Applied Mechanics and Materials Vols. 166-169 (2012) pp 2337-2340
Online available since 2012/May/14 at www.scientific.net
© (2012) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMM.166-169.2337
Displacement-based Lateral Stiffness Design for Multi-storey
Structures Subject to Earthquake Motions
Feng Wang1, a, Hongnan Li2,b and Tinghua Yi2,c
1
2
College of Architecture & Civil Engineering , Dalian Nationalities University, Dalian 116600, China;
Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China.
a
b
c
[email protected], [email protected] , [email protected],
Keywords: Displacement-based design; Storey drift; Equivalent SDOF system; Lateral Stiffness.
Abstract. The determination of structural stiffness for the currently seismic design method depends
on subjective experience of designers which is not rational and economical. A method that uses
displacements as the basis for the stiffness design procedure is then presented: (1)By means of
preliminary design, the initial elastic structure are obtained and the 1th mode shape, period etc are
then calculated by modal analyses; (2) The target period and lateral equivalent stiffness of structure
are determined according to target displacement used in seismic code; (3)The two periods for initial
designed structure and target structure are compared and the lateral stiffness is adjusted to make the
displacement responses of the structural weak members meet the limited displacements by adjustment
parameter. An example is implemented for demonstrating the process and verifying the accuracy of
the procedure.
Introduction
Seismic structures must be capable of absorbing and dissipating energy in a non-degrading manner for
many cycles of substantial deformation[1]. Damage to structural components can be controlled if
displacements can be limited to predetermined values for a specified level of earthquake shaking. If
damage control is the essence, procedure of seismic design to control displacements of structure are
necessary[2]. The direct displacement-based seismic design is being advocated as a rational method,
in which displacements are considered at the start of the design process with attention focused on
deformations to provide a strength of structure that meets the requirements for the several limit
states[3,4]. However, the design of structural lateral stiffness in preliminary design stage is dependent
on subjective experience of designer which is not rational and economical. So, a displacement-based
lateral stiffness design procedure for structures is presented in the paper.
Theories.
Inelastic displacement. The relationship between inelastic maximal storey drift ∆up and elastic
maximal storey drift ∆ue of structure , based on the code for seismic design of building[5],is as follow:
∆u p = η p ∆ue ,
(1)
where ηp is inelastic amplification parameter, as given in the code for seismic design of building.
Elastic displacement. A N-storey structure is modeled as a MDOF system with N degrees of
freedom. The displacement-based stiffness design method is applied to this system by first
transforming it into an equivalent SDOF system [6]. The lateral displacement vector of the MDOF
system, in terms of the 1th vibration modal shape, is assumed as:
{u (t )} = {φ1}q (t ) = {φ1}D(t )γ 1 ,
(2)
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Progress in Structures
in which {u (t )} is the vector for lateral displacement; {φ1} is the normalized 1-th modal vector; γ 1 is
participation factors of the 1-th mode shape; q (t ) = D(t ) ⋅ γ 1 . From Eq.2, the relationship between
maximal roof displacement ue, r and maximal storey drift ∆ue is expressed:
ue,r = ∆ue φ1,i ,
(3)
where φ1,i is a component value of the 1th modal shape. From Eq. 2 and Eq.3, a relationship is
represented as:
ur ,lim = Dlimγ 1 ,
(4)
in which ur ,lim and Dlim are the limited maximal roof displacement of the elastic MDOF system and
limited maximal displacement of elastic equivalent SDOF system, respectively. Entering the elastic
displacement response spectra with known target displacement , the fundamental period (or vibration
frequency) can be read.
Lateral Stiffness adjusting. Defining α as the parameter of lateral stiffness adjustment, the
relationship of lateral stiffness between initial structure and adjusted structural stiffness is expressed
as: α×initial structural stiffness=adjusted structural stiffness. Displacement-based stiffness design is
derived from equivalent SDOF system theories which is based on the assumption of responding with
fixed modal shape, as written in Eq.2, so the modal shape of adjusted structures must be
approximately consistent with initial structures. As shown in Eq.5, the stiffness matrix and equivalent
frequency of structure are adjusted by parameter α , but the modal shape of structure is not changed
which meets the prior discussion.






