buildings
Article
Development of a New Response Spectrum Analysis Approach
for Determining Elastic Shear Demands on Shear-Dominated
Steel Building Frames
Junlin Li 1,2 and Wei Wang 1,2, *
1
2
*
State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China
Department of Structural Engineering, Tongji University, Shanghai 200092, China
Correspondence: weiwang@tongji.edu.cn
Abstract: This research is focused on improving the conventional response spectrum analysis (CRSA)
method for the elastic shear demands estimation of shear-dominated steel building frames. An
alternative approach named the improved response spectrum analysis (IRSA) method is proposed
and validated in this paper. A simplified procedure to capture the dynamic features of a continuous
shear beam (CSB) with stepped stiffness is first presented, and then validated. The CSB is employed in
IRSA to replace the original eigenvalue analysis in CRSA to provide the modal parameter estimation
for the considered system. A modified SRSS (MSRSS) mode superposition based on a genetic
algorithm is then proposed and employed in IRSA. Based on the analyses conducted in this research,
it is found that using first three modes in MSRSS to execute mode superposition could provide a
great estimation of the elastic shear demands distribution. The amplification of weighting coefficients
for the second and third mode contribution indicates the underestimation of the high mode effect
in CSRSS. Further, response history analyses (RHA) are performed on two demonstration building
frames to evaluate the improvement of the IRSA. The results indicate that IRSA provides a more
precise estimation on the elastic shear force demand distribution in shear-dominated steel building
frames under seismic effects compared with that which was achieved by CRSA.
Citation: Li, J.; Wang, W.
Development of a New Response
Spectrum Analysis Approach for
Keywords: response spectrum analysis; elastic shear demands; continuous shear beam; modified
SRSS mode superposition
Determining Elastic Shear Demands
on Shear-Dominated Steel Building
Frames. Buildings 2023, 13, 258.
https://doi.org/10.3390/
buildings13010258
Academic Editors: Flavio Stochino
and Chiara Bedon
Received: 22 September 2022
Revised: 9 January 2023
Accepted: 11 January 2023
Published: 16 January 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1. Introduction
Response spectrum analysis (RSA) is a linear analysis method that is used to estimate
the structural response under dynamic excitations [1]. In this method, the peak response of
a considered multiple-degrees-of-freedom system (MDOF) is combined from those of the
single-degrees-of-freedom systems (SDOFs), each of which has a vibration period that is
equal to that associated with a certain vibration mode of the MDOF system. The modal
contributions from those SDOFs are then combined using a certain criterion. Considering
the nonlinearity of a structure during major earthquakes, the elastic demand derived from
the modal composition needs to be further deducted. The RSA has been widely accepted
by structural engineers in the design process due to its practicability. Many prevailing
codes for seismic designs, such as ASCE/SEI 7-16 [2], treat it as an effective way to estimate
the story shear demands for a considered system during design-basis earthquakes.
Many mode superposition methods have been proposed and studied since the RSA
was proposed. Among these mode superposition methods, the square root of the sum of
the squares (SRSS) method is widely used in mode superposition by structural engineers.
This method is actually a statistical method based on the hypothesis that an earthquake is
a stationary Gaussian process, and the cross-correlation between the modal responses is
negligible [3]. The SRSS method proposes that the peak response of the considered system
can be estimated as the square root of the sum of the squares of the peak response from each
Buildings 2023, 13, 258. https://doi.org/10.3390/buildings13010258
https://www.mdpi.com/journal/buildings
Buildings 2023, 13, 258
2 of 17
mode of vibration. To date, this method is recommended in numerous codes for seismic
design [2,4]. From the structural point of view, using SRSS mode superposition in the RSA
provides an alternative approach to include the contribution from the high-mode effect.
Nevertheless, some research has pointed out the detrimental defects of the SRSS method, i.e.,
the cross-correlation between the modal responses is neglected. Hence, the CQC method
was developed to include the effect of cross-correlation between the modal responses [5].
Chopra summarized the application scope of each method and proposed that adopting
modal combination rules in the RSA to conduct an elastic analysis may generally lead to the
inadequate results, especially in upper story of the system [1]. The maximum deviation of
the considered research was up to 25%. Moreover, the researcher also indicated that using
the mode superposition to estimate the response of the structure under a single ground
motion record may lead to much more deviations. This is caused by the assumptions of
the random vibration theory behind the derivations for each method. The similar defect
of RSA also exists while performing a nonlinear analysis. To more accurately capture
the nonlinear demands during major earthquakes, some modified RSA were proposed to
adjust the contribution of the higher mode effect. Khy and Chintanapakdee et al. proposed
a modified response spectrum analysis to capture the shear force demand for tall RC shear
wall buildings [6]. Sullivan, Priestley et al. proposed a substitute structure method to
include the higher mode effect for ductile structures [7]. Pennucci, Sullivan et al. proposed
that the higher mode effect should be considered based on the ductile state [8]. Moreover,
some spine systems were developed to eliminated the higher mode effect under nonlinear
deformation [9,10].
In this research, the authors intend to propose an improved response spectrum analysis (IRSA) method in two dimensions to estimate the elastic base shear demands in the
preliminary design stage on steel building frames, whose earthquake-induced lateral displacements are dominated by shear and which are denoted as shear buildings hereafter.
Note that this preliminary study is focused on normal two-dimensional steel building
frames with well-separated natural frequencies. The cross-correlation between the modal
responses is minor and neglected. Thus, the SRSS method, instead of CQC method, is
adopted in this study. A simplified model was first proposed to obtain the modal properties
of the considered shear building frames, which are needed in the IRSA. Specifically, in the
model, a considered shear building can be represented by a shear-dominated cantilever
with varied shear stiffness. As will be described in detail in the following sections, the
simplified model eliminates the needs for the eigenvalue analysis of the original shear
building in finite element software. As such, the simplified model is particularly attractive
in the preliminary design stage when many design parameters may be varied. Aside from
the simplified model for gleaning the modal properties, this paper establishes a new rule to
combine the shear demands of the story associated with different vibration modes, which
is denoted as the modified SRSS method (MSRSS). The advantage of the MSRSS method
over the other existing approaches and design code recommendations are demonstrated
through the parametric analyses of representative shear building examples. Finally, the
authors summarize the application scope and limitation of the study and point out the
direction of further research.
2. A Simplified Model for Shear Building
While the earthquake-induced lateral displacement of a building structure combines
the contributions of the shear and flexural demands, according to past investigations [2,11],
a cantilever member can be used to approximate the modal property of the building.
Generally speaking, finite element (FE) model analysis could provide a precise prediction about the modal parameters for the RSA. However, the modeling process will
cost great deal of time in the preliminary design stage. Additionally, adjusting the design
parameters will incur a large workload to revise the FE model. Thus, in this study, an algorithm based on a cantilever member is adopted, which can also be involved in parametric
analyses. When a building frame is categorized as a “shear building”, it can be represented
Buildings 2023, 13, 258
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by a cantilever member with proper shear stiffness and mass distributions. This section
introduces how to build a continuous shear beam model with stepped stiffness based on
the uniform shear beam model.
2.1. Continuous Shear Beam with Stepped Stiffness
Referring to prior research [12], the differential equation of a shear beam along its
height is shown as Equation (1).
ρ
∂
∂2 u(z, t)
∂u(z, t)
−
GA
(
z
)
=0
∂z
∂z
∂t2
(1)
where u is displacement, t is time, z is height from the ground, and GA is the shear stiffness.
The mass distribution ρ is assumed to be uniform along the height in this model. When we
substitute x = z/H and GA(x) = GA0 S(x) into Equation (1), Equation (2) can be obtained. x
is the normalized height, S(x) is the shear stiffness ratio of the section of x height to that
of base, H is the height of the structure, and GA0 is the shear stiffness at the base of the
shear beam.
ρ
∂2 u( x, t) GA0 ∂u( x, t) d[S( x )] GA0
∂2 u( x, t)
−
S
(
x
)
=0
−
∂x
dx
∂t2
H2
H2
∂x2
(2)
When we decompose the variables, u(x, t) can be expressed as Equation (3), where φ(x)
is the mode shape which determines the relative distribution of displacements along the
height, and the function q(t) defines the way it varies over time.
u( x, t) = φ( x )q(t)
(3)
By substituting Equation (3) into Equation (2), Equation (2) can be decomposed into
Equations (4) and (5). Equation (4) defines the free vibration of an undamped SDOF system
with circular vibration frequency ω.
d2 q ( t )
+ ω 2 q(t) = 0
dt2
S( x )
d2 [φ( x )] d[S( x )] d[φ( x )]
ρH 2 2
+
+
ω φ( x ) = 0
dx
dx
GA0
dx2
(4)
(5)
It is Equation (5) which precisely governs the vibration mode shape and the corresponding circular vibration frequency ω. This equation also indicates that the dynamic
features of the shear beam system can only be affected by the stiffness distribution function
S(x) with the given parameters ρ, H, and GA0 .
For most building structures, the inter-story stiffness is not uniform along the building’s height. Generally speaking, the inter-story stiffness of a lower story is higher than
that of an upper story. For a multi-story building structure, adjacent stories often possess
the same beams and columns. Thus, the stiffness distribution of a multi-story building is
usually stepped, and it decreases progressively from the base to the roof of the building,
as shown in Figure 1. The stiffness distribution function S(x) for this distribution type is
correspondingly stepped as shown. This distribution makes S(x) remain as a constant in
a specific length of a shear member. Thus, for the nth segment of the shear-dominated
cantilever member, Equation (5) can be transformed into Equation (6), where Sn is the
stiffness distribution coefficient of nth segment of the member. Note that Equation (6) is
also remain valid for the member with uniform properties.
Sn
d2 [φ( x )]
ρH 2 2
+
ω φ( x ) = 0
2
GA0
dx
(6)
where A1,n and A2,n are constants of the nth segment of the member, which depend on
the
4 of 17
boundary conditions. βn is a constant associated with the stiffness and mass of the nth
segment of the member. βn in Equation (8) can be expressed as below.
Buildings 2023, 13, x FOR PEER REVIEW
Buildings 2023, 13, 258
ρ H 2 2 S1β12
2
=
=
d 2 βφ
 ρωH 2 2
n ( x)
SnGA
ω Sφn ( x ) =
Sn
0
+0
2
dx
GA0
4 of 17
(8)
(6)
The general solution of Equation (6) can be easily obtained as Equation (7).
=
φn ( x ) A1,n sin( β n x) + A2,n cos( β n x)
(7)
where A1,n and A2,n are constants of the nth segment of the member,
depend on the
GAwhich
0S ( x)
boundary conditions. βn is a constant associated with the stiffness and mass of the nth
segment of the member. βn in Equation (8) can be expressed as below.
x
ρ H 2 2 S1β12
=
=
β
ω
SnGA0
Sn
2
n
GA0
(8)
Figure1.1.Illustration
Illustrationof
ofthe
thestiffness
stiffnessdistribution
distributionalong
alongbuilding
buildingheight.
height.
Figure
Thegeneral
boundary
conditions
of each (6)
segment
the shear-dominated
cantilever
The
solution
of Equation
can beof
easily
obtained as Equation
(7). member
should be considered. As presented in Equation (9), the displacement of the lower end of
S ( x(n−1)th
)
φn (consistent
x ) = A1,n sin
( β nthat
x ) +ofAthe
( β n xend
) GA
(7)
2,n cos
the nth segment should be
with
upper
of 0the
segment
(the displacement of the base should be zero, φ1(0) = 0), as illustrated in Figure 2.
where A1,n and A2,n are constants of the nth segment of the member, which depend on the
x
boundary conditions. βn is a constant associated with the stiffness and mass of the nth
segment of the member. βn in Equation (8) can be expressed as below.
S1 β21
ρH 2
β2n =
ω2 =
Figure 1. Illustration of the stiffness distribution
Sn GA0along building
Sn height.
GA0
(8)
The
Theboundary
boundaryconditions
conditionsof
ofeach
eachsegment
segmentof
ofthe
theshear-dominated
shear-dominatedcantilever
cantilevermember
member
should
be
considered.
As
presented
in
Equation
(9),
the
displacement
of
the
lower
should be considered. As presented in Equation (9), the displacement of the lowerend
endof
of
the
nth
segment
should
be
consistent
with
that
of
the
upper
end
of
the
(n
−
1)th
segment
the nth segment should be consistent with that of the upper end of the (n−1)th segment
(the
be
(thedisplacement
displacement
of
thebase
baseshould
should
bezero,
zero,φφ11(0)
(0)== 0),
0), as
as illustrated
illustrated in
in Figure
Figure 2.
2.
Figure
2. Illustrationof
ofthe
boundary
conditions.
Moreover, the shear between the two contacted faces in the adjacent segments should
be consistent, as Equation (10) (the shear force at the top should be zero, Snφ’n(1) = 0).
φn −1 ( xn −1 ) = φn ( xn −1 )
(9)
Sn −1φn′−1 ( xn −1 ) = Snφn′ ( xn −1 )
(10)
By substituting Equation (8) into Equations (9) and (10), for each segment, the equation
matrix
for boundary
conditions
can be established as Equation (11).
Figure
Illustration
theboundary
boundary
conditions.
Figure
2.2.Illustration
ofofthe
conditions.
Moreover,the
theshear
shearbetween
betweenthe
thetwo
twocontacted
contactedfaces
facesininthe
theadjacent
adjacentsegments
segmentsshould
should
Moreover,
beconsistent,
consistent,as
asEquation
Equation(10)
(10)(the
(the
shear
force
the
top
should
zero,
n(1)==0).
0).
be
shear
force
atat
the
top
should
bebe
zero,
SnSφn0φn’(1)
(
)
(
φn−φ
φnφ(nxnx−n1−1)
1 (nx
−1n−x1n)−1= =
)
S n −1 φ 0 n −1 ( x n −1 ) = S n φ 0 n ( x n −1 )
(9)
(9)
(10)
(10)
By substituting Equation (8) into Equations (9) and (10), for each segment, the equation
matrixByfor
boundary conditions
caninto
be established
as and
Equation
(11).
substituting
Equation (8)
Equations (9)
(10), for
each segment, the equation matrix for boundary conditions can be
established
as
Equation
(11).


