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\begin{document}

\title{Seismic Design of Moment Resisting Steel Framed Buildings.
An Investigation of the European Approaches}



\author[1]{Davide Rodigari}%
\affil[1]{Dipartimento ABC. Politecnico di Milano}%


\vspace{-1em}



  
  \date{May 19, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
The research is focused on the investigation of the design methods of moment resisting steel framed structures (MRF) in seismic areas. The Eurocode 8, the main reference for earthquake design in Europe, allows
four design approaches. Two of these (i.e. the lateral force method and the modal response spectrum), are based on linear finite element analysis, while the other methods (i.e. the pushover and the time history) are based on non-linear analysis. In the current design practice, linear analysis are preferred because they lead to a simpler structural modelling and to easy interpretation of the results. It is a common idea that Eurocode 8 should guarantee the same structural performance independently from the analysis method used. In order to state the validity of this idea it is useful to estimate with non-linear analysis the performance of an extended sample of structures designed with linear methods. In accordance to the research purpose, this paper shows and explains the preliminary results obtained analysing a single MRF.%
\end{abstract}%



\sloppy



\textbf{Keywords}

Seimic Design; Eurocodes; Behaviour Factor; Moment Resisting Steel
Frames; Capacity Design.

\subsection*{INTRODUCTION}
A moment frame is an integrated vertical and horizontal load-carrying system. The resistance of this structural system is granted by a combined action of columns and beams joined by connections. The analysis of a MRF is not easy because the structural scheme is statically indeterminate and global resistance and deformability are defined by the
complex interaction between beams and columns. At the same time, MRFs are largely used because they allow layouts with wide free spaces (Figure 1). Moreover, MRFs seem to exhibit superior performances to earthquakes {[}30{]}. This is thank to the ductility of steel beams that allow withstanding to severe events with sustainable structural costs.
For this reason in the 1960s and in the 1970s the cyclic behaviour of steel members was investigated and the acquired knowledge was introduced in building codes in terms of the capacity design principle {[}18{]}, Nowadays seismic design methods used in the profession for the design of MRFs are based on elastic analysis and the seismic input is defined by the response spectrum (RS) {[}31{]} depending on the construction site hazard {[}9{]}. All structural non-linearities are approximately considered by reducing the elastic force by an amount depending on the structural scheme used. This reducing quantity is called behavior factor (q) in European seismic design code (EC8) {[}15{]} and strength reduction factor (R) in the American code (ASCE-10) {[}1{]}. It is expected that the seismic response estimated by linear analysis should be very close to the response obtained by non-linear analysis. If this assumption is not true, it means that the evaluation of structural safety is strongly affected by the analysis methods adopted. This would imply a reconsideration of the actual seismic design approaches used in the routine design.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}
\caption{{Willis Tower's (443 m) Moment Resisting Steel Frames.%
}}
\end{center}
\end{figure}

\subsection*{EUROPEAN DESIGN APPROACH}

According to EC8, seismic resistant structures should show clear and
direct forces transmission paths. Another principle is that lateral
loads resisting systems shall run without interruption from the
foundation to the top of the building. This avoids concentration of
ductility demand and it reduces the uncertainty in seismic analysis due
to complicate layouts.
Earthquake resistant steel buildings can be designed according to low,
medium or high dissipation behaviour. In low dissipation behaviour
(DCL), no ductility details (e.g. dissipative bracings, plastic
hinges,\ldots{}) are foreseen in design codes. Instead, if medium (DCM)
or high (DCH) ductility behaviour are chosen, ductility detailing
allowing dissipation must be provided in specific zones of the
structure. MRF dissipative zones should be located in beams' plastic
hinges, where seismic energy is dissipated by cycling bending. During
hysteretic behaviours, local buckling should not affect the overall
stability of the structure. Therefore, class 1 and class 2 cross
sections are preferred in ductility design because of their high plastic
rotation capacity. The developing of plastic hinges in beams is
guaranteed by applying the capacity design principle of ``strong
column-weak beam''. According to this criterion, in the design procedure
columns resistance is checked considering internal actions amplified by
means of a factor (i.e. the beam overstrength factor).Design codes indicate linear analysis as the main method for structural
design. All the non-linearities are represented by the behaviour factor
``q''. The value of ``q'' depends on structural regularity and on the
amount of ductility level, as shown in Figure 2.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}
\caption{{Eurocode 8 Behavior Factor for MRF Steel Building%
}}
\end{center}
\end{figure}

