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%BJ \journal{Comput. Methods Appl. Mech. Engrg.}

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\begin{document}

% %BJ \begin{frontmatter}

% \title{Energy Dissipation Analysis for Inelastic  Reinforced Concrete and Steel Beams}


% %BJ 
% \begin{center}
% {\large Energy Dissipation Analysis for Inelastic Reinforced Concrete and Steel Beams}
% \\
% {Han  Yang,    
% Yuan  Feng,  
% %Dragan  Kova{\v c}evi{\' c}, 
% Hexiang Wang and
% %David  McCallen, and 
% Boris Jeremi{\'c} }
% \\
% {UCD, LBNL}
% \end{center}
% %BJ 

% \begin{center}
% version: \today, \hhmm
% \end{center}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \newpage
% \setcounter{tocdepth}{4}
% \tableofcontents

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \newpage
% \begin{abstract}
% %
% Presented     is    a   thermodynamics based methodology  for  computing  energy
% dissipation in inelastic beam-column elements.
% %
% Theoretical  formulation  for  energy storage and dissipation in uniaxial steel fiber
% and  concrete fiber models  is  derived  from  the  principles of thermodynamics, in
% conjunction with a few assumptions on energy transformation and dissipation.
% %
% Proposed    methodology    is  implemented  in MS-ESSI Simulator and illustrated
% through  a  number  of numerical examples on beam-columns and frame models under
% various loading conditions.
% %
% It   is  shown  that  the  consideration  of  plastic free energy in addition to
% plastic work, is necessary to correctly evaluate energy dissipation in nonlinear
% beam-column elements.
% %
% Results  of  energy  analysis indicates that the difference between plastic work
% and  plastic  dissipation  could be significant, and that the ratio between them
% evolves with time.
% %
% \end{abstract}



% %BJ 
% %BJ \begin{keyword}
% %BJ Seismic  energy  dissipation  \sep  Structural elements \sep Uniaxial fiber \sep  
% %BJ Computational  geomechanics  \sep Thermodynamics \sep Plastic free energy
% %BJ \end{keyword}
% %BJ 
% %BJ \end{frontmatter}



\begin{frontmatter}

\title{Energy Dissipation Analysis for Inelastic Reinforced Concrete and Steel Beam-Columns}
% 
%  \begin{center}
%  {Energy Dissipation Analysis for Inelastic Reinforced Concrete and Steel Beam-Columns}
%  
%  
%  
%  Han Yang, Yuan Feng, Hexiang Wang, Boris Jeremi{\'c}
%  
%  UCD, LBNL
%  \end{center}
%  

%% or include affiliations in footnotes:
\author[mymainaddress]{Han Yang}
%\corref{mycorrespondingauthor}}
%\ead{hhhyang@ucdavis.edu}

% \author[mymainaddress]{Sumeet Kumar Sinha}

\author[mymainaddress]{Yuan Feng}

\author[mymainaddress]{Hexiang Wang}

% \author[mysecondaddress]{David B McCallen}

\author[mymainaddress,mysecondaddress]{Boris Jeremi{\'c}\corref{mycorrespondingauthor}}
\ead{jeremic@ucdavis.edu}

\cortext[mycorrespondingauthor]{Corresponding authors}

\address[mymainaddress]{Department of Civil and Environmental Engineering, University of California, Davis, CA, USA}
\address[mysecondaddress]{Earth Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{abstract}
%
Presented     is    a   thermodynamics based methodology  for  computing  energy
dissipation in inelastic beam-column elements.
%
Theoretical  formulation  for  energy storage and dissipation in uniaxial steel fiber
and  concrete fiber models  is  derived  from  the  principles of thermodynamics, in
conjunction with a few assumptions on energy transformation and dissipation.
%
Proposed    methodology    is  implemented  in MS-ESSI Simulator and illustrated
through  a  number  of numerical examples on beam-columns and frame models under
various loading conditions.
%
It   is  shown  that  the  consideration  of  plastic free energy in addition to
plastic work, is necessary to correctly evaluate energy dissipation in nonlinear
beam-column elements.
%
Results  of  energy  analysis indicates that the difference between plastic work
and  plastic  dissipation  could be significant, and that the ratio between them
evolves with time.
%

% 
% 
% Presented      is    a   thermodynamics based methodology  for 
% computing energy dissipation in inelastic beam-column elements.
% Theoretical  formulation  for energy storage and dissipation in
% uniaxial  steel fiber and concrete fiber models is derived from
% the  principles  of  thermodynamics,  in conjunction with a few
% assumptions  on energy transformation and dissipation. Proposed
% methodology is implemented in MS-ESSI Simulator and illustrated
% through  a  number  of  numerical  examples on beam-columns and
% frame models under various loading conditions. It is shown that
% the consideration of plastic free energy in addition to plastic
% work,  is necessary to correctly evaluate energy dissipation in
% nonlinear  beam-column  elements.  Results  of  energy analysis
% indicates  that the difference between plastic work and plastic
% dissipation  could  be  significant, and that the ratio between
% them evolves with time.