A1,1
 A2,1 


 A1,2 



P( β 1 ) · 
(11)
 A2,2  = 0
 ... 


A 
1,n
A2,n
Sn −1φn′−1 ( xn −1 ) = Snφn′ ( xn −1 )


P ( β1 ) ⋅  A2,2  =
0
 ... 


 A1,n 
A 
 2,n 
Buildings 2023, 13, 258
(11)
5 of 17
where P(β1) is the coefficient matrix. To derive a nontrivial solution for A1,n and A2,n, the
determinant of P(β1) should be zero as Equation (12). Additionally, β1,i for the ith mode
where
the coefficient
matrix.
To derive
nontrivial
solution
for A1,n The
and miniA2,n ,
can beP(β
solved
it. Note that
the solution
for aEquation
(12)
is not unique.
1 ) is using
the
determinant
should
be zero
as Equation
(12).
ith
mum
solution β1,1ofisP(β
for1 )the
first mode,
while
the greater
β1,i Additionally,
is the solutionβfor
thethe
higher
1,i for
mode
can be solved using it. Note that the solution for Equation (12) is not unique. The
ith mode.
minimum solution β1,1 is for the first mode, while the greater β1,i is the solution for the
det ( P ( β1 ) ) = 0
(12)
higher ith mode.
det(P( β 1 )) = 0
(12)
Given the β1,i for the ith mode, the circular
frequency ωi and βn,i can be obtained from
Equation
theith
mode
shape
n,i for the
ith mode
Given(8).
the Finally,
β1,i for the
mode,
the φ
circular
frequency
ω ican
andbeβnobtained.
,i can be obtained from
Equation (8). Finally, the mode shape ϕn ,i for the ith mode can be obtained.
2.2. Validation of the Approach
2.2. Validation
of the
To validate
theApproach
equations derived in Section 2.1 for the periods and the mode shapes,
To validate the
equations
derived
in Section
2.1 for
the periods The
and DSB
the mode
shapes,
a demonstration
shear
beam (DSB)
in FEM
software
is established.
is discretized
ainto
demonstration
shear
beam
(DSB)
in FEM
software
is established.
The DSB
is discretized
1000 elements
which
only
possess
shear
stiffness.
The mass is evenly
assigned
at two
into
1000
elements
which
only
possess
shear
stiffness.
The
mass
is
evenly
assigned
at
end nodes of each element. The distribution of the shear stiffness of the DSB and the contwo
end
nodes
of
each
element.
The
distribution
of
the
shear
stiffness
of
the
DSB
and
the
tinuous shear beam (CSB) are illustrated in Figure 3a. The lateral stiffness of upper part is
continuous
shearstiffness
beam (CSB)
are illustrated
in Figure
The lateral the
stiffness
20% base shear
less than
that of lower
part. 3a.
Additionally,
mass of
is upper
evenlypart
disistributed
20% base
shear
less
that Figure
of lower
Additionally,
the mass
evenly
along
thestiffness
height of
thethan
system.
3bpart.
presents
the comparisons
foristhe
first
distributed
thefor
height
of systems.
the system.
Figure
3b three
presents
theshapes
comparisons
for the
three modalalong
shapes
the two
Note
that the
modal
are all normalfirst
three
modal
shapes
for
the
two
systems.
Note
that
the
three
modal
shapes
all
ized, and the maximum displacement for each modal shape of each system is 1.0. Itare
could
normalized,
and
the
maximum
displacement
for
each
modal
shape
of
each
system
is
1.0.
It
be clearly noticed that the results from CSB match these from DSB, validating the accuracy
could
be
clearly
noticed
that
the
results
from
CSB
match
these
from
DSB,
validating
the
of the derived equations.
accuracy of the derived equations.
(a)
1st mode from FEM ω1=1.319
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
Displacement φ1(x)/φ1(1)
Normalized height x
Normalized height x
S ( x)
1st mode from CSB ω1=1.320
0.8
1.0
1.0
1.0
1.0
2nd mode from CSB ω2=3.284
0.8
Normalized height x
x
2nd mode from FEM ω2=3.281
0.6
0.4
0.2
0.0
-1.0
-0.5
0.0
0.5
Displacement φ2(x)/φ2(1)
1.0
0.8
0.6
0.4
0.2
0.0
-1.0
3rd mode from CSB ω3=5.362
3rd mode from FEM ω3=5.358
-0.5
0.0
0.5
1.0
Displacement φ3(x)/φ3(1)
(b)
Figure3.3. Validation
Validation of
ofCSB
CSBmodel.
model. (a)
(a) Stiffness
Stiffness distribution.
distribution. (b)
(b) Comparisons
Comparisons for
for the
the first
first three
three
Figure
modal shapes.
modal shapes.
Though, the
the consistency
consistency between
between CSB
CSB and
and FEM
FEM is
is significant
significant in
in Figure
Figure 3b,
3b, and
and the
the
Though,
discrepancy
between
CSB
and
the
discretized
model
still
exists,
especially
when
there
are
discrepancy between CSB and the discretized model still exists, especially when there are
not
that
many
degrees
of
freedom.
Note
that
the
building
frames
are
actually
discretized
not that many degrees of freedom. Note that the building frames are actually discretized
systems.Thus,
Thus,totovalidate
validatethe
theapplicability
applicability
CSB,
FEM
analyses
with
different
numsystems.
ofof
CSB,
thethe
FEM
analyses
with
different
numbers
bers
of
element
were
performed.
Figure
4
presents
the
comparison
of
modal
parameters
of element were performed. Figure 4 presents the comparison of modal parameters between
between
FEM.
Threepossessing
FEMs possessing
five,
ten,hundred
one hundred
numerical
elements
CSB
and CSB
FEM.and
Three
FEMs
five, ten,
one
numerical
elements
are
are analyzed.
It should
noted
that
discrepancydoes
doesincrease
increasewith
withthe
thedecrease
decreasein
in
analyzed.
It should
be be
noted
that
thethe
discrepancy
degrees of freedom. However, Figure 4 presents satisfactory discrepancies in the modal
parameters between CSB and FEM, and even the degree of freedom is five.
Buildings 2023, 13, x FOR PEER REVIEW
6 of 17
degrees of freedom. However, Figure 4 presents satisfactory discrepancies in the modal
6 of 17
parameters between CSB and FEM, and even the degree of freedom is five.
Buildings 2023, 13, 258
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.0
Normalized height x
0.8
1.0
CSB
FEM-100
FEM-10
FEM-5
Normalized height x
Normalized height x
1.0
0.8
0.6
0.4
CSB
FEM-100
FEM-10
FEM-5
0.2
0.0
-1.0
Displacement φ1(x)/φ1(1)
-0.5
0.0
0.5
0.8
0.6
0.4
CSB
FEM-100
FEM-10
FEM-5
0.2
0.0
-1.0
1.0
-0.5
0.0
0.5
1.0
Displacement φ3(x)/φ3(1)
Displacement φ2(x)/φ2(1)
Figure 4. The influence of the number of degrees of freedom.
Figure 4. The influence of the number of degrees of freedom.
ModifiedSRSS
SRSSMethod
Methodfor
forImproved
ImprovedResponse
ResponseSpectrum
SpectrumAnalysis
Analysis
3.3.Modified
Asmentioned
mentionedbefore,
before,prior
priorresearch
researchhas
hasreported
reportedthat
thatadopting
adoptingthe
theconventional
conventional
As
SRSS
the RSA
RSA may
maylead
leadtotoinadequate
inadequateresults
results
[1].
Additionally,
as
SRSS(CSRSS)
(CSRSS)method
method in
in the
[1].
Additionally,
as recrecommended
ASCE/SEI
the RSA
should
include
a minimum
number
of
ommended in in
ASCE/SEI
7–167–16
[2], [2],
the RSA
should
include
a minimum
number
of modes
modes
to obtain
a combined
mass participation
of at90%
least
of the actual
mass.
to obtain
a combined
modalmodal
mass participation
of at least
of90%
the actual
mass. Not
all
Not
all of
the modes
have
to be considered,
willlead
also to
lead
the reduction
the
of the
modes
have to
be considered,
whichwhich
will also
thetoreduction
of theofshear
shear
demands.
To mitigate
the limitation
of the
CSRSS,
MSRSS
method
was
developed
demands.
To mitigate