For high ductility level, ``q''  depends on the amount of the ratio
\selectlanguage{greek}α\selectlanguage{english}\textsubscript{u}/\selectlanguage{greek}α\selectlanguage{english}\textsubscript{1}, which represents the structural
overstrenght due to redundancy. This number varies from a minimum of 1.1
up to 1.6. As anticipated, EC8 allows four seismic analysis methods. The lateral
force method and the modal response spectrum, based on linear finite
element analysis, and the pushover and the time history, based on
non-linear analysis.The Lateral Force Method (LFM) considers the seismic action as an
horizontal load at floor level. The total base shear is defined by the
product of effective mass ( \(m\cdot\lambda\) ) and spectral design
acceleration of the first vibration mode (\(S_{d}\)) , as shown in Eq. 1.

\[
{\left(1\right)\text{F}}_{b} = S_{d}\cdot\left( T_{1} \right)\cdot m\cdot \lambda
\]

Then, the base shear is redistributed at each level proportionally to
floor height and mass. This approach can be easily adopted if the
structure is regular and the response is dominated by one mode.
When there is no regularity, linear analysis should be performed with
Modal Response Spectrum method (MRSA). This is a dynamic analysis where
all the modes that have a participating mass higher than 5\% must be
considered. The sum of the effective modal mass should be at least 90\%.
The response of the structure (E) is obtained by a statistical
combination of the modal responses, adopting for example the CQC
approach (Eq.2).

\[
\left(2 \right)\text{\ E} = \left( \sum_{j}^{}{\sum_{i}^{}{\rho_{\text{ij}}\cdot\ E_{i}\cdot\ E_{j}}} \right)^{\frac{1}{2}}
\]

Regarding non-linear static analysis, the Pushover (PSV) is a method
that defines the capacity curve by pushing the non-linear model of the
structure first with a uniform load pattern and then with a modal
horizontal load scheme. This curve describes the relation between the
total base shear and the displacement at the top of the structure. The
curve is then reduced by a transformation factor (\Gamma) that relates it to an
equivalent single degree of freedom system. This factor depends on the
mass distribution and on the first mode normalized eigenvector (\phi)  as in Eq.3.

\[
 \left( 3 \right) \Gamma=\frac{\phi^T M r}{\phi^T M \phi}   
\]

Bilinearizing the curve by means of an equal energy criterion, the
period of the equivalent sdof is determined by the slope of the first
branch of this transformed curve. Known the equivalent period, the
target displacement is defined in the Acceleration Displacement Response
Spectra (ADRS) {[}16{]}. At target displacement, section rotation demand
and internal actions are evaluated, and then structural checks are
performed. The most advanced seismic analysis is the Non Linear Time History
Analysis (NLTH). Structural response is obtained by direct integration
of the equation of motion (Eq. 4).

\[
  \left( 4 \right)    M\ddot u + C \dot u +F_s(u,\dot u)=- M L \ddot u_g (t)
\]

Earthquake is described by many spectrum-compatible accelerograms that
could be natural {[}20{]} or artificial. If seven accelerograms are
used, the average of the response peak values is considered in the
structural check. If less than seven accelerograms are used, the highest
value should be assumed.

\subsection*{BEHAVIOUR FACTOR}\
Non linear effects are taken into account reducing the design spectrum
by the behaviour factor. This values represents the energy structural
dissipation capacity, and it includes the contribution of ductility,
overstrength and redundancy. The value of the q-factor is defined by
ECCS {[}11{]} as in Eq. 5.

\[
\left( 5 \right)\ q = \frac{a_{u}}{a_{y}}
\]

Where \(a_{u}\) is the peak ground acceleration causing the collapse of
the structure and \(a_{y}\) is the peak ground acceleration causing the
attainment of the elastic limit of the structure. These two quantities
can be determined by incremental NLTH analysis. The design codes avoid
this work defining the value of the behaviour factor as shown in Table
1. In the past extended studies were done on ``q'' {[}17{]} and many
authors have proposed different methods {[}25{]} to evaluate this value
through dynamic analysis {[}8{]}. An alternative procedure, consisting
in the use of the pushover analysis, derives from a document
\emph{ATC-19} {[}5{]} written by American Building Council. This
document describes a method that allows the estimation of the strength
reduction factor ``R'' (analogue of q) by the use of pushover analysis.
R is defined as a contribution of the three key components: \(\ \)the
ductility factor \(R_{\mu}\),\(\ \) the redundancy factor \(R_{R}\) and
the overstrength factor \(R_{S}\) (Eq. 6):

\[
\left( 6 \right)\  R = R_{\mu}\cdot\ R_{R}\cdot\ R_{S} = \frac{d_{u}}{d_{y}}\cdot\frac{V_{y}}{V_{1}}\cdot \frac{V_{1}}{V_{d}}\text{\ .}
\]