\end{abstract}


\begin{keyword}
Energy dissipation \sep
Inelastic beam column
%Uniaxial fiber material \sep 
%Thermodynamics \sep 
%Plastic free energy  \sep
%Computational mechanics  
\end{keyword}

\end{frontmatter}


%\linenumbers


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%
%\setcounter{tocdepth}{4}
%\tableofcontents
%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

Mechanical  energy  in  soil  structure interaction (SSI) systems is dissipated
during  the  irreversible  dissipative process of energy transformation in which
entropy of the system increases.
%
Energy  dissipation has been used, directly or indirectly, as a key parameter to
evaluate damage in elastic-plastic materials.
%
A  common misconception about plastic work and energy dissipation due to plasticity
has  been noticed in a number of recent publications 
\citep{Uang1990,Leger1992,Symans1998,Soong2002,Symans2008,Wong2008,Nehdi2010}   
in which violations of the second law of thermodynamics is observed.
%
As  presented in an earlier paper \citep{Yang2017a}, the correct formulation for
energy analysis on
elastic-plastic  solids  has been derived from the second law of thermodynamics.
The  theoretical  and  computational  framework has been verified through system
energy  balance  in a series of numerical studies on elastic and elastic-plastic
material models.
%
The  purpose  of  this paper is to present a methodology for correctly evaluating
energy    dissipation  in nonlinear fiber beam-column structural elements.
% which
% is crucial in determining the
% safety and economy of a SSI system.


%



%
Early  work  reported  by  \citet{Farren1925} and \citet{Taylor1934} showed that
plastic free energy could be significant in metals, thus should not be neglected
without reasoning.
%
The  ratio  of  plastic  work  converted  into  heat, usually referred to as the
Quinney--Taylor   coefficient,   was   measured   to   be  between  0.6  to  1.0
\citep{Belytschko1991,Zhou1996,Dolinski2010,Osovski2013}.
%
\citet{Mason1994}  pointed out that  the Quinney--Taylor coefficient is
both  strain  and strain rate dependent but could be assumed to be a constant in
most cases.
%
A   constitutive   model   for      metals  was  presented  by 
\citet{Rosakis2000},   \citet{Hodowany2000},   \citet{Ravichandran2002}   based  on
thermoplasticity. Presented model can  model the evolution of energy dissipation
and has been validated through experiments.
%
\citet{Semnani2016}  presented  a  thermoplastic  framework  that  could predict
strain localization in transversely isotropic materials.
%



%
Despite  of the existence of sophisticated theories that are capable of modeling
the  evolution  of  energy  dissipation, including those mentioned earlier, most
constitutive  relationships  used  to  model  structural elements do not involve
thermodynamics or thermoplasticity.
%
One  commonly  used  finite  element  (FE)  technique  to  model inelastic frame
structures,  is to use nonlinear beam-column finite element, and nonlinear fiber
sections.
%
In  this  approach,  a  beam-column  element is analyzed using a number of cross
sections, at locations of integration points.
%
Such  cross  sections  are divided into a number of uniaxial fibers with various
constitutive models, for steel and/or concrete for example.
%
This  model  have  been  proved  to  be able to capture nonlinear stress--strain
behaviors of structural elements under axial loading and/or pure bending.
%



%
Problems arise when such elements are used to calculate energy dissipation.
%
As    observed    in   many   publications   
\citep{Kwan2001,Zhu2006,Gajan2011,Wang2012,Zhang2013a,Nikbakht2014}, 
energy  dissipation    analysis       was     performed   using   
hysteretic stress--strain and/or force--displacement response of the elements.
%
Hysteretic  stress--strain  and/or  force--displacement responses corresponds to
plastic work.
%
Plastic work is not the same as plastic energy dissipation.
% This  indicates  that plastic work was confused with plastic energy dissipation,
% which is the common misconception pointed out earlier.
%
It    is    also  important  to  point  out that various damage indices that are
used to evaluate seismic performance of frame structures are derived from energy
dissipation.
%
It  is  then   noted that such  damage indices are not valid  if the fundamental
computation of energy dissipation is incorrect.
%








%
It  has  been shown by \citet{Dafalias2002,Feigenbaum2007,Yang2017a}
that  the difference between plastic work and plastic dissipation is the plastic
free  energy,  or  cold  work,  which  can  be calculated from material internal
variables (or state variables).
% , like radius of yield surface or back stress.
%
This  computation  can  be   performed  on  solids  modeled  with 
elastic-plastic constitutive relations in which internal variables are 
updated at every increment.
%
On the other hand, constitutive relationships used to model nonlinear structural
elements based on fiber cross section are mainly based on empirical fitting of 
experimental results 
\citep {Spacone1996a,Spacone1996b,Lee1998,Popovics1973,Mander1988,Chang1994,Waugh2009,Kolozvari2015}.
%
The  parameters used  in these models are different than internal variables that
are used in elastic-plastic  constitutive models for solids.
%
In  order  to apply rational mechanics for computing energy dissipation, a  new 
methodology is needed. 
%
This  new  methodology,    based  on thermodynamics, should be able to correctly
evaluate  energy storage and dissipation in structural elements, while using the
same fiber material models for steel and concrete.
%