the limitation
of the
CSRSS,
the the
MSRSS
method
was
developed
for
for
use
in
the
RSA
in
this
research.
The
following
first
briefly
presents
the
RSA
use in the RSA in this research. The following first
presents the RSAprocedure
procedure
with
withCSRSS.
CSRSS.Then,
Then,MSRSS
MSRSSisisproposed
proposedbased
basedon
onadjusting
adjustingthe
theweighting
weightingfactors
factorsfor
foreach
each
modal
modalcontribution.
contribution.
3.1.
3.1.Response
ResponseSpectrum
SpectrumAnalysis
Analysiswith
withCSRSS
CSRSS
For
an
elastic
building
frame,
the
For an elastic building frame, thedisplacement
displacementcan
canbe
beexpressed
expressedas
asEquation
Equation (13)
(13)inin
the RSA, where N is the number of considered modes; Γn is the participation factors of the
the RSA, where N is the number of considered modes; Гn is the participation factors of the
nth mode given by Equation (14); ι can be calculated according to Equation (15); Dn (t) is
nth mode given by Equation (14); ι can be calculated according to Equation (15); Dn(t)
the displacement response history of the SDOF system, representing the nth mode.
is the displacement response history of the SDOF system, representing the nth mode.
N
N
Γ nnφD
u ( t=
u(t) =
) Γ∑n φ
nn((tt))
∑
nD
(13)
(13)
T
φnTφmι
n mι
Γn =Γ n =
T
T
φn mφ
φn mnφn
(14)
(14)
n=1 n =1
N
ι = ∑ ΓN n φn
=
nι=1 ∑ Γ nφn
(15)
(15)
n=1
The maximum effective seismic force on each floor can be given as follows:
The maximum effective seismic force on each floor can be given as follows:
f n = fΓn=mφ
nA
Γm
φ nA
n
n
n
n
(16)
(16)
where
n nisisthe
whereAA
thedesign
designspectrum
spectrumacceleration
accelerationassociated
associatedwith
withthe
thenth
nthmode.
mode.The
Thedesign
design
shear
force
demand
associated
with
the
ith
story
and
nth
mode
F
can
be
then
,n
i
shear force demand associated with the ith story and nth mode Fi,n can be thenobtained
obtained
by
byaccumulating
accumulatingthe
theeffective
effectiveseismic
seismicforce
forcealong
alongthe
thebuilding
buildingheight.
height.The
Thedesign
designshear
shear
force
for
the
ith
story
F
can
be
obtained
by
combining
F
using
the
SRSS
method
as
,n
i
force for the ith story Fi ican be obtained by combining Fi,n using
the SRSS method
as EquaEquation (17).
tion (17).
!0.5
N
Fi =
2
0.5
∑ FNi,n
2 
Fi n==1∑ Fi ,n 
 n =1 
(17)
(17)
3.2. The MSRSS Method
Since the CSRSS method was proposed and applied in structural engineering [3], it has
been widely accepted by engineers. From the structural point of view, the CSRSS method
indicates that each modal vibration of the system is a stationary stochastic process, and it
provides an alternative approach to include the high mode effect of the considered system.
3.2. The MSRSS Method
3960×19=75,240
36,600
6100
6100
6100
6100
3650×2=7300
6100
5490
6100
6100
6100
6100
6100
Buildings 2023, 13, 258
Since the CSRSS method was proposed and applied in structural engineering [3], it
has been widely accepted by engineers. From the structural point of view, the CSRSS
of 17
method indicates that each modal vibration of the system is a stationary stochastic7 process, and it provides an alternative approach to include the high mode effect of the considered system. Nevertheless, some researchers pointed out that using the CSRSS method
may lead to inadequate
results, especially
in that
upper
story
ofCSRSS
a multi-story
as deNevertheless,
some researchers
pointed out
using
the
methodsystem,
may lead
to
scribed
above
[2].
inadequate results, especially in upper story of a multi-story system, as described above [2].
Toquantitatively
quantitativelyaddress
addressthe
the
limitation
of the
CSRSS
method,
a 20-story
moTo
limitation
of the
CSRSS
method,
a 20-story
steel steel
momentment-resisting
(benchmark
structure)
designed
forLos
theAngeles,
Los Angeles,
California
reresisting
frame frame
(benchmark
structure)
designed
for the
California
region,
gion, which
hasconsidered
been considered
past investigations
for different
purposes
which
has been
in pastin
investigations
for different
researchresearch
purposes
[13], is
[13], is revisited
in this research.
5 shows
the plan
floorof
plan
the benchmark
building
revisited
in this research.
FigureFigure
5 shows
the floor
theofbenchmark
building
and
and
the
elevation
of
the
selected
lateral
force
resisting
frame.
As
shown,
the
structural
the elevation of the selected lateral force resisting frame. As shown, the structural system
system
m (5by
bays)
bym
36.60
m (6 bays)
in plan.
bay
is 6.10
in length
in both
is
30.50 is
m30.50
(5 bays)
36.60
(6 bays)
in plan.
EachEach
bay is
6.10
m inmlength
in both
of
of the
directions.
The
lateral
force
resisting
framesare
arearranged
arrangedasasexterior
exteriorbays,
bays,while
while the
the
the
directions.
The
lateral
force
resisting
frames
interiorbays
baysare
aresteel
steelframes
frameswith
withsimple
simpleconnections.
connections.
The
floor-to-floor
height
a typinterior
The
floor-to-floor
height
forfor
a typical
story
is 3.96
m, and
for the
levellevel
andand
basement
levels,
theythey
are are
5.495.49
andand
3.653.65
m,
ical story
is 3.96
m, and
for ground
the ground
basement
levels,
respectively.
More
detailed
information
can be
in prior
research
[13]. The
m, respectively.
More
detailed
information
canfound
be found
in prior
research
[13].selected
The seframe
numerically
modelledmodelled
as a frame
in Opensees
to perform
theperform
eigenvalue
lectedisframe
is numerically
asmodel
a frame
model in[14]
Opensees
[14] to
the
analysis
andanalysis
the RHA.
that this
research
focuses
onfocuses
the mode
of the
eigenvalue
andNote
the RHA.
Note
that this
research
on superposition
the mode superposielastic
and the
nonlinearity
the materialofisthe
notmaterial
considered
in this
research.inThus,
tion ofmodes,
the elastic
modes,
and the of
nonlinearity
is not
considered
this
the
columns
andthe
beams
are modelled
using
Timoshenko
beamTimoshenko
elements. The
lumped
research.
Thus,
columns
and beams
areelastic
modelled
using elastic
beam
elemass
at The
eachlumped
floor formass
the considered
frame
is 276
ton. Additionally,
theton.
rigid
diaphragm
ments.
at each floor
for the
considered
frame is 276
Additionally,
assumption
is adopted
in this FEismodel.
This
model
is established
to obtain
the modal
the rigid diaphragm
assumption
adopted
in this
FE model.
This model
is established
to
properties
and
quantify
the
shear
demands
using
the
RHA
under
each
earthquake
ground
obtain the modal properties and quantify the shear demands using the RHA under
each
motion.
Thisground
is followed
byThis
obtaining
shearby
demands
using
the
equivalent
lateral
force
earthquake
motion.
is followed
obtaining
shear
demands
using
the equiv(ELF)
procedure
and
the
RSA
[2].
alent lateral force (ELF) procedure and the RSA [2].
6100
30,500
Figure 5.
5. Floor
Floor and
and elevation
elevation of
of the
the demonstration
demonstration frame.
frame.
Figure
Buildings 2023, 13, x FOR PEER REVIEW
The first
firstthree
threemode
modeshapes
shapesofofthe
theconsidered
consideredsystem
