart et al (1999))
consisting generally of radiation and material
damping components. Veletsos & Nair (1975)
established the following relationship for
the system damping based on equivalence of
maximum deformations of the two oscillators in Figure 2:
ζ̃ = ζ̃ 0 +
ζ
(T̃/T)3
(11)
The plots of Equation (11) against the period
ratio are given in Figure 3 for the commonly
assumed fixed-base structural damping
(FBSD) of 5% and a number of foundation
damping (FD) values ranging from 3% to
20%. Such ranges of foundation damping ratios have been reported in the past
(Stewart et al 2003).
The plots show that the overall effective
damping of the flexible-base system is larger
than the fixed-base damping (FBSD = 5%)
with the exception of the rare case of the
foundation damping itself being very low
(smaller than 5%), and the period ratio being
large. For any given foundation damping, the
system damping gradually decreases with
increasing period ratio due to the decreasing
Journal of the South African Institution of Civil Engineering • Volume 56 Number 1 April 2014
57
4
E
damping ratio including both material/hysteretic and geometric damping is estimated
at 10% for the purpose of this study, even
though larger damping ratios are reported
for such a class of soil in the literature
(Stewart et al 2003). The corresponding
period lengthening of the SDOF system
due to SSI is also conservatively estimated
at 10% so that T/T̃ reaches up to 1.10. With
the effective system damping calculated
from Equation (11) or read from Figure 3 as
13.76%, the corresponding design spectral
curve is determined as per the provisions
of EC 8 (2004) by scaling down the sitedependent spectral curve using a scaling
factor to account for the modified damping.
The factor is given by:
D
C
3
B
Se/ag
A (Rock site)
2
1
0
0
1
2
3
4
T (s)
η(EC8) = 10/(5 + ζ̃ ) ≥ 0.55
Figure 5 EC 8 2004 design spectra for different site conditions for a damping ratio of 5%
(after EC8 2004)
contribution of the structural damping with
increasing period ratio. It should, however,
be noted that the effective damping may not
generally be taken less than the structural
damping of 5% (BSSC 2004, BSSC 2010).
INFLUENCE OF INERTIAL SSI
ON DESIGN SPECTRA
In this equation ζ̃ is the effective system
damping in percentile that accounts for both
structural and foundation damping. The plot
is given in Figure 6(a) by the dashed curve,
which indicates that a significant reduction
in the design base shear of up to 30% could
be achieved for structures with a fundamental period larger than 0.2 seconds. Many
classes of buildings belong to this period
range. Most actual cases are expected to plot
on or above the dashed curve.
Similarly, a little larger maximum
limit for the foundation damping of 15% is
assumed for the much softer Site Class D
with corresponding period lengthening of up
to 15%. The effective damping calculated as
18.3% resulted in the dashed curve shown in
Figure 6(b). A larger reduction in the design
base shear than in Site Class C seems attainable in this case. It is, however, important to
point out that current code provisions for SSI
cap the maximum permissible base-shear
reduction to 30% (BSSC 2004).
The modified spectral curves for the two
site classes are compared in Figure 7 against
A more direct insight into the influence
of SSI on code-specified design spectra can
be obtained by considering the EC8 (2004)
Type 1 design spectra specified for five different site soil classes shown in Figure 5 for a
structural damping ratio of 5%. The various
site soil classes are defined in the code (EC8
2004). The amplification potential of the
site soils is evident from the spectral curves.
These spectra are incorporated into the provisions of the recently revised South African
seismic code, with the exception of the spectrum for Site Class E (SANS 10160-4).
Let us consider the two soft site soil
classes of C and D characterised by an average shear-wave velocity of 180 to 360 m/s
and less than 180 m/s, respectively, over the
upper 30 m depth in accordance with EC 8
(2004). The corresponding design spectra for
the two site classes are presented separately
in Figures 6(a) and 6(b) together with the
spectrum for Site Class A – rock site.
Based on the definition of Site Class
C, the maximum attainable foundation
4
4
3
3
Se/ag
Se/ag
The influence of the lengthened period and
the modified damping on a smoothened
response spectrum is shown schematically in
Figure 4. The figure shows that, for a fixedbase period of up to around 0.3 seconds,
SSI has the effect of increasing the spectral
response of the structure. However, for the
most common case of building structures
having a fundamental natural period larger
than about 0.3 seconds, SSI has the effect of
reducing the spectral response and thereby
reducing the design base shear force (compare ordinates of the two curves corresponding to the pairs of T and T̃ on either sides of
T ≈ 0.3 s).