%
During  the  recent  few  decades,  a number of studies have been conducted with
focus  on  energy  analysis  of  SSI  systems 
\citep{Uang1990,Leger1992,Kalkan2007a,Symans2008,Gajan2011,
Moustafa2011,Mezgebo2017,Deniz2017}.
%
Despite    different formulations used,  the calculations of energy  dissipation
due  to  hysteretic  damping  (material elasto-plasticity) in these publications
were  all  performed without consideration of plastic free energy, which leads to
the violation of the second principle of thermodynamics.
%
In  other  words, results show negative incremental energy dissipation, which is
equivalent to energy production.
%
It    is    worth   pointing  out  that  such problem can be found in many other
publications in the last few decades.
% % starting from \citet{Uang1990}
% 
%  oversight  is  not rare, especially in
% literature of civil and geotechnical engineering.
%





%
In  order  to  correctly  evaluate  energy dissipation in nonlinear beam-column 
elements  modeled  using fiber  sections, the thermo-mechanics framework must
be applied to the uniaxial constitutive models used for fibers.
%
Focus  of this paper is on proper modeling of different forms of energy storage
and  dissipation  in  uniaxial  material  models.
% that follows the second law of
%thermodynamics.
%
Presented     is    a   theoretical    and    computational    formulations  for
computing energy dissipation in uniaxial concrete and steel fiber models.
%
A    series  of  FE  simulations  are  carried  out  using the MS-ESSI Simulator
\cite{Real_ESSI_Simulator} to illustrate
the energy behavior of structural frame systems.
%
The  method  is  verified by comparing the input work and the energy storage and
dissipation in the system.
%
The   difference  between  accumulated  plastic  work  and  accumulated  plastic
dissipation, which can be significant in many cases, is addressed.
%
Finally,  conclusions  on  plastic energy dissipation in structural elements are
drawn from the verified results.
%




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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theoretical and Computational Formulations}


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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Thermomechanical Framework}
\label{Thermomechanical_Framework}

%
The    theory   of  continuum thermo-mechanics has been discussed in a number of
earlier publications by \citet{local-44} and \citet{Rosakis2000}, from which the
fundamental framework of this study is derived.
%
General  equations of elastoplasticity and thermodynamics are modified using few
plausible assumptions to accommodate use of existing fiber material models.
%
Small  deformation  theory  is  assumed,  so  that  the  {small  strain  tensor}
$\epsilon_{ij}$ is used to describe deformation of a material.
%
It is noted that all equations in this paper are expressed in index notation.
%


%
The  general  thermomechanical  process  is governed by momentum balance and the
first and second law of thermodynamics.
%
The  localized  version  of  the  first  law  of  thermodynamics (energy balance
equation) is given in the form:
%
\begin{equation}
\sigma_{ij} \dot{\epsilon}_{ij} + q_{i,i} + \rho r = \rho \dot{e}
\label{equation_first_thermodynamics}
\end{equation}
%
where  $\sigma_{ij}$  is  Cauchy stress, the term $\sigma_{ij} \epsilon_{ij}$ is
called the {stress power}, $q_{i}$
are  the  components  of the {heat flux vector}, $\rho$ is the {mass density} of
the material, $r$ is the {heat supply} per unit volume, and $e$ is the {internal
energy} per unit volume.
% %
% Note that in this paper all stresses are defined as effective stresses.
% %
% In order to avoid confusion, the common notation ($\sigma_{ij}^{'}$) will not be
% used.
%
Standard  definition  of  stress  from  mechanics of materials, i.e. positive in
tension, is used.
%


%
The  localized  version  of  the  second  law of thermodynamics (Clausius--Duhem
inequality) is expressed as:
%
\begin{equation}
\rho \dot{\eta} - (\frac{q_{i}}{\theta})_{,i} - \frac{1}{\theta} \rho r \ge 0
\label{equation_second_thermodynamics}
\end{equation}
%
where  $\eta$  is  the  {entropy}  per unit volume and $\theta$ is the {absolute
temperature}.
%

%
Substituting     the     heat     supply     per     unit    volume    $r$    in
Equation~\ref{equation_second_thermodynamics}    with    the   expression   from
Equation~\ref{equation_first_thermodynamics}, and introducing the rate of change
of {internal dissipation} per unit volume $\Phi$ gives:
%
\begin{equation}
\rho \theta \dot{\eta} - \rho \dot{e} + \sigma_{ij} \dot{\epsilon}_{ij} + \frac{1}{\theta} q_{i} \theta_{,i} 
= \Phi + \frac{1}{\theta} q_{i} \theta_{,i} 
\ge 0
\label{equation_second_thermodynamics2}
\end{equation}
%
%
Note  that  the  internal  dissipation can have many sources, including material
plasticity,  viscous  coupling,  and  other forms of energy
dissipation.
%