system
from
model
presented
The
from
FEFE
model
areare
presented
in
Figure
6, and
thethe
corresponding
mode
periods
areare
alsoalso
presented
in the
figure.
in Figure
6, and
corresponding
mode
periods
presented
in the
figure.
16
Story N
Story N
16
20
1st mode Τ1 = 3.897s
12
8
20
2nd mode Τ2 = 1.352s
16
Story N
20
12
8
12
8
4
4
4
0
0.0
0
-1.0
0
-1.0
0.2
0.4
0.6
0.8
Displacement φ1(N)/φ1(20)
1.0
8
-0.5
0.0
0.5
Displacement φ2(N)/φ2(20)
1.0
3rd mode Τ3 = 0.789s
-0.5
0.0
0.5
1.0
Displacement φ3(N)/φ3(20)
Figure 6. The modal
of the
considered
frame.
Figureinformation
6. The modal
information
of benchmark
the considered
benchmark frame.
As recommended in ASCE/SEI 7–16 [2], the RSA should include a minimum nu
of modes to obtain a combined modal mass participation of at least 90% of the actual
Therefore, the first three modes of the benchmark frame are used in the CSRSS mod
Stor
Stor
Stor
8
8
8
4
4
4
0
0.0
0
-1.0
0
-1.0
0.2
0.4
0.6
0.8
1.0
-0.5
0.0
0.5
1.0
-0.5
0.0
0.5
1.0
Displacement φ3(N)/φ3(20)
Displacement φ2(N)/φ2(20)
Displacement φ1(N)/φ1(20)
Buildings 2023, 13, 258
3rd mode Τ3 = 0.789s
8 of 17
Figure 6. The modal information of the considered benchmark frame.
As recommended
recommended in
in ASCE/SEI
ASCE/SEI 7–16
As
7–16 [2],
[2], the
the RSA
RSA should
should include
include aa minimum
minimum number
number
of modes
modal
mass
participation
of atofleast
90% of
theof
actual
mass.
of
modesto
toobtain
obtaina acombined
combined
modal
mass
participation
at least
90%
the actual
mass.
Therefore,
thethree
first three
modes
the benchmark
frame
are used
in CSRSS
the CSRSS
mode
Therefore,
the first
modes
of theofbenchmark
frame
are used
in the
mode
susuperposition.
Twenty
typical
earthquake
ground
motions,
named
LA01-LA20,
in AnLos
perposition. Twenty
typical
earthquake
ground
motions,
named
LA01-LA20,
in Los
Angeles
from
SAC
steel
project[15]
[15]are
areemployed
employedfor
forthe
thecomparison.
comparison. The
The shear
shear force
geles from
thethe
SAC
steel
project
force
demands
of
the
considered
frame
from
the
varied
calculation
methods
under
the
20
ground
demands of the considered frame from the varied calculation methods under the 20
motions
(LA1-20)(LA1-20)
are summarized
in Figure in
7. Figure
Note that
the CSRSS
curve
in each
subfigure
ground motions
are summarized
7. Note
that the
CSRSS
curve
in each
represents
the shear demands
from the RSA
the with
CSRSS
The MSRSS
subfigure represents
the shear demands
from with
the RSA
thesuperposition.
CSRSS superposition.
The
mode
will be discussed
in depthinlater.
MSRSSsuperposition
mode superposition
will be discussed
depth later.
5
20,000
0
30,000
5
0
Shear force F (kN)
15
5
10,000
20,000
10
0
30,000
0
10,000
20,000
15
Story N
0
15
10,000
20,000
0
30,000
15
10
0
0
0
0
15
20,000
30,000
10
RHA-LA18
ELF-LA18
CSRSS-LA18
MSRSS-LA18
20
15
10
5
0
0
0
0
20,000
Shear force F (kN)
30,000
0
10,000
20,000
Shear force F (kN)
30,000
0
10,000
20,000
30,000
20,000
30,000
Shear force F (kN)
10,000
20,000
RHA-LA15
ELF-LA15
CSRSS-LA15
MSRSS-LA15
20
15
10
0
30,000
0
10,000
20,000
30,000
Shear force F (kN)
RHA-LA19
ELF-LA19
CSRSS-LA19
MSRSS-LA19
10
5
10,000
0
15
5
10,000
5
20
5
0
0
Shear force F (kN)
Story N
10
RHA-LA17
ELF-LA17
CSRSS-LA17
MSRSS-LA17
20
10,000
10
0
RHA-LA14
ELF-LA14
CSRSS-LA14
MSRSS-LA14
Shear force F (kN)
Story N
Story N
15
30,000
Shear force F (kN)
RHA-LA16
ELF-LA16
CSRSS-LA16
MSRSS-LA16
20
20,000
15
30,000
10
0
10,000
20,000
15
0
0
10,000
20
5
30,000
30,000
5
5
20,000
20,000
RHA-LA10
ELF-LA10
CSRSS-LA10
MSRSS-LA10
Shear force F (kN)
RHA-LA13
ELF-LA13
CSRSS-LA13
MSRSS-LA13
20
10,000
20
5
0
0
Shear force F (kN)
10
5
10,000
0
30,000
RHA-LA9
ELF-LA9
CSRSS-LA9
MSRSS-LA9
5
0
20,000
20
Shear force F (kN)
10
10,000
Shear force F (kN)
10
0
30,000
RHA-LA12
ELF-LA12
CSRSS-LA12
MSRSS-LA12
20
Shear force F (kN)
Story N
15
Story N
Story N
Story N
10
0
30,000
RHA-LA8
ELF-LA8
CSRSS-LA8
MSRSS-LA8
Shear force F (kN)
RHA-LA11
ELF-LA11
CSRSS-LA11
MSRSS-LA11
15
20,000
5
Shear force F (kN)
20
10,000
20
5
0
0
10
5
Shear force F (kN)
Story N
10
0
0
30,000
RHA-LA7
ELF-LA7
CSRSS-LA7
MSRSS-LA7
20
Story N
Story N
15
20,000
15
5
Shear force F (kN)
RHA-LA6
ELF-LA6
CSRSS-LA6
MSRSS-LA6
20
10,000
10
Story N
10,000
15
Story N
0
10
Story N
0
15
RHA-LA5
ELF-LA5
CSRSS-LA5
MSRSS-LA5
20
Story N
5
10
RHA-LA4
ELF-LA4
CSRSS-LA4
MSRSS-LA4
20
Story N
10
RHA-LA3
ELF-LA3
CSRSS-LA3
MSRSS-LA3
20
Story N
15
Story N
Story N
15
RHA-LA2
ELF-LA2
CSRSS-LA2
MSRSS-LA2
20
Story N
RHA-LA1
ELF-LA1
CSRSS-LA1
MSRSS-LA1
20
RHA-LA20
ELF-LA20
CSRSS-LA20
MSRSS-LA20
20
15
10
5
0
Shear force F (kN)
10,000
20,000
30,000
Shear force F (kN)
0
0
10,000
20,000
30,000
Shear force F (kN)
Figure
Shear force
Figure 7.
7. Shear
force demands
demands of
of the
the considered
considered benchmark
benchmark frame
frame from
from the
the ELF,
ELF,RSA,
RSA,and
andRHA.
RHA.
The
in Figure
The results
results comparison
comparison in
Figure 77 indicates
indicates that
that the
the RSA
RSA method
method with
with the
the CSRSS
CSRSS
superposition
generally
underestimates
the
shear
demands
compared
with
that
of
the
RHA.
superposition generally underestimates the shear demands compared with that of
the
Moreover,
the shear
of the upper
storiesstories
are underestimated
in many
cases when
RHA. Moreover,
thedemands
shear demands
of the upper
are underestimated
in many
cases
the
CSRSS
method
is used
in theinRSA,
suggesting
that that
the high-mode
effect
may
notnot
be
when
the CSRSS
method
is used
the RSA,
suggesting
the high-mode
effect
may
precisely captured by the CSRSS mode superposition, which may lead to inadequate design
be precisely captured by the CSRSS mode superposition, which may lead to inadequate
parameters. These observations indicate the need of an improved method for combining the
design parameters. These observations indicate the need of an improved method for commodal shear demands. As for the ELF procedure, the deviation between the ELF and RHA
bining the modal shear demands. As for the ELF procedure, the deviation between the
is much more significant along the building’s height, which also indicated the limitation of
the lateral force distribution used in the ELF.
As mentioned before, the CSRSS method combines the responses of different modes in
an equal manner, as Equation (17) suggests. However, the results discussed above indicate
that the CSRSS method fails to adequately capture the contributions of the high modes.
Thus, the MSRSS method that equips the shear demand from the ith mode with a weighting
coefficient An is proposed below:
N
Fi =
∑ ( An Fi,n )
n =1
!0.5
2
(18)
Buildings 2023, 13, 258
9 of 17
To clarify the improvement of the MSRSS method and choose the applicable weighting
coefficients, an Error indicator is defined as Equation (19) to represent the deviation between
the story shear demands from the RHA and the RSA with MSRSS. It can be noticed that
Error can also represent the deviation between the shear demands from the RHA and the
RSA with CSRSS by setting An to be equal to 1.0.
0.5 2 0.5