Where \(V_{y}\), \(V_{1}\), \(V_{d}\) are, respectively: the base shear
at conventional yielding, the base shear at first yielding and the
design base shear; \(d_{y}\) and \(d_{u}\) are the top displacements
relative to first yielding and to hinge rupture.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}
\caption{{Response Spectra of nine Italian site.%
}}
\end{center}
\end{figure}

The value of R obtained by this method can be compared to the behaviour
factor proposed by EC8. Therefore, designing a structure by means of a
linear method adopting the behaviour factor proposed by the code, and
then applying the non-linear ATC-19 procedure in order to obtain the
value of R, the two quantities can be compared. With this purpose, a DCM MRF has been designed. Beams are pre-designed
performing vertical load checks, while columns are defined according to
EC8 capacity design formula (Eq.7):
\[
\left(7 \right) M_{C,pl,Rd} \ge 1.3 \sum M_{B,pl,Rd}
\]
In Eq.7, \(M_{C,pl,Rd}\ ,\ M_{B,pl,Rd}\ \)  are respectively the resisting
moment of columns and beams.
A linear seismic analysis MRSA is then performed and all the structural
checks are done considering the nine italian {[}10{]} response spectra
represented in Figure 3. As the structure design is verified, a
non-linear model of the structure can be created and the capacity curve
is determined pushing it horizontally.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/image4/image4}
\caption{{ATC-19 procedure on MRF Structure: (a) Structural Layout (b) Modal Capacity Curve%
}}
\end{center}
\end{figure}

The structure analyzed in this paper is the one-bay four-levels DCM MRF
represented in Figure (4a). According to Table 1 the ``q'' value for
seismic analysis was assumed equal to 3.2 because the structure is
irregular in elevation due to a mass variation higher then 25\% between
the floors. In correspodance to the first hinge activation, the first
yelding point is identified. Ropture is defined at the point at which
the first section reaches a plastic rotation of 30 mrad. This value of
rotation is commonly considered as a general reference for the ultimate
rotation capacity of steel sections. The conventional yielding point is
obtained by a bilinear approximation (in terms of energy criteria) of
the capacity curve (Figure 4(b)). The design base shear values, needed
for the definition of \(R_{S}\), are nine, each one releted to one of
the earthquake that are considered in the structural check.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}
\caption{{Strenght Reduction Factors obtained by ATC compared to behaviour factor.%
}}
\end{center}
\end{figure}

The results of the procedure are represented in Figure 5 where for each
acceleration considered it is shown the R value obtained. The dashed
curve represents the behaviour factor (constant, according to EC8).
Maximum variation between q and R is at least 5\% (TRF Spectra) and up
to 95\% (CLO Spectra). This wide variability leads to reflections about
the use of a constant value that is fixed independently from the seismic
event considered.

\subsection*{INVESTIGATION PROCEDURE}
The investigation procedure used is in this research is in the following
described. It allows to evaluate the effectiveness of design methods
based on ``q''. This scheme is composed by five phases, each ones
briefly described and schematically represented (Figure 6):
\begin{itemize}
\item
  Preliminary design: the structure is preliminary designed by simple
  hand calculation in order to satisfy the main checks relative to
  vertical load carrying capacity and capacity design.
\item
  Linear analysis LFM and MRS: Linear analysis are performed on a fem
  model of the preliminary designed structure. To the MRF is applied a
  sample of nine earthquakes in both life safety and damage limit state
  {[}6{]}, {[}19{]}. The internal actions and displacements obtained
  from the analysis are used for structural steel checks according to
  EC3 (i.e. members resistance, columns stability, global stability
  {[}4{]} {[}24{]} and inter-storey displacement {[}3{]}, {[}7{]},
  {[}13{]}, {[}12{]}). Analysis are performed by SAP2000 fem program
  {[}27{]} and EC8 checks are done with Excel. If any element or
  displacement is not verified, the structural model is updated until
  checks are satisfied.
\item
  Pushover analysis: a non-linear model of the structure is created in
  Opensees {[}23{]}. A bilinear steel material curve with kinematic
  hardening is used. Structural elements are modelled as beams with
  fibres hinges {[}28{]}. Geometrical non-linearities are also included
  in the model (P-\selectlanguage{greek}Δ \selectlanguage{english}effects), while beam-to-columns connections are
  supposed rigid {[}14{]}, {[}26{]}. The structure is pushed according
  to EC8 by a modal and then uniform horizontal force pattern, in order
  to define the capacity curve. Targets displacements are determined by
  a Matlab routine {[}22{]}, that extracts and saves the corresponding
  internal actions and hinge rotations.
\item
  Non Linear Time history Analysis: The same model used for the pushover
  analysis is used for the performance of non-linear dynamic analysis. A
  5\% damping is proportionally assumed to current stiffness matrix
  {[}21{]}. Three artificial spectrum compatible signals are used to
  represents each response spectrum. The signals are generated according
  to EC8 by the program SIMQKE {[}29{]}. Results are post-processed by
  Matlab in order obtain the most unfavourable combination of internal
  actions and the maximum rotation demand for each structural element.
\item
  Final results: all the data from PSV and NLTH are imported from Matlab
  to Excel and structural verifications are performed. The results in
  terms of internal actions, rotations and structural checks obtained
  from each analysis methods can be now compared.
\end{itemize}
\begin{quote}
\includegraphics[width=4.71667in,height=4.52961in]{/tmp/figures/media/image6.png}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/Immagine-2-BIS/Immagine-2-BIS}
\caption{{Flow chart of the investigation procedure.
{\label{855180}}%
}}
\end{center}
\end{figure}