%
The  {Helmholtz  free  energy}  per  unit volume $\psi$, which is referred to as
{free energy} in this paper, is defined as:
%
\begin{equation}
\psi = e - \theta \eta
\label{equation_Helmholtz}
\end{equation}
%
The second law of thermodynamics can be expressed in terms of free energy $\psi$ as:
%
\begin{equation}
\Phi + \frac{1}{\theta} q_{i} \theta_{,i} = 
- \rho \dot{\psi} - \rho \dot{\theta} \eta + \sigma_{ij} \dot{\epsilon}_{ij} + \frac{1}{\theta} q_{i} \theta_{,i}
\ge 0
\label{Second_law_of_Thermodynamics}
\end{equation}
%
The rate of internal dissipation per unit volume $\Phi$ can be written as:
\begin{equation}
\Phi = \sigma_{ij} \dot{\epsilon}_{ij} - \rho \dot{\psi} - \rho \dot{\theta} \eta
\label{equation_internal_dissipation}
\end{equation}
%


%
At  this  point,  a  few  assumptions  are  introduced to simplify the governing
equations.
%
According          to         \citet{Feigenbaum2007},      \citet{Collins97},   
\citet{Collins2002a},   \citet{Collins2002b},   it   can  be  assumed  that  the
deformation  of  beam-column  elements under earthquake loading is approximately
isothermal,  which indicates that the temperature field $\theta$ is constant and
uniform.
%
This  approximation  is  reasonable  considering the fact that seismic energy is
mostly  carried  by  the  low-frequency  components of earthquake ground motion,
which allows the heat generated in the material to be largely dissipated.
%
With this assumption, the rate of internal dissipation $\Phi$ is simplified into
the form:
%
\begin{equation}
\Phi = \sigma_{ij} \dot{\epsilon}_{ij} - \rho \dot{\psi} \ge 0
\label{equation_internal_dissipation_simplified}
\end{equation}
%


%
Next,  all  material  models  studied  in this paper are assumed to be decoupled,
which  means  that  the  (small) strain tensor can be additively decomposed into
elastic and plastic parts:
%
\begin{equation}
\epsilon_{ij} = \epsilon_{ij}^{el} + \epsilon_{ij}^{pl}
\label{equation_strain_decomposition}
\end{equation}
%
\citet{local-44}  and  \citet{Collins97}  showed  that  this  assumption  can be
deduced if the instantaneous elastic moduli of a material are independent of the
internal variables.
%
Under  the  assumption of decoupled material, the free energy $\psi$ can also be
decomposed into elastic and plastic parts:
%
\begin{equation}
\psi = \psi_{el} + \psi_{pl}
\label{equation_free_energy}
\end{equation}
%
where  the  elastic  part  of  the  free energy $\psi_{el}$ is also known as the
{elastic strain energy}.
%
Elastic strain energy is defined in incremental form as:
%
\begin{equation}
\dot{\psi}_{el} = \sigma_{ij} \dot{\epsilon}_{ij}^{el}
\label{equation_strain_energy}
\end{equation}
%
By           substituting          Equation~\ref{equation_strain_decomposition},
Equation~\ref{equation_free_energy},  and  Equation~\ref{equation_strain_energy}
into   Equation~\ref{equation_internal_dissipation_simplified},   the   rate  of
internal  dissipation  $\Phi$  can  be expressed in terms of the rate of plastic
free energy $\dot{\psi}_{pl}$:
%
\begin{equation}
\Phi 
= 
\sigma_{ij} \dot{\epsilon}_{ij} - \sigma_{ij} \dot{\epsilon}_{ij}^{el} - \rho \dot{\psi}_{pl}
\ge 0
\label{equation_internal_dissipation_final}
\end{equation}
%


%
Equation~\ref{equation_internal_dissipation_final}    represents    two    basic
principles  that  should  always  be upheld in any energy analysis for decoupled
material undergoing isothermal process:
%
\begin{itemize}
 
  \item  The stress power that is input into a material body by external loading
  is  transformed  into elastic strain energy, plastic free energy, and material
  internal  dissipation.  All  forms  of  energy  must be considered to maintain
  energy balance of the material body. This principle ensures that the first law
  of thermodynamics holds.
  
  \item     The  rate  of  change  of  material  internal  dissipation  (plastic
  dissipation)  is nonnegative at any time. In other words, accumulated internal
  dissipation  can  not  decrease during any time period. This principle ensures
  that the second law of thermodynamics holds.