Fi,m,RHA − ∑ ( An Fi,m,n,RSA )
 
i
 1 mGM 
 
 1 FL
n =1

Error = 
 

∑
∑
m
i
F
 
i,m,RHA

 GM m=1  FL i=1


N
2
(19)
Fi,m,RHA is the calculated shear demand of the ith story of the considered frame under
the mth ground motion from the RHA of the frame model; Fi,m,n,RSA is the corresponding
shear force demand of nth mode under the mth ground motion from the RSA of the
simplified model of the benchmark building (i.e., the cantilever member model); N is the
number of modes considered in the RSA method; iFL is the total story number of the frame;
mGM is the number of ground motion records in this research. Note that the factor Error
is the average deviation in terms of the different ground motions and stories. Thus, Error
could reflect the shear demand discrepancy between the RSA method and the RHA method.
To clarify the participation of the high-mode effect on the inter-story shear force demands in the representative benchmark frame, the weighting coefficients An in Equation (19)
are adjusted to achieve the minimum Error. Note that searching for a group of An values
to achieve the minimum Error is a project with many iterations. Therefore, a genetic algorithm (GA) is adopted in this study to find the minimum Error and the corresponding
weighting coefficients. The genetic algorithm (GA) is a search-based optimization technique based on the principles of genetics and natural selection. This method has been
widely in optimization design to decrease the number of iterations. Note that Fi,m,RHA in
Equation (19) is obtained by OpenSees using the RHA method, Fi,m,n,RSA is obtained using
the RSA as mentioned before. The undecided variables in Equation (19) are only weighting
coefficients An .
Figure 7 shows the results from the RSA method with the MSRSS and RHA methods,
respectively. The figure clearly indicates that modifying the weight factor for the higher
modes significantly increases the adequacy of the shear force demand estimation in the
RSA. Compared with the results associated with the CSRSS method, the results from the
MSRSS agree more with the results from the RHA.
Note that three modes are used to conduct the optimal process of the weight coefficients in the RSA-MSRSS. The number of modes in this optimal process is also deemed
as a variable. The authors also present the resulting Error of the CSRSS and MSRSS with
different considered mode numbers in Figure 8. The figure indicates that the Error of
MSRSS is lower than that of CSRSS by considering the weighting coefficients for each mode.
Additionally, it can be clearly noticed that the Error would decrease when we increased the
number of considered modes N, which is in accordance with our cognition. Nevertheless,
the indicator will not decrease remarkably with the increase in N when N is higher than
three. Moreover, the first three modes in the RSA for the benchmark building achieved a
combined modal mass participation of more than 90% of the system reactive mass, which
is required in ASCE/SEI 7–16 [2].
The weighting coefficients An for the first three modes are given in Table 1. The group
of coefficients exhibit some meaningful laws. The coefficient for the first mode is 1.000, and
for the second and third modes, they are 1.276 and 1.663, respectively.
10 of 17
0.5
MSRSS
CSRSS
0.4
Error
Buildings 2023, 13, 258
Nevertheless, the indicator will not decrease remarkably with the increase in N when N is
higher than three. Moreover, the first three modes in the RSA for the benchmark building
achieved a combined modal mass participation of more than 90% of the system reactive
mass, which is required in ASCE/SEI 7–16 [2].
0.3
0.2
0.1
0.0
0
2
4
6
8
10
Mode number
Figure
8. Resulting
of optimal
process
a different
number
of modes.
Figure
8. Resulting
Error Error
of optimal
process
usingusing
a different
number
of modes.
Table 1. Weighting coefficients An for the first three modes.
The
weighting coefficients An for the first three modes are given in Table 1. The group
of coefficients exhibit
someCoefficients
meaningful laws. The coefficient for the Value
first mode is 1.000,
Weighting
and for the second and third modes, they are 1.276 and 1.663, respectively.
A1
A2
Table 1. Weighting coefficients
A3 An for the first three modes.
1.000
1.276
1.663
Weighting Coefficients
Value
4. Validation
A1 of the IRSA Method
1.000
2
1.276
The A
improved
response spectrum analysis with the
modified SRSS mode superposition
includes A
the
method is used to obtain the elastic
3 contents described in Sections 2 and 3. The
1.663
story shear demand for the design of shear-dominated building frames.
4. Validation of the IRSA Method
4.1. Design Procedure of IRSA
The improved response spectrum analysis with the modified SRSS mode superposiCompared
with described
the conventional
RSA
method,
IRSA is
uses
a to
continuous
tion includes
the contents
in Sections
2 and
3. Thethe
method
used
obtain theshear
beam
mode
to
provide
dynamic
parameters
without
conducting
an
eigenvalue
analysis
elastic story shear demand for the design of shear-dominated building frames.
using numerical software. Moreover, adopting the MSRSS mode superposition in IRSA
may achieve
more
adequate shear demand distributions along the building’s height. The
4.1. Design
Procedure
of IRSA
step-by-step design procedure is presented in Figure 9.
Compared
the conventional
RSA method,
IRSAestimation
uses a continuous
shear and
Firstly,with
the engineers
are supposed
to make the
a rough
on the stiffness
beammass
modedistribution.
to provide dynamic
parameters
without
conducting
an
eigenvalue
analysis
Then, the method proposes that they use the continuous shear beam
usingwith
numerical
software.
the MSRSS
mode
superposition
IRSA The
stepped
stiffnessMoreover,
to provideadopting
the dynamic
features
of the
considered in
system.
may achieve
more
adequate
shear
demand
distributions
along
the
building’s
height.
Theforce
dynamic parameters will be then used for conducting the RSA to obtain the shear
step-by-step
design
is presented
9.
demands
alongprocedure
the building’s
height in
forFigure
each mode.
This is followed by the mode superposition by MSRSS to obtain the final shear force demand for each story. Note that
the method is proposed for linearly elastic structures with shear deformation, and so the
systems exhibiting nonlinear behavior and flexural-type deformation are not included in
this research.
Two demonstration building frames are revisited to assess the improvement of the
IRSA method that is presented above. The two frames possess different story numbers
and elevations. The ELF procedure, the RSA method with different mode superpositions, and the RHA method are performed, respectively, using multiple ground motions
to conduct the validations. The following section presents the basic information of the
Buildings 2023, 13, 258
11 of 17
Buildings 2023, 13, x FOR PEER REVIEW
of the
17
demonstration buildings, a detailed procedure based on the proposed approach,11
and
comparison results.
Estimate stiffness and
mass distribution
Start
Satisfied
End
Revise the story shear
force demands
Resistance check
by RHA
Calculate dynamic
features using CSB model
Conduct RSA to obtain
modal shear force demand
Design the elements of
the considered system
Calculate shear force demand
of ith story using MSRSS mode
superposition
Dissatisfied
Figure 9.
9. Design
Design procedure
procedure of
of IRSA.
IRSA.
Figure
4.2. Demonstration
Buildingsare supposed to make a rough estimation on the stiffness and
Firstly, the engineers
massThe
distribution.
Then, the steel
method
proposes that they
useinitially
the continuous
shear
beam
two demonstration
moment-resisting
frames
produced
in the
Caliwith
stepped
stiffness
to
provide
the
dynamic
features
of
the
considered
system.
The
dyfornia Strong Motion Instrumentation Program (CSMIP) are reconsidered in this research.
namic
parameters
will be
then(denoted
used for as
conducting
thedesigned
RSA to obtain
thebased
shearon
force
deOne
frame
is a six-story
frame
6F) that was
in 1976
the 1973
mands
along the building’s
for each
This
is followed
byasthe
mode
UBC
requirements
[16]. The height
other frame