2π 2
2π 2
2π 2
 [ K ] − ( ) [ M ]  {φ1} = 0 ⇒  [α K ] − α ( ) [ M ] {φ1} = 0 ⇒  [ K ] − ( ) [ M ] {φ1} = 0 ,
T1
T1
T1






(5)
where [K] and T1 are initial stiffness matrix and the 1-th periods, respectively; [ K ] and T1 are the
stiffness matrix and the1-th period of the adjusted structure.
Procedure.
A direct displacement-based lateral stiffness design procedure is outlined as a sequence of steps:
(1) Preliminary Design. The preliminary design (or conceptual design) is implemented, basing on
which the first modal shape, corresponding period and participation factors are calculated.
(2) Lateral Stiffness design for strong earthquake. 1) The limited inelastic storey drift for strong
earthquake is obtained as ∆u p = [θ p ]h . According to Eq.1, the limited elastic storey drift for the same
level of earthquake can be calculated by ∆ue ,lim = γ 1[θ p ]h η p , where [θ p ] is limited storey deformation
angle for strong earthquake, used in seismic code; h is floor height; γ 1 is the safety factor, and γ 1 < 1 .
2) The limited roof displacement for strong earthquake is calculated by Eq.3.
3) The limited displacement of equivalent SDOF system can be obtained by Eq.4.
4) Enter the elastic displacement design spectra for strong earthquake with known Dlim to read Teq.
If it is reinforced concrete structure, considering stiffness reduction caused by concrete cracking, the
fundamental period is reduced as T1 = 0.85 ⋅ Teq [5].
5) The T1* of preliminary designed structure is compared with T1 , and parameter of stiffness
adjustment is calculated by: α1 = (T1* T1 ) .
2
Applied Mechanics and Materials Vols. 166-169
2339
(3) Lateral Stiffness design for weak earthquake. The steps 2) and 3) of this process are identical
to those in the previously-described process for strong earthquake, and steps 1), 4) and 5) are replaced
by the following steps.
1) The limited storey drift for weak earthquake is calculated by ∆ue ,lim = γ 2 [θ e ]h , in which [θe ] is
limited storey deformation angle for weak earthquake, used in seismic code; γ 2 is the safety factor,
and γ 2 < 1 .
4) Enter the elastic displacement design spectra for weak earthquake with known Dlim and ξ1 to
read Teq. The fundamental period of structure is obtained by T1 = Teq .
5) The parameter of stiffness adjustment for weak earthquake is calculated by α 2 = (T1* T1 ) .
2
(4) Determination of parameter α. The parameter α is determined by α = max(α1 , α 2 ) , basing on
which the sectional sizes of frame members are adjusted.
Example.
In order to demonstrate the processes and prove the rationalities of the proposed method, a nine-storey
reinforced concrete frame structure will be designed. The expected sizes in height are 3.6m for each
floors, and in plane are shown in Fig.1.
Fig.1 Sketch of the building
Design process. The sectional sizes of columns are 700mm×700mm for 1th~3th storey, 650mm
×650mm for 4th~6th storey and 600mm×600mm for 7th~9th storey. The reduction factor of period
is assumed as 0.7. The first period and corresponding participation factors are given as T1*=0.72sec.
and Γ1=1.32, respectively.
Consulting the seismic code, the [θp] is determined as 1/50, ηp and γ1 is defined as 2.0 and 1/1.5,
respectively. By calculating, Teq=0.98sec , T1 = 0.85 ×Teq =0.90sec and α1≈0.64; The [θe] is
determined as 1/550 and γ1=1/1.3. By calculating, T1 =Teq =0.86sec and the parameter of stiffness
adjustment is α2≈0.71.
The parameter of stiffness asjustment equals to 0.71 by α = min(α1 , α 2 ) . The adjusting designs for
sectional sizes of structural members, based on parameter α , are shown in Table1.