2
1
(12)
2
1
0
0
0
1
2
3
4
0
1
2
T (s)
Site Class A
Site Class C
Foundation damping 10%
Site Class A
Site Class D
(a)
(b)
Figure 6 Comparison of design spectra of EC8 2004 with those modified for SSI for (a) Site Class C, and (b) Site Class D
58
3
4
T (s)
Journal of the South African Institution of Civil Engineering • Volume 56 Number 1 April 2014
Foundation damping 15%
2
2
Se/ag
3
Se/ag
3
1
1
0
0
0
1
2
3
0
1
2
T (s)
3
T (s)
Site Class A, EC8-1994
Foundation damping 10%
Site Class A, EC8-1994
Site Class B, EC8-1994
Foundation damping 15%
Site Class C, EC8-1994
(a)
(b)
Figure 7 Comparison of modified design spectra against EC8 1994 design spectra for (a) Site Class B, and (b) Site Class C
η(NEHRP) = (5/ζ̃ )0.4
(13)
The plots of Equations (12) and (13) are
compared in Figure 8, which shows that the
reduction proposed by the NEHRP document (BSSC 2004) is slightly larger than that
of EC 8 (2004).
Finally, to be emphasised is the fact that
the foundation damping and the period
lengthening are key factors that affect the
amount of spectral reduction due to SSI.
It is, however, important to note that the
reductions demonstrated in the above plots
are based on assumed ranges of foundation
damping and period lengthening for the
purpose of this study, even though these are
based on reasonable engineering judgment
and reported cases (Stewart et al 2003).
1.0
0.9
Scale factor
the corresponding site-dependent design
spectra specified by the older version of EC8
(1994). In Figure 7(a), the design spectra for
Site classes A and B of EC8-1994 are compared with the EC8-2004 design spectrum
for Site Class C modified for SSI. Similarly,
the spectra for site classes A and C of EC8
(1994) are compared in Figure 7(b) against
the EC8 (2004) design spectrum for Site
Class D modified for SSI.
It is interesting to note from the plots
that the design spectra, and thus the design
base shear, as per EC8 (2004) modified for
inertial SSI effects can even be significantly
lower than the values specified by the older
EC8-1994 spectra for the corresponding soil
classes over a significant range of fundamental period. The reduction is particularly
significant in long-period structures.
The factor in the National Earthquake
Hazard Reduction Program (NEHRP) document – a resource document for most seismic codes in the USA – for scaling down the
site-dependent spectral curves corresponding to Equation (12) is given by (BSSC 2004):
0.8
0.7
0.6
0.5
5
10
15
20
25
Effective system damping (%)
EC8
NEHRP 2003
Figure 8 Comparison of spectral scale factors for effective system damping as stipulated by
EC 8 2004 and NEHRP 2003 (BSSC 2004)
Hence, the actual gains must be established
by the design engineer on a case by case
basis, and no generalisation is warranted on
the basis of the presented material alone.
Nevertheless, the plots in Figures 6 to 8
demonstrate that the magnitude of spectral
amplification by soil sites could be substantially offset by inertial SSI effects. If properly
employed, SSI provisions could have the potential of leading to a significant financial saving
in many cases, as is evident from the plots.
However, to be remembered is also the
other important effect of inertial SSI that
increases the lateral displacement of the
building. This effect must be taken into
account when considering ductility issues,
secondary effects like P–Δ and possibilities
of pounding with contiguous structures
– considerations that are important in the
design of tall buildings regardless of whether
reduction in base-shear is achievable or not.
EXAMPLES
In order to illustrate the use of code provisions of SSI in seismic design of buildings
Table 1 Site soil data
γvs2
g
(kN/m 2)
vs
vs0
G
G0
vs
(m/s)
G
(kN/m 2)
0.40
88 807
0.95
0.90
209
79 926
0.45
41 284
0.64
0.47
96
19 403
Soil
type
vs0
(m/s)
γ
(kN/m3)
v
D
220
18
E
150
18
G0 =
Journal of the South African Institution of Civil Engineering • Volume 56 Number 1 April 2014
59
Table 2 Building data
Table 3 Computed building data
Building
type
Structural
system
Storeys
hn
(m)
Total mass
(ton)
Cr
x
1
framed
6
18
3 600
0.0466
0.90
2
dual
11
33
6 600
0.0488
0.75
3
dual
16
48
9 600
0.0488
0.75
4
dual
26
78
15 600
0.0488
0.75
Building
type
Fixed base Natural Structural
period
frequency stiffness
Ta (sec)
ω (sec–1)
k (kN/m)
1
0.63
9.973
250 657
2
0.67
9.378
406 305
3
0.89
7.060
334 926
4
1.28
4.909
263 125
Table 4 Computed system data
Building
type
ah
Rm
vsT
aθ
kh
kN
( m × 10 6)
kθ
(kNm × 10 6)
Soil
Soil
Soil
Soil
Soil
D
E
D
E
D
E
D
E
D
E
1
1
1
0.096
0.209
0.93
0.81
5.523
1.384
665.58
153.52
2
1
1
0.090
0.196
0.94
0.82
5.523
1.384
672.73
155.42
3
1
1
0.068
0.148
0.97
0.85
5.523
1.384
694.20
161.10
4
1
1
0.047
0.103
1.00
0.93
5.523
1.384
715.67
176.27
and of the potential benefits, the site-soil
classification and SSI procedures proposed
in NEHRP are employed (BSSC 2004, 2010).