is amode.
13-story
frame
(denoted
13F)
thatsuperpowas built
sition
MSRSS
obtain
the final
shear force
demand
forUBC
each
story. Note Figure
that the10
in
1975,by
which
wastoalso
designed
in accordance
with
the 1973
requirements.
method
is
proposed
for
linearly
elastic
structures
with
shear
deformation,
and
so
the m
sysshows the floor plans of the demonstration buildings. The six-story frame is 36.60
by
tems
exhibiting
nonlinear
behavior
and
flexural-type
deformation
are
not
included
in
this
36.60 m in plan. The selected frame possesses six bays (6 m × 6.10 m) in one direction.
research.
The
thirteen-story frame is 48.80 m by 48.80 m in plan. The selected frame possesses five
building
frames
revisited
assess lateral
the improvement
of the
bays Two
(5 m demonstration
× 9.76 m) in one
direction.
Asare
shown,
two to
exterior
moment resisting
IRSA method
that is presented
above.
Theeach
two horizontal
frames possess
different
numbers
frames
are symmetrically
arranged
along
direction.
The story
interior
frames
and elevations.
Thegravity
ELF procedure,
the RSA
method
with different
mode superpositions,
were
designed as
frames, which
consist
of simple
shear connections
only. The
and the RHA
method
are performed,
respectively,
using multiple
ground
motions
elevations
of the
demonstration
buildings
are presented
in Figure
11. The
heightto
ofconeach
duct
the
validations.
The
following
section
presents
the
basic
information
of
the
demonstory and the sectional information of each column and beam are presented in the figure.
stration
buildings,
a reactive
detailedmasses
procedure
based
onfor
the
approach,
comThe
tributary
seismic
of each
floor
6Fproposed
and 13F are
235.6 tonand
andthe
270.3
ton,
parison
results.
respectively. More detailed information about the demonstration buildings can be found in
prior research [17]. Note that the two building frames are revisited to demonstrate that the
4.2. Demonstration
Buildings 2023, 13, x FOR PEER REVIEW
of the
17
RSA
which adoptsBuildings
the MSRSS method could provide relatively adequate results12for
6@6100=36,600
5@9760=48,800
two demonstration
steel
moment-resisting frames initially produced in the CalshearThe
demands
of the building
stories.
ifornia Strong Motion Instrumentation Program (CSMIP) are reconsidered in this research. One frame is a six-story frame (denoted as 6F) that was designed in 1976 based on
the 1973 UBC requirements [16]. The other frame is a 13-story frame (denoted as 13F) that
was built in 1975, which was also designed in accordance with the 1973 UBC requirements. Figure 10 shows the floor plans of the demonstration buildings. The six-story frame
is 36.60 m by 36.60 m in plan. The selected frame possesses six bays (6 × 6.10 m) in one
direction. The thirteen-story frame is 48.80 m by 48.80 m in plan. The selected frame possesses five bays (5 × 9.76 m) in one direction. As shown, two exterior lateral moment resisting frames are symmetrically arranged along each horizontal direction. The interior
frames were designed as gravity frames, which consist of simple shear connections only.
The elevations of the demonstration buildings are presented in Figure 11. The height of
each story and the sectional information of each column and beam are presented in the
figure. The tributary seismic reactive masses of each floor for 6F and 13F are 235.6 ton and
270.3 ton, respectively. More detailed information about the demonstration buildings can
be found in prior research [17]. Note that the two building frames are revisited to demonstrate that the RSA which adopts the MSRSS method
could provide relatively adequate
5@9760=48,800
6@6100=36,600
results for the shear demands of the building stories.
Figure10.
10.Floor
Floorplan
planof
ofthe
the6F
6F and
and 13F
13F buildings.
buildings.
Figure
5@9760=48,800
×246 W14×167
W27×84
W33×118
W33×118
W33×130
5@9760=48,800
6@6100=36,600
12 of 17
Figure 10. Floor plan of the 6F and 13F buildings.
5@9760=48,800
W27×84
W27×102
W30×116
W33×118
W33×130
W33×141
W33×152
53,033
W33×141
12@4013
W33×130
W33×152
W33×152
W33×152
W33×152
W36×230
W33×194
4420
W14×500
W14×136
W24×84
5@3960=19,800
W24×84
W33×118
4877
6@6100=36,600
5330
W14×95
W24×68
W14×426 W14×398 W14×314 W14×287 W14×246 W14×167
W27×84
W14×184
Buildings 2023, 13, 258
Figure11.
11.Elevation
Elevationofofthe
the6F
6Fand
and13F
13Fbuildings.
buildings.
Figure
4.3.
4.3.Computer
ComputerModeling
Modelingand
andSeismic
SeismicExcitations
Excitations
The
systems
shown
in
Figure
11
The systems shown in Figure 11are
arenumerically
numericallymodelled
modelledininOpenSees
OpenSeestotoconduct
conductthe
the
RHA
[14].
Further,
in
order
to
validate
the
application
scope
of
the
method,
one
RHA [14]. Further, in order to validate the application scope of the method, onespecial
special
case
casebased
basedon
on13F
13Fisisestablished
establishedasas13F-W,
13F-W,ininwhich
whichthe
thebending
bendingstiffness
stiffnessofofthe
thebeams
beamsand
and
columns
below
the
eighth
floor
are
reduced
by
50%.
This
case
is
supplemented
to
columns below the eighth floor are reduced by 50%. This case is supplemented tovalidate
validate
the
of of
thethe
method
in the
with with
an irregular
stiffness
distribution
along
theapplication
application
method
in structures
the structures
an irregular
stiffness
distribution
the building’s height.
along the building’s height.
All of the frame members are modelled using the displacement-based beam-column
All of the frame members are modelled using the displacement-based beam-column
elements. The sectional parameters are consistent with those of prior research to achieve the
elements. The sectional parameters are consistent with those of prior research to achieve
same dynamic features. Since the method is proposed for linearly elastic shear structures,
the same dynamic features. Since the method is proposed for linearly elastic shear structhe material nonlinearity of the systems is not considered in the OpenSees model. Thus, all
tures, the material nonlinearity of the systems is not considered in the OpenSees model.
of the elements are modelled with an elastic material. It should be noted that the elastic
Thus, all of the elements are modelled with an elastic material. It should be noted that the
modulus of steel in the 6F model is increased by 18% to achieve the same periods, as
elastic modulus of steel in the 6F model is increased by 18% to achieve the same periods,
recommended in prior research [17]. As proposed in prior research, the Raleigh damping
as recommended in prior research [17]. As proposed in prior research, the Raleigh dampmodel of 3% is used for the two models. A leaning column pinned to the ground is included
ing model of 3% is used for the two models. A leaning column pinned to the ground is
in the model to capture the P-∆ effect of the system. The leaning columns are modeled using
included in the model to capture the P-∆ effect of the system. The leaning columns are
the truss element with tremendous axial stiffness. The tributary gravity loads are applied
modeled
using
the truss
element
with tremendous
axial
stiffness.
The tributary
gravity
to the leaning
column
on the
floor levels.
Additionally,
a rigid
diaphragm
is adopted
in
loads
are
applied
to
the
leaning
column
on
the
floor
levels.
Additionally,
a
rigid
diathis model.
phragm
is adoptedmodel
in thisofmodel.
The computer
each frame is analyzed using 100 earthquake records, as recommended in the ATC 63 project [18]. One hundred ground motions contain multifarious
spectral contents. Forty-four far-field and fifty-six near-fault ground motions are included.
Note that the selected records may not closely match the design spectrum. Abundant
spectral contents in these ground motions are especially adopted to eliminate the influence
of the earthquake records.
4.4. Discussions of Analysis Results
The IRSA is conducted following the procedure that is described above. Firstly, the
continuous shear beams are established to make an estimation on the dynamic features of
the three building frames. The mode frequencies and mode shapes of first three modes
from the CSB method and the eigenvalue analysis in OpenSees are presented in Figure 12. It