\end{itemize}
%


%
%  Note  that  material internal dissipation can have many sources. 
%  %
%  Our  interest  is the  energy  dissipation  caused  by  material  plasticity, so
%  the  term  {plastic  dissipation} will be used instead.
%  %
%  Th
%   which indicates no other
%  source of energy
%  dissipation  is present in the examples that are being analyzed in the remaining
%  part of this paper.
%  %

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Plastic Free Energy}

The  physical  nature  of  plastic  free  energy is associated with the material
micro-structure.
%
For  particulate materials, plastic free energy will be accumulated or
released  if  there  is  evolution of particle arrangement (micro-fabric).
%
Evolution  of  particle  arrangement  happens  as  soon  as the material is
loaded.
%
For   other  materials, for example metals,  
micro-structures  is represented by the shape and arrangement of the crystals,
whose evolution will result in change in plastic free energy.
%
Detailed  explanations  of  the evolution of plastic free energy can be found in
publications by \citet{Besseling1994}, \citet{Collins2002b}, and \citet{Yang2017a}.
%


%
Using Equation~\ref{equation_internal_dissipation_final}, the energy dissipation
of  any  elastic-plastic  material  under  isothermal  loading  process  can  be
calculated,  provided that  all the terms on the right hand side of the equation
are known.
%
For  most  elastic-plastic  constitutive models, the stress tensor $\sigma_{ij}$
and  the  elastic  strain  tensor  $\epsilon_{ij}^{el}$  are being calculated as
simulation progresses.
%
The  challenging  task  is to evaluate the plastic free energy term $\psi_{pl}$,
whose  formulation  depends  on  the internal variables used in the constitutive
model.
%


%
For  a decoupled elastic-plastic material model that exhibits both isotropic and
kinematic  hardening,  the  plastic free energy is decomposed into isotropic and
kinematic parts, that are calculated separately and then summed up.
%
The  formulation  of  plastic  free energy for this type of material was given by
\citet{Feigenbaum2007}:
%
\begin{equation}
\psi_{pl} = \psi_{pl}^{iso} + \psi_{pl}^{kin} = 
\frac{1}{2 \rho \kappa_1} k^2 + \frac{1}{2 \rho a_1} \alpha_{ij} \alpha_{ij}
\label{equation_plastic_free_energy}
\end{equation}
%
where  $\psi_{pl}^{iso}$  and $\psi_{pl}^{kin}$ are the {isotropic and kinematic
parts  of  the  plastic  free energy}, respectively, $k$ is the {radius of yield
surface},  $\alpha$  is the {back stress}, $\kappa_1$ and $a_1$ are non-negative
material constants.
%
Note  that  Equation~\ref{equation_plastic_free_energy}  can  be used for a wide
range  of  constitutive models with various yield functions, including von Mises
and  Drucker-Prager  yield  criteria whose energy behavior has been studied and
presented by \citet{Yang2017a}.
%
Such  materials  are usually used to model solids (soil and mass concrete).
%


%
On   the  other hand,  frame  structures  are  usually modeled using beam-column
elements in combination with fiber sections and uniaxial material models.
%
In this case, Equation~\ref{equation_plastic_free_energy} does not apply.
%
It  is  noted  that most uniaxial constitutive models that are used for concrete
and   steel  modeling  \citep{Menegotto1973,Filippou1983,Yassin1994},  were  not
developed with thermodynamics based energy dissipation in mind.
%
Therefore,    material  model  definitions for concrete and steel were appraised
using  thermodynamics framework \citep{Yang2017a} in order to correctly evaluate
energy storage and dissipation in these materials.
%
% One such approach is presented below.






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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Energy Dissipation in Beam-Column Element}

%
Beams    and  columns  are modeled with nonlinear, displacement-based beam-column
element, that is available within the MS-ESSI Simulator.
%
In  order  to  incorporate  confined/unconfined concrete and steel reinforcement
in  beam-column element, fiber sections are assigned with corresponding material
model uniaxial fibers.
%
An example model is shown in Figure~\ref{figure_fiber_section}.
%
\begin{figure}[!htbp]
\centering
\includegraphics[width=\columnwidth]{Figures/Fiber_Section.pdf}
\caption{\label{figure_fiber_section}
Schematic of a bottom-fixed column modeled with concrete and steel fibers.}
\end{figure}
%
Model represents a bottom-fixed, cantilever    reinforced    concrete    column
%
Figure~\ref{figure_fiber_section} also shows constant beam-column cross section,
as well as constitutive response of concrete and steel fibers.
%
This    model    is  analyzed later, it is presented here in order to illustrate
nonlinear  model  for  a  beam-column  element  with  fiber  cross  section, and
individual fiber constitutive response.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Uniaxial Steel Fiber}