EC8 (2004) does not have provisions for SSI.
Four different idealised reinforcedconcrete buildings of height ranging from 5
to 25 storeys founded on the site soils types
of D and E, according to the NEHRP classification system, are considered. Thus, the
influence of SSI on eight different cases of
dynamic system is studied simultaneously.
The buildings are assumed to be located at
sites characterised by a design peak ground
acceleration (PGA) of 0.1 g, where g is the
gravitational acceleration. Most sites in South
Africa, where seismic design is required, are
assigned a PGA of 0.1 g. According to the
current response-spectra based on NEHRP
seismic hazard mapping (BSSC 2004, 2010), a
site of such seismicity can be represented by a
short-period normalised spectrum Ss of about
0.25 and an intermediate-period (1 second)
spectrum S1 of 0.1. Note that the US codes
are no longer using PGA for seismic hazard
characterisation.
A small-strain shear-wave velocity of 220
m/sec and 150 m/sec, a Poisson’s ratios of 0.4
and 0.45 are assigned to the two site classes
D and E respectively, whereas an effective
soil unit weight of 18 kN/m3 is assumed for
both. The characteristics of the site soils are
summarised in Table 1, in which the smallstrain shear modulus is also computed from
the direct relationship with the small-strain
shear-wave velocity.
The actual shear-wave velocity and shear
modulus corresponding to the large strains
sustained during strong earthquakes at any
given site are smaller and depend on the
60
actual strain level, which in turn depends on
the intensity of the anticipated earthquake
shaking as represented by the seismicity of
the site. The pertinent NEHRP provisions
specify the ratios of va/vs0 and G/G0 as per
the seismicity of sites. These recommended
ratios are given in columns 6 and 7 of Table
1, and the reduced values of the two dynamic
properties are provided in the last columns
of Table 1. It can be noted that the reduction is larger in the softer soil E due to the
expected larger strains.
The data pertaining to the building
structures are given in Table 2. All four
building types considered are supported by
a 20 m by 30 m rectangular raft foundation,
have an additional basement storey and have
a uniform story height of 3 m including the
basement floor. The radius of the equivalent
circular foundation for the horizontal and
rocking (around the longer side of the rectangular foundation) degrees of freedom are
computed from Equation (9) as 13.82 m and
12.63 m respectively.
The fundamental period is estimated
using the relationship provided in the code:
Tα = Cr hn x
(14)
The constants Cr and x, which depend on
the structural system, are also provided in
Table 2 as proposed by the provisions of the
code. The height hn is the total height of the
building measured from the foundation level.
The periods computed using Equation (14)
are presented in Table 3. A uniformly distributed permanent gravity load of 10 kN/m2
is assumed on each floor for the subsequent
computation of the building mass. The
structural stiffness is computed using the natural period and the effective mass m̄ obtained
by reducing the total mass by a factor of 0.7 as
recommended by the code using the relationship in Equation (15).
k = 4πm̄/T 2
(15)
The structural stiffnesses computed in
this manner are given in the last column of
Table 3.
The dynamic foundation stiffnesses are
dependent on the soil type, foundation shape,
foundation embedment and structural properties. Neglecting the effect of foundation
embedment due to the shallow depth, the
stiffnesses are calculated using the relationships in Equation (16):
kh =
8
8
GRhαh; kθ =
GR 3 α
2–v
3(1 – v) θ θ
(16)
The coefficients αh and αθ are generally
frequency-dependent dynamic modifiers
applied on the respective static stiffness given
by Equation (8) for the horizontal and rocking
motion, respectively. Whereas the modifier
αh may be taken as unity for all practical
purposes, the modifier αθ must be established
depending on the ratio Rmvs/Tα (BSSC 2004,
2010). Both the modifiers and the stiffnesses
are computed and provided in Table 4.
Once the stiffnesses are established, the
system (effective) period is calculated from
Equation (10). The computed values are
given in Table 5. The foundation damping
β̃ 0 is dependent on the aspect ratio h̄/R of
the building, the period ratio T̃/T, and the
seismicity of the site, where h̄ is the effective
height taken equal to 0.7hn. It is determined
in accordance with graphs provided in the
code document. Then the effective system
damping is determined as per Equation (11),
in which a structural damping of 5% is
assumed for concrete structures as usual.
The foundation damping and the system
damping are given in the last columns of
Table 5. Note that for computed values of
the effective damping that are less than 5%, a
minimum damping of 5% is taken according
to the recommendations of NEHRP.