Table 1. Adjusting designs of structural section sizes
Structural
member
Column
Beam
Floor
Preliminary design
Adjusting
Adjusted design
Sectional sizes Moment of
[mm2]
Inertia, I [m4]
I×0.71
Sectional
Moment of
sizes [mm2] Inertia , I [m4]
1th~3th
700 × 700
0.0200
0.0142
650 × 650
0.0149
4th~6th
650 × 650
0.0149
0.0106
600 × 600
0.0108
7th~9th
600 × 600
0.0108
0.0077
550 × 550
0.0076
1th~9th
250 × 600
0.0045
0.0032
250 × 550
0.0035
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Progress in Structures
Dynamic analysis. For proving the accuracy of this method, the seismic acceleration history records
are chosen with the El-Centro 180 records to analyze the dynamic responses of the adjusted structure
subjected to strong earthquake and weak earthquake, respectively. The short side of the structural
plane is taken as the direction of earthquake action. The maximal lateral floor displacement patterns
of structure subjected to strong earthquake and weak earthquake are represented in Fig.2 which shows
that the displacement responses of adjustment designed structure, based on the proposed method,
meet the limit state of seismic code.
(a) PGA= 0.07g
(b) PGA=0. 4g
Fig.2 The maximal floor displacement angle for El-Centro 180 records
Conclusions
(1) Displacements play a major role for stiffness design stage which is beneficial for controlling
displacements over the entire design process. Target displacement criteria are selected for the
serviceability and ultimate limit states and thus damage control is achieved directly.
(2) The storey drift criteria are adopted as controlled target of displacement which make it easier
to combine this procedure with seismic code.
(3) The concepts of lateral stiffness design are presented which substitute the traditional design
methods, based on subjective experience of designer.
(4) The parameter of stiffness adjustment is adopter in this paper which avoids the repeated
calculation and simplified the design procedure.
Acknowledgements
This work was financially supported by the project of the National Science Foundation for
Distinguished Young Scholars of China (51108067), project of the State Key Program of National
Natural Science of China (90815026), project of the Education Department of Liaoning (L2010098)
and project of the Mistry of Housing and Urban-Rural Development of the People’s Republic of
China (2010-K3-55).
References
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[2] M.S. Medhekar, D.J.L. Kennedy: Engineering Structures.Vol. 22(2000), p. 201-209.
[3] M. Fragiadakis, M. Papadrakakis: Earthquake Engineering and Structural Dynamics.Vol.37
(2008), p. 825-844.
[4] P. Franchin, P.E. Pinto: Journal of Structural Engineering, ASCE. 2011.
[5] GB50011-2010 Code for seismic design of buildings. Beijing, China, Architecture & Buiding
press, 2010 (in Chinese).
[6] T.Tjhin, M. Aschheim: Journal of Structural engineering, Vol.131(2005), p. 517-522.
Progress in Structures
10.4028/www.scientific.net/AMM.166-169
Displacement-Based Lateral Stiffness Design for Multi-Storey Structures Subject to Earthquake
Motions
10.4028/www.scientific.net/AMM.166-169.2337
DOI References
[1] A.M. Chandler, N.T.K. Lam: Engineering Structures. Vol. 23(2001), pp.1525-1543.
http://dx.doi.org/10.1016/S0141-0296(01)00070-0
[2] M.S. Medhekar, D.J.L. Kennedy: Engineering Structures. Vol. 22(2000), pp.201-209.
http://dx.doi.org/10.1016/S0141-0296(98)00092-3
[3] M. Fragiadakis, M. Papadrakakis: Earthquake Engineering and Structural Dynamics. Vol. 37 (2008),
pp.825-844.
http://dx.doi.org/10.1002/eqe.786
[6] T. Tjhin, M. Aschheim: Journal of Structural engineering, Vol. 131(2005), pp.517-522.
http://dx.doi.org/10.1061/(ASCE)0733-9445(2005)131:3(517)