%
The  uniaxial  steel  material  model  examined  in  this study was developed by
\citet{Menegotto1973}  and extended by \citet{Filippou1983}.
% 
This  uniaxial  steel model is capable of capturing  the  nonlinear  hysteretic 
behavior and isotropic strain-hardening effect of steel.
%
The  uniaxial  stress--strain    response    of  steel   material    is    shown
in   Figure~\ref{figure_steel_model},  along  with  explanation of material
parameters.
%
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.8\columnwidth]{Figures/Steel_Model.pdf}
\caption{\label{figure_steel_model}
Constitutive model for uniaxial steel fiber (after \citet{Menegotto1973}).}
\end{figure}
%
The model, as presented by \citet{Menegotto1973}, takes on the form:
%
\begin{equation}
\sigma^{*} = b \epsilon^{*} + \frac{(1-b) \epsilon^{*}}{(1 + \epsilon^{* R})^{1/R}}
\label{equation_steel_model}
\end{equation}
%
with
%
\begin{equation}
\epsilon^{*} = \frac{\epsilon - \epsilon_{r}}{\epsilon_{0} - \epsilon_{r}}
; \quad
\sigma^{*} = \frac{\sigma - \sigma_{r}}{\sigma_{0} - \sigma_{r}}
\label{equation_steel_model2}
\end{equation}
%
where  $b$  is the {strain-hardening ratio}, $\epsilon_{r}$ and $\sigma_{r}$ are
the  {strain  and  stress  at  the point of strain reversal}, $\epsilon_{0}$ and
$\sigma_{0}$  are the {strain and stress at the point of intersection of the two
asymptotes},  $R$  is  the  {curvature  parameter} that governs the shape of the
transition curve between the two asymptotes.
%
Note that this model is defined for uniaxial material, in which the stresses and strains
are scalars instead of tensors.
%
Therefore,    beam-column    finite    element  that uses this model for section
modeling is defined for pure bending and pure compression, tension.
%



%
The   expression   for   the   curvature   parameter   $R$   is   suggested   by
\citet{Menegotto1973}:
%
\begin{equation}
R = R_{0} - \frac{c_{R_{1}} \xi}{c_{R_{2}} + \xi}
\label{equation_steel_model3}
\end{equation}
%
where  $R_{0}$  is  the  value  of  the  curvature  parameter $R$ during initial
loading,  $c_{R_{1}}$ and $c_{R_{2}}$ are degradation parameters that need to be
experimentally determined.
%
The parameter $\xi$, that is updated after strain reversal, is defined as:
%
\begin{equation}
\xi = \left| \frac{(\epsilon_{m} - \epsilon_{0})}{\epsilon_{y}} \right|
\label{equation_steel_model4}
\end{equation}
%
where  $\epsilon_{m}$  is the maximum (or minimum) strain at the previous strain
reversal  point,  depending  on  the  loading  direction of the material. If the
current  incremental  strain is positive, the parameter $\epsilon_{m}$ takes the
value of the maximum reversal strain.
%
Parameter $\epsilon_{y}$ is the {monotonic yield strain}.
%


%
In   order   to   capture  isotropic  hardening  behavior,  \citet{Filippou1983}
introduced    stress    shift    mechanism    into   the   original   model   by
\citet{Menegotto1973}.
%
Note  that  the  hardening  rate  in compression and tension can be different by
choosing different hardening parameters for compression and tension.
%
The proposed relation takes the form:
%
\begin{equation}
\frac{\sigma_{st}}{\sigma_{y}} = a_{1} \left( \frac{\epsilon_{max}}{\epsilon_{y}} - a_{2} \right)
\label{equation_steel_model5}
\end{equation}
%
where  $\sigma_{st}$  is  the  shift  stress  that determines the shift of yield
asymptote,  $\epsilon_{max}$  is the absolute maximum strain at strain reversal,
and  $a_{1}$  and  $a_{2}$  are  hardening  parameters  in compression that are
experimentally determined.
%
In  the  case  of  tension,  the  hardening  parameters  $a_{1}$  and $a_{2}$ in
Equation~\ref{equation_steel_model5}   are  replaced  by  $a_{3}$  and  $a_{4}$,
respectively.
%
Parameters    $a_{3}$  and $a_{4}$ are also determined by experiment.
%


%
The  energy  computation  procedure  for  this  uniaxial steel model is shown in
Figure~\ref{figure_steel_fiber}, and it follows  the thermomechanical framework
established earlier in this paper.
%
\begin{figure}[!htbp]
 \centering
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Steel_Fiber_1.pdf}} 
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Steel_Fiber_2.pdf}}
 \caption{\label{figure_steel_fiber}
 Energy computation of uniaxial steel fiber: (a) Monotonic loading branch;
 (b) Cyclic loading branch.}
\end{figure}
%


%
Note   that   the   only   difference   between  the  monotonic  loading  branch
(Figure~\ref{figure_steel_fiber}(a))    and    the    cyclic    loading   branch
(Figure~\ref{figure_steel_fiber}(b)) is that the strain reversal point $c$ is at
the origin $o$ in the monotonic case.
%
The  following explanation of the proposed energy computation method applies
to both monotonic and cyclic loading scenarios.
%