The seismic response coefficient corresponding to the fixed-base period T is given
Journal of the South African Institution of Civil Engineering • Volume 56 Number 1 April 2014
by (for an elastic response and a normaloccupancy building):
Cs =
2
2 FvS1
F S ≤
3 a s 3 T
Table 5: Computed system effective period and effective damping
(17)
The coefficients Fa and Fv are the site amplification factors for the short-period and the
intermediate-period regions of the design
spectrum, respectively. They are determined
from tables provided in the design code.
Fa assumes the values of 1.6 and 2.5 for
soils D and E, whereas Fv takes the values 2
and 3.2, respectively. Similarly, the seismic
response coefficient C˜s corresponding to the
flexible-base period T˜ is determined from
Equation (17).
Finally, the effective system damping and
the modified seismic response coefficient are
employed together with the structural damping and the fixed-base seismic coefficient
to calculate the reduction in base shear due
to SSI using Equation (18). The results are
presented in the last column of Table 6 as
percentages of the base shear of the fixedbase system.
∆V
C̃ æ β æ0.4 é
× 100%
= 0.7 é1 – s
ë
V
Cs è β̃ è ë
ê
ç ç ê
(18)
The results obtained demonstrate that a
significant amount of reduction in design
base shear can be achieved if SSI provisions
of design codes are properly used. In these
particular examples a reduction in base
shear of 7% to 39% is achieved. However, it
is important to point out that the series of
NEHRP documents (BSSC 2004, 2010) limit
the maximum base-shear reduction due to SSI
to a maximum of 30% as shown in brackets
in the last column of Table 6. Obviously, the
resulting cost saving in general could be of
significant proportion, especially in mediumheight buildings. The percentage saving
increases with decreasing stiffness of the soil.
However, the increasing trend of reduction
in base shear with increasing building height
seen in Table 6 is not expected to continue
with further increase in the number of storeys
outside the range considered, as the influence
of SSI generally decreases with increasing
slenderness of the building in taller buildings.
CONCLUSIONS AND
RECOMMENDATIONS
The material presented in this paper demonstrated the importance of inertial SSI,
which has the beneficial effect of reducing
design spectral values or base shear in most
building structures, but also increasing their
lateral deformation. It was observed that
effects of SSI increase with decreasing stiffness of the site soil. This effect of soils is in
addition to their amplification potential and
Building
type
T̃
T
T̃
(sec)
Soil
Soil
h̄
R
D
E
D
E
1
1.051
1.200
0.662
0.756
2
1.181
1.640
0.792
3
1.267
1.894
4
1.464
2.375
β̃0
(%)
β̃
(%)
Soil
Soil
D
E
D
E
0.912
1.0
6.0
5.30
8.89
1.099
1.671
1.8
8.0
4.83*
9.13
1.128
1.686
2.431
1.3
4.2
3.76*
4.94*
1.874
3.040
3.951
1.1
3.0
2.69*
3.37*
* For computed values of the effective damping less than 5%, the minimum damping is taken the same as the structural damping, i.e. 5%.
Table 6 The seismic coefficients and the base-shear reduction
Building
type
Cs
C̃s
∆V
V
(%)
Soil
Soil
Soil
D
E
D
E
D
E
1
0.26
0.37
0.24
0.31
7
23
2
0.24
0.35
0.20
0.21
12
37 (max = 30)
3
0.18
0.26
0.14
0.14
16
32 (max = 30)
4
0.13
0.18
0.09
0.08
20
39 (max = 30)
tends to compensate for part of the seismic
base shear demand associated with response
amplification.
According to the state of the art, the actual
amplification potential of site soils is much
more than stipulated in older design codes
like EC8 (1994) and SABS (1989). It has been
shown in this paper that the cost implications
due to site amplifications, which in some
cases could be prohibitive, may be significantly offset if SSI provisions are introduced
in design codes. The necessary procedures are
available in recent code provisions such as the
NEHRP series (BSSC 2004, 2010).
Recent research has shown that codespecified relationships in design codes for
computing the period lengthening, the effective damping and the reduction in base shear
are meanwhile calibrated using recorded and
measured data from the near past such that
these provisions can give reliable results. In
fact, it can be said that the state of current
knowledge and confidence attained with regard
to seismic SSI is comparable with that of the
amplification potential of site soils. The examples considered in the paper demonstrated that
substantial savings could indeed be achieved
by accounting for seismic SSI effects. It is thus
suggested that engineers are encouraged to use
such provisions for a potentially economical
structural design until these provisions make
their way into the local code.
As a final note, it should be recalled
that SSI has also the effect of increasing
the total lateral deformation of buildings,
which will in turn have an impact on the
ductility requirements of the structure and
on secondary effects like P–Δ. This aspect of
SSI should also be duly accounted for in the
design process, especially for tall structures.
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