can be clearly indicated that the CSB method can provide a satisfactory estimation of the
modal parameters. The theoretical analysis results agree strongly with the results from the
eigenvalue analysis of the refined numerical model, especially for the first mode. It seems
that the results of 6F for higher modes are not very precise. The difference mainly derived
from the assumption in the CSB method that the considered system could be simplified
into a continuous shear member. The discreteness of a building structure will certainly lead
to deviation between the results from two methods, as mentioned before. Additionally, the
the CSB method and the eigenvalue analysis in OpenSees are presented in Figure 12. It can
be clearly indicated that the CSB method can provide a satisfactory estimation of the modal
parameters. The theoretical analysis results agree strongly with the results from the eigenvalue analysis of the refined numerical model, especially for the first mode. It seems that the
results of 6F for higher modes are not very precise. The difference mainly derived from the
13 of 17
assumption in the CSB method that the considered system could be simplified into a continuous shear member. The discreteness of a building structure will certainly lead to deviation between the results from two methods, as mentioned before. Additionally, the normalnormalization
the mode
makes
it much
remarkable
in the
figure.
However,
ization of theofmode
shapeshape
makes
it much
moremore
remarkable
in the
figure.
However,
the
the
higher
system
more
precise
estimation
modal
features
could
be achieved.
higher
thethe
system
is, is,
thethe
more
precise
estimation
on on
modal
features
could
be achieved.
6
6
4
4
4
2
Story N
6
Story N
Story N
Buildings 2023, 13, 258
2
2
Mode 1 from CSB T1=1.434s
Mode 1 from FEM T1=1.224s
0
0.0
0.2
0.4
0.6
0.8
1.0
0
-1.0
-0.5
0.0
0.5
1.0
0
-1.0
(a)
12
12
12
10
10
10
6
4
Mode 1 from CSB T1=2.056s
2
0
0.0
8
6
4
Mode 1 from FEM T1=2.353s
0.2
0.4
0.6
0.8
1.0
0
-1.0
Displacement φ1(N)/φ1,max
0.5
6
1.0
0
-1.0
(b)
14
,
10
10
6
4
0
0.0
Mode 1 from FEM T1=3.156s
0.2
0.4
0.6
0.8
Displacement φ1(N)/φ1,max
6
4
Mode 1 from CSB T1=2.777s
2
Story N
12
10
Story N
12
8
1.0
0
-1.0
0.0
0.5
Displacement φ2(N)/φ2,max
(c)
0.5
1.0
6
Mode 3 from CSB T3=0.551s
2
Mode 2 from FEM T2=0.991s
-0.5
0.0
8
4
Mode 2 from CSB T2=0.902s
2
Mode 3 from FEM T3=0.499s
-0.5
14
12
8
Mode 3 from CSB T3=0.450s
Displacement φ3(N)/φ3,max
Displacement φ2(N)/φ2,max
,
14
0.0
1.0
8
2
Mode 2 from FEM T2=0.833s
-0.5
0.5
4
Mode 2 from CSB T2=0.737s
2
0.0
14
Story N
14
8
Mode 3 from FEM T3=0.269s
-0.5
Displacement φ3(N)/φ3,max
14
Story N
Story N
Mode 2 from FEM T2=0.438s
Displacement φ2(N)/φ2,max
Displacement φ1(N)/φ1,max
Story N
Mode 3 from CSB T3=0.300s
Mode 2 from CSB T2=0.516s
1.0
0
-1.0
Mode 3 from FEM T3=0.596s
-0.5
0.0
0.5
1.0
Displacement φ3(N)/φ3,max
Figure 12. Comparison of dynamic features. (a) Comparison of dynamic features for 6F. (b) ComFigure 12. Comparison of dynamic features. (a) Comparison of dynamic features for 6F. (b) Compariparison of dynamic features for 13F. (c) Comparison of dynamic features for 13F-W.
son of dynamic features for 13F. (c) Comparison of dynamic features for 13F-W.
Then, the dynamic parameters of the three demonstration building frames are adopted
to conduct the spectral analysis. The story shear demand of each mode of the considered
systems under each ground motion can be easily obtained. The MSRSS methods is then
performed to estimate the shear demands of the building stories along the height of the
building. It should be noted that the weighting coefficients An in the MSRSS method are
the same as those described in Section 3.2. For comparison purposes, the RSA that consider
the CSRSS method are also conducted to obtain the shear demands of the building stories.
Note that the modal parameters used in the CSRSS are in accordance with the parameters
used in the MSRSS. The shear force demands of the three demonstration buildings under
typical ground motion are presented in Figure 13. It can be clear noted that the MSRSS
provides a relatively more adequate prediction of the shear force demands.
Additionally, the results from the two methods are substituted into Equation (19), and
thus, the deviation of the results from the two RSA methods and the RHA method can
be obtained, as presented in Table 2. The results clearly indicate that, the MSRSS mode
superposition could achieve an improved estimation of the shear force demands on the
steel building frames compared with that of the CSRSS mode superposition. The Error
values from the MSRSS for 6F, 13F, and 13F-W are 8.38%, 11.49%, and 11.93%, which from
the CSRSS are 9.68%, 13.86%, and 15.05%, respectively.
6
RSA-MSRSS
RHA
RSA-CSRSS
Story N
5
4
3
2
1
0
1500
3000
4500
Shear force F (kN)
13
12
11
10
9
8
7
6
5
4
3
2
1
RSA-MSRSS
RHA
RSA-CSRSS
0
2000
4000
6000
Story N
Buildings 2023, 13, 258
Story N
that consider the CSRSS method are also conducted to obtain the shear demands of the
building stories. Note that the modal parameters used in the CSRSS are in accordance with
the parameters used in the MSRSS. The shear force demands of the three demonstration
buildings under typical ground motion are presented in Figure 13. It can be clear noted
that the MSRSS provides a relatively more adequate prediction of the shear force14deof 17
mands.
8000
13
12
11
10
9
8
7
6
5
4
3
2
1
RSA-MSRSS
RHA
RSA-CSRSS
0
1000
Shear force F (kN)
2000
3000
4000
Shear force F (kN)
Figure 13. Shear force demands of the two demonstration buildings under typical ground motion.
Figure 13. Shear force demands of the two demonstration buildings under typical ground motion.
Additionally, the results from the two methods are substituted into Equation (19),
Table 2. Deviation of the results between RSA and RHA.
and thus, the deviation of the results from the two RSA methods and the RHA method
can be obtained, as presented in Table 2. The results clearlyError
indicate
(%) that, the MSRSS mode
Mode Superposition
superposition
could achieve an improved6Festimation of the13F
shear force demands
13F-Won the
steel building frames compared with that of the CSRSS mode superposition. The Error
MSRSS
8.38
11.49
11.93
values from the MSRSS for 6F, 13F, and 13F-W are 8.38%, 11.49%, and 11.93%, which from
CSRSS
9.68
13.86
15.05
the CSRSS are 9.68%, 13.86%, and 15.05, respectively.
Rate of improvement
13.4
17.1
20.7
Table 2. Deviation of the results between RSA and RHA.
To make the improvement of the IRSA clear, the factor logarithmic Error ratio (LER) is
Error (%)
defined
Equation (20). This factor represents the accuracy of using a certain method to
ModeasSuperposition
6F
13F
13F-W
estimate the elastic shear demands compared using IRSA with an MSRSS superposition.
MSRSS
8.38
11.49
As defined, the LER value of 0 suggests that the Error from a certain method11.93
is equal to
CSRSS
9.68
13.86
15.05 to a
the Error from
the IRSA with MSRSS superposition.
A higher
LER value corresponds
RateError
of improvement
17.1
20.7
larger
compared with that from 13.4
the MSRSS superposition.
Error/Error
log(clear,
(20)
) logarithmic Error ratio (LER)
To make the improvement ofLER
the =
IRSA
the MSRSS
factor
is defined as Equation (20). This factor represents the accuracy of using a certain method
Figurethe
14 elastic
presents
the LER
of thecompared
ELF and CSRSS
for 6F,with
13F, an
andMSRSS
13F-Wsuperposiunder each
to estimate
shear
demands
using IRSA
earthquake
ground
motion.
The
results
reveal
that
the
Errors
of
the
MSRSS
are
smaller
than
tion. As defined, the LER value of 0 suggests that the Error from a certain method is equal
those
of
the
ELF
and
CSRSS
in
most
of
the
cases.
to the Error from the IRSA with MSRSS superposition. A higher LER value corresponds to
Based
the datawith
shown
Figure
14, a normal
distribution is considered to be
a larger
Erroron
compared
thatin
from
the MSRSS
superposition.
appropriate for LER (H0 ). The parameters and the K-S test results for the normal distribution
of all four datasets are presented in Table
3. The goodness-of-fit
tests are conducted at