\subsection*{PRELIMINARY RESULTS}
The preliminary results obtained by the application of the previously
descripted procedure are herein as: the total base shear, the element
internal action, the element rotation and the collapse mechanism.
\begin{itemize}
\item
  \textbf{Total base shear}. The maximum base shear obtained by the
  preliminary application procedure are reported in Figure 7. It could be
  seen that relevant differences are between linear and
  non linear analysis results.
\end{itemize}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/image7/521135}
\caption{{Total Base Shear by EC8 Methods%
}}
\end{center}
\end{figure}

This differences could be remarked in Figure 6 where the effective
structural performace described by the capacity curve are compare with
the target point of the four methods. Lower differences are displayed in terms of targetdisplacement; by the
fact all the analysis gives values quite closer.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/image8/final---Copia}
\caption{{Target Point on Uniform Capacity Curve: Lateral Force Method (a), Modal Response Spectrum Analysis (b), Pushover (c) and Non Linear Time History (d).%
}}
\end{center}
\end{figure}

\begin{itemize}
\item
  \textbf{Internal actions and rotations.} Considering two elements at
  the lower floor, the maximum internal action and rotation are reported
  in Figure 9. The comparison between design quantities obtained by linear analysis and non linear analysis results shows that both
  internal actions and rotation {[}2{]} values are very close between
  the methods.
\end{itemize}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/image9/image9}
\caption{{Maximum elements actions and rotations by EC8 methods%
}}
\end{center}
\end{figure}

\begin{itemize}
\item
  \textbf{Collapse mechanism}. Evaluation plastic hinges activation
  sequence, as in Figure 10, is possible to evaluate if the collapse
  mechanism, designed by the application of the capacity design rules,
  it is effectively designed.
\end{itemize}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/image10/image10}
\caption{{ATC-19 procedure on MRF Structure: (a)Structural Layout, (b) Modal Capacity Curve%
}}
\end{center}
\end{figure}

It is shown that the collapse mechanism design by linear analysis
method, is coherent with the results obtained by non-linear analysis.
The safety margin, that it could be estimated as the ratio between the
target displacement at spectra TRF and the maximum top displacement, is
equal to 0.66. Indeed the structure has a margin of 34\% until the first
section reach rupture.

\subsection*{REMARKS AND CONCLUSION}
The aim of the research is an extended investigation on the
effectiveness of linear approaches for seismic design regarding MRF
steel structures. These approaches have been widely used for years in
the professional practice and still today remain the main methods. An
investigation procedure concerning the European seismic design
approaches of MRF has been described. The preliminary results obtained
by applying this procedure to one-bay four-floors structure lead to some
significant considerations:

\begin{itemize}
\item
  The values of the strength reduction factor ``R'' obtained by the
  application of ATC-19 non-linear procedure are far from the value of
  the behaviour factor assumed according to the EC8 elastic design. This
  is mainly due to the fact that the behaviour factor does not depend on
  the design base shear, i.e. it is independent from the seismic input
  considered;
\item
  Base shear values obtained by linear analysis methods are quite far to
  those computed by non-linear analysis.
\item
  Internal actions obtained by linear analysis methods are quite closer
  to those computed considering non-linear analysis.
\item
  The results of the non-linear methods show that the structure --
  designed with linear analysis- has an higher capacity than seismic
  demand, with a safety margin up to 34\% in correspondence of the first
  rupture.
\item
  The collapse mechanism detected by non-linear analysis is consistent
  with capacity design principle.
\end{itemize}

In the light of these observations, it is clear that linear design procedure based on LFM and MRSA are not able to quantify explicitly seismic performance, since the analysed structure is overdesigned. On the contrary, non-linear approaches can foresee the structural seismic performance more precisely, but they are time-consuming and so they are rarely applied in the design practice.In order to generalize these statements and to give general validity to the above described observations, the procedure adopted will be applied to a wide sample of MRF structures. This will be object of the incoming research.

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\selectlanguage{english}
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\end{document}