%
Firstly, the elastic strain energy density $E_{S}$ is defined in accordance with
the classic assumption that it is only a function of current stress state of the
material:
%
\begin{equation}
E_S = E_S (\sigma) = \frac{1}{2 E_0} \sigma^{2}
\label{equation_steel_SE}
\end{equation}
%
Graphically,  the  elastic  strain  energy  density  of  the  material  shown in
Figure~\ref{figure_steel_fiber}  at  states $a$ and $b$ are the triangular areas
$afd$ and $bge$, respectively.
%
Then the incremental form of Equation~\ref{equation_steel_SE} is simply:
%
\begin{equation}
d {E}_{S} = \frac{1}{E_0} \sigma d \sigma
\label{equation_steel_dSE}
\end{equation}
%
% Note that in this material model, the strain is not being decomposed into elastic and plastic parts, which is a fashion that will be in the energy formulations.
%


%
Next,  the incremental plastic dissipation density $D_P$ from state $a$ to state
$b$ is assumed to be the triangular area $abc$:
%
\begin{equation}
d {D}_{P} = \frac{1}{2} [(\sigma - \sigma_{r}) d \epsilon - (\epsilon - \epsilon_{r}) d \sigma]
\label{equation_steel_dDP}
\end{equation}
%
This      assumption   ensures  that  the  incremental  plastic  dissipation  is
non-negative,  satisfying one of the two basic principles of thermodynamics.
%



One  special case to consider is when the material exhibits no cyclic softening, in 
other words, material micro-structure is not evolving.
%
In  this  case a perfectly  overlapping  stress--strain  loops will be observed.
%
In   this   case   only,   the      energy      dissipation    calculated  using
Equation~\ref{equation_steel_dDP}    for   one cyclic will be represented by the
area of the hysteresis loop. 
%
In thermodynamics, the area of hysteresis loop is equal to  the  plastic  work, 
rather than plastic dissipation, however in the case of non-evolving material structure, 
plastic work becomes equal to plastic dissipation. 
%
It is important to stress that this is true only in this case.
% %
% But  in  this  special  case of no cyclic softening, which means no evolution of
% material  state  and thus no development of plastic free energy after a complete
% loading  cycle,  the  plastic  work  equals  to  the  plastic dissipation in the
% material in one loading cycle.
% %


%
For  a  general  case, where the material does exhibit cyclic softening, plastic
free  energy  density  $E_P$  is graphically represented by the areas of polygon
$adoca$ and polygon $beocb$ at states $a$ and $b$, respectively.
%
The plastic free energy calculated using this assumption is given by:
%
\begin{equation}
E_P = \frac{1}{2} \left[\sigma \left( \epsilon - \frac{\sigma}{E_0} - \epsilon_{r} \right) + \sigma_{r} \epsilon \right]
\label{equation_steel_PF}
\end{equation}
%
The incremental form of Equation~\ref{equation_steel_PF} is given as:
%
\begin{equation}
d E_P = \frac{1}{2} \left[ \left(\sigma + \sigma_{r} \right) d \epsilon + \left( \epsilon - \frac{1}{E_0} \sigma - \epsilon_{r} \right) d \sigma \right]
\label{equation_steel_dPF}
\end{equation}
%


%
Adding    Equations~\ref{equation_steel_dSE},    \ref{equation_steel_dDP},   and
\ref{equation_steel_dPF}, the incremental form of energy balance is written as:
%
\begin{equation}
d E_S + d E_P + d D_P = \sigma d \epsilon
\label{equation_steel_energy_balance}
\end{equation}
%
where the increment of three energy components add up to the increment of stress
power during any loading step.
%
% Note that Equation~\ref{equation_steel_energy balance} has the same form with Equation~\ref{equation_internal_dissipation_final}, indicating that the energy formulation for the uniaxial steel fiber material ensure the first law of thermodynamics.
%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Uniaxial Concrete Fiber}

%
The  uniaxial  concrete  material model used in this study is based on the model
proposed  by  \citet{Yassin1994}.
% 
This model is  capable  of modeling
the nonlinear hysteretic behavior and damage effects in concrete.
%
The  material  parameters and stress--strain response of this material are shown
in Figure~\ref{figure_concrete_model}.
%
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.7\columnwidth]{Figures/Concrete_Model.pdf}
\caption{\label{figure_concrete_model}
Constitutive model for uniaxial concrete fiber (after \citet{Yassin1994}).}
\end{figure}
%