LER = log  Error
(20)

different significance levels (shown in Figure
If DMSRSS
≤ Dlimit , the null hypothesis (H0 ) is
 15).Error
accepted. As presented in Table 3 and Figure 15, the distributions pass the K-S goodnessFigure 14 presents the LER of the ELF and CSRSS for 6F, 13F, and 13F-W under each
of-fit tests in all of the cases. The mean values for the four cases presented in Table 3
earthquake ground motion. The results reveal that the Errors of the MSRSS are smaller
clearly indicate that the Errors from the MSRSS are much smaller than those from the ELF
than
of the ELF and CSRSS in most of the cases.
and those
CSRSS.
Table 3. Distribution parameters and K-S test results for normal distribution.
6F
Mean
SD
Dlimit-0.01
Dlimit-0.15
D
H0
13F
13F-W
ELF
CSRSS
ELF
CSRSS
ELF
CSRSS
0.464
0.300
0.066
0.200
0.447
0.307
0.083
0.243
0.403
0.292
0.094
0.254
0.113
A
0.084
A
0.112
A
0.163
0.114
0.084
A
0.092
A
0.107
A
Mean
SD
Dlimit-0.01
Dlimit-0.15
D
H0
Buildings 2023, 13, 258
0.464
0.300
0.066
0.200
0.084
A
0.092
A
0.447
0.307
0.083
0.243
0.163
0.114
0.107
0.113
A
A
0.403
0.292
0.094
0.254
0.084
15 of 17
A
0.112
A
log (Error/ErrorMSRSS)
2
1
0
-1
-2
6F-ELF
6F-CSRSS
0
20
40
Ground motion
60
80
100
2
log (Error/ErrorMSRSS)
Buildings 2023,
1 13, x FOR PEER REVIEW
2 of 18
0
log (Error/ErrorMSRSS)
negligible [3]. The SRSS method proposes that the peak response of the considered system
can be estimated as the square root of the sum of the squares of the peak response from
each mode of vibration. To date, this method is recommended in numerous codes for seis-1
mic design [2,4]. From the structural point of view, using SRSS mode superposition in the
13F-ELF
RSA provides an alternative approach to include the contribution from the high-mode
13F-CSRSS
effect. Nevertheless, some research has pointed out the detrimental defects of the SRSS
-2
0
20
40
80
method,
i.e., the cross-correlation
between the60modal responses is neglected.
Hence, the 100
CQC method was developed Ground
to include
the effect of cross-correlation between the modal
motion
2
responses [5]. Chopra summarized the application scope of each method and proposed
that adopting modal combination rules in the RSA to conduct an elastic analysis may generally lead to the inadequate results, especially in upper story of the system [1]. The max1
imum deviation of the considered research was up to 25%. Moreover, the researcher also
indicated that using the mode superposition to estimate the response of the structure under a single ground motion record may lead to much more deviations. This is caused by
0
the assumptions of the random vibration theory behind the derivations for each method.
The similar defect of RSA also exists while performing a nonlinear analysis. To more accurately capture the nonlinear demands during major earthquakes, some modified RSA
-1
were proposed to adjust the contribution of the higher mode effect. Khy and Chintana13F-W-ELF
pakdee et al. proposed a modified response spectrum analysis to capture the shear
force
13F-W-CSRSS
demand for tall RC shear wall buildings [6]. Sullivan, Priestley et al. proposed a substitute
-2
0
20
40
80
structure
method to include
the higher mode60effect for ductile structures
[7]. Pennucci, 100
Sullivan et al. proposed that Ground
the higher
mode effect should be considered based on the
motion
ductile state [8]. Moreover, some spine systems were developed to eliminated the higher
mode
effect between
under
nonlinear
deformation
[9,10].
Figure
14. Error
comparison
between
RSA-MSRSS
ELF and RSA-CSRSS.
Figure 14. Error
comparison
RSA-MSRSS
with
ELF andwith
RSA-CSRSS.
Figure 15. Results for K-S
test
for logarithmic
Error to
ratio
between
RSA-MSRSS
and other
In this
research,
the authors intend
propose
an improved
response spectrum
anal-methods.
ysis (IRSA) method in two dimensions to estimate the elastic base shear demands in the
Buildings 2023, 13, 258
16 of 17
5. Concluding Remarks
This paper proposes that the conventional response spectrum analysis method may
lead to inadequately designed shear forces for steel building frames, and it develops an
alternative response spectrum analysis method for the elastic shear force demand estimation
of shear-dominated steel building frames in the preliminary design stage. Following the
proposed procedure, the shear force demands along the building’s height can be captured
more precisely. RHA are performed for comparing the improvements of the proposed
design approaches. Based on the analyses conducted in this research, the following major
conclusions can be made:
•
•
•
•
•
The continuous shear beam model with a stepped stiffness could achieve a precise
estimation of the dynamic parameters. Following the proposed procedure, the modal
periods and modal shapes can be easily and accurately captured without conducting
an eigenvalue analysis using numerical software.
The conventional response spectrum analysis method, which uses limited modes in
the SRSS method will underestimate the shear demands of a steel building frame.
Moreover, the underestimation of the shear demands on the top of the considered
system is more significant.
It was found that adjusting the weighting coefficient for each modal shear demand
could improve the adequacy of the RSA method in determining the elastic shear
force in the steel building frames. The optimal weighting coefficient set derived from
the genetic algorithm validates the fact that adding a weighting coefficient that is
greater than 1.0 to high modal shear items could result in the better estimation of
shear force demands. Moreover, using the first three modal shear demands to conduct
mode superposition can achieve a satisfactory estimation, and so, more modes are
not necessary.
The proposed IRSA are performed on two demonstration building frames. Results
validate the superiority of the proposed method and the adequacy of the proposed
weighting coefficients.
This study is conducted mainly on a regular building frame model in two dimensions.
The cross-correlation between the modal responses is minor and negligible in this
study. Thus, the application scope is limited. Much more irregular scenarios should
be included to explore the method to improve the adequacy of conventional RSA.
Moreover, since the cross-correlation between the modal responses of a structure in
three dimensions is significant, the relevant research based on CQC is also promising.
Author Contributions: Conceptualization, J.L. and W.W.; methodology, J.L. and W.W.; software, J.L.;
validation, J.L.; formal analysis, J.L.; investigation, J.L.; resources, W.W.; data curation, J.L.; writing—
original draft preparation, J.L.; writing—review and editing, W.W.; visualization, J.L.; supervision,
W.W.; project administration, W.W.; funding acquisition, W.W. All authors have read and agreed to
the published version of the manuscript.
Funding: The financial supports from the National Natural Science Foundation of China (NSFC)
with Grant Nos. 52078366, 51820105013 and National Key Research and Development Program of
14th Five-Year Plan of China with Grant No. 2022YFC3801904 are gratefully acknowledged. This
study is also supported by the Top Discipline Plan of Shanghai Universities-Class I with Grant No.
2022-3-YB-18.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
2.
Chopra, A.K. Dynamics of Structures: Theory and Applications to Earthquake Engineering, 4th ed.; Prentice Hall: Upper Saddle River,
NJ, USA, 2012; pp. 562–595.
ASCE. Minimum Design Loads for Buildings and Other Structures; American Society of Civil Engineers: Reston, VA, USA; Structural
Engineering Institute: Weston, FL, USA, 2016.
Buildings 2023, 13, 258
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
17 of 17
Rosenblueth, E. A Basis for Aseismic Design. Ph.D. Thesis, University of Illinois, Urbana, IL, USA, 1951.
GB 50011-2010; Code for Seismic Design of Buildings. Ministry of Housing and Urban-Rural Development of the People’s
Republic of China (MHURD-PRC): Beijing, China, 2010. (In Chinese)
Newmark, N.M.; Rosenblueth, E. Fundamentals of Earthquake Engineering, 1st ed.; Prentice Hall: Englewood Cliffs, NJ, USA, 1971.
Khy, K.; Chintanapakdee, C.; Warnitchai, P.; Wijeyewickrema, A.C. Modified Response Spectrum Analysis to Compute Shear
Force in Tall RC Shear Wall Buildings. Eng. Struct. 2019, 180, 295–309. [CrossRef]
Sullivan, T.J.; Priestley, M.J.N.; Calvi, G.M. Estimating the Higher-Mode Response of Ductile Structures. J. Earthq. Eng. 2008, 12,
456–472. [CrossRef]
Pennucci, D.; Sullivan, T.J.; Calvi, G.M. Inelastic Higher-Mode Response in Reinforced Concrete Wall Structures. Earthq. Spec.
2015, 31, 1493–1514. [CrossRef]
Hu, S.; Wang, W. Self-Centering Energy-Absorbing Rocking Core System with Friction Spring Damper: Experiments, Modeling
and Design. Eng. Struct. 2020, 225, 111338. [CrossRef]
Ke, K.; Chen, Y.; Zhou, X.; Yam, M.C.H. Experimental and Numerical Study of a Brace-Type Hybrid Damper with Steel Slit Plates
Enhanced by Friction Mechanism. Thin Wall Struct. 2023, 182, 110249. [CrossRef]
Timoshenko, S.P. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag.
1921, 41, 744–746. [CrossRef]
Andrés, A.R.; Miranda, E. Assessment of Effects of Reductions of Lateral Stiffness along Height on Buildings Modeled as Elastic
Cantilever Shear Beams. J. Earth. Eng. 2016, 22, 553–568.
Ohtori, Y.; Christenson, R.E.; Spencer, B.F. Benchmark Control Problems for Seismically Excited Nonlinear Buildings. J. Struct.
Eng. 2005, 130, 366–385. [CrossRef]
Mazzoni, S.; McKenna, F.; Scott, M.H.; Fenves, G.L. OpenSees Command Language Manual. Pac. Earthq. Eng. Res. (PEER) Cent.
2006, 264, 137–158.
Somerville, P.; Smith, N.; Punyamurthula, S.; Sun, J. Development of Ground Motion Time Histories for Phase 2 of the FEMA/SAC
Steel Project; SAC Background Document; Report submitted to the Report SAC/DB-97/04; SAC Joint Venture: Sacramento, CA,
USA, 1997.
ICBO. Uniform Building Code; International Conference of Building Official: Wittie, CA, USA, 1973.
Kunnath, S.K.; Nghiem, Q.; El-Tawil, S. Modeling and Response Prediction in Performance-Based Seismic Evaluation: Case
Studies of Instrumented Steel Moment-Frame Buildings. Earthq. Spec. 2004, 20, 883–915. [CrossRef]
FEMA-695; Quantification of Building Seismic Performance Factors. Federal Emergency Management Agency: Washington, DC,
USA, 2009.
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