%
The  monotonic envelope curve of this model in compression is based on the model
of \citet{Kent1971} and later generalized by \citet{Scott1982}.
%
For  a  given  strain  $\epsilon_{c}$,  the  compressive stress $\sigma_{c}$ and
corresponding tangent stiffness $E$ are given as:
%
\begin{align}
\epsilon_{c} &\le \epsilon_{cs}  
~~\mbox{;}
&
\sigma_{c} &= f_{cs} \left[ 2 \left( \frac{\epsilon_{c}}{\epsilon_{cs}} \right) - \left( \frac{\epsilon_{c}}{\epsilon_{cs}} \right)^{2} \right] 
~~\mbox{;} 
&
E &= E_{c} \left( 1 - \frac{\epsilon_{c}}{\epsilon_{cs}} \right)
\\
\epsilon_{cs} < \epsilon_{c} &\le \epsilon_{cu} 
~~\mbox{;} 
&
\sigma_{c} &= \frac{\epsilon_{c} - \epsilon_{cs}}{\epsilon_{cu} - \epsilon_{cs}}(f_{cu} - f_{cs}) + f_{cs}
~~\mbox{;} 
&
E &= \frac{f_{cu} - f_{cs}}{\epsilon_{cu} - \epsilon_{cs}}
\\
\epsilon_{c} &> \epsilon_{cu} 
~~\mbox{;} 
&
\sigma_{c} &= f_{cu}
~~\mbox{;} 
&
E &= 0
\label{equation_concrete_envelope}
\end{align}
%
where  $f_{cs}$  is  the  maximum compressive strength of the concrete material,
$\epsilon_{cs}$  is the concrete strain at compressive strength, $f_{cu}$ is the
ultimate  (crushing)  strength  of the concrete material, $\epsilon_{cu}$ is the
concrete  strain  at  ultimate  strength,  and  $E_{c}$  is the initial concrete
tangent stiffness that can be calculated using the equation:
%
\begin{equation}
E_{c} = \frac {2 f_{cs}}{\epsilon_{cs}}
\label{equation_concrete_initial_stiffness}
\end{equation}
%
All material parameters should be determined experimentally.
%


%
The  cyclic  behavior  of   concrete  model  in  compression  is  shown  in
Figure~\ref{figure_concrete_model}.
%
One  assumption  of this model is that all reloading lines intersect at a common
point,  where the stress $\sigma_{r}$ and strain $\epsilon_{r}$ are given by the
following expressions:
%
\begin{align}
\epsilon_{r} &= \frac{f_{cu} - \lambda E_{c} \epsilon_{cu}}{E_{c} (1 - \lambda)}
\\
\sigma_{r} &= E_{c} \epsilon_{r}
\end{align}
%

%
\noindent
After  unloading  from  a point on the compressive monotonic envelope, the model
response is bounded by two lines that are defined by:
%
\begin{align}
\sigma_{max} &= \sigma_{m} + E_{r} (\epsilon_{c} - \epsilon_{m})
\\
\sigma_{min} &= 0.5 E_{r} (\epsilon_{c} - \epsilon_{t})
\end{align}
%
where
%
\begin{align}
E_{r} &= \frac{\sigma_{m} - \sigma_{r}}{\epsilon_{m} - \epsilon_{r}}
\\
\epsilon_{t} &= \epsilon_{m} - \frac{\sigma_{m}}{E_{r}}
\end{align}
%
and  $\sigma_{m}$ and $\epsilon_{m}$ are the stress and strain at the unloading
point   on   the   compressive   monotonic   envelope,   respectively.   If  the
unloading--reloading  cycle  is  incomplete,  the  material  response  will be a
straight      line      with      slope      $E_{c}$,      as      shown      in
Figure~\ref{figure_concrete_model}.
%


%
The tensile behavior of concrete model takes into the account tension stiffening and the
effects of initial cracking.
%
Details  of monotonic and cyclic behavior of concrete model under tensile stress
are given by \citet{Yassin1994}.
%



% %% %
% %The energy computation for this concrete fiber
% %%
Since  there  are different loading/unloading branches in this model, the energy
computation needs to be considered separately for each branch.
%
One  energy  component that remains the same in all loading cases is the elastic
strain energy density $E_S$, that is a function of current stress only:
%
\begin{equation}
E_S = E_S (\sigma) = \frac{1}{2 E_c} \sigma^{2}
\label{equation_concrete_SE}
\end{equation}
%
The incremental form of Equation~\ref{equation_concrete_SE} is:
%
\begin{equation}
d {E}_{S} = \frac{1}{E_c} \sigma d \sigma
\label{equation_concrete_dSE}
\end{equation}
%



%
In       order  to  calculate plastic dissipation, a few assumptions are made in
order  to  ensure  that the  energy  behavior of concrete material  follows 
thermodynamics, as illustrated in Figure~\ref{figure_concrete_fiber}:
%
\begin{itemize}

\item      Majority      of   energy   is   dissipated  during  first loading in
compression    and/or    tension   (Figures~\ref{figure_concrete_fiber}(a)   and
\ref{figure_concrete_fiber}(d)).

\item Subsequent cycles of loading, on an already damaged concrete, do 
not dissipate much energy (Figures~\ref{figure_concrete_fiber}(b) and \ref{figure_concrete_fiber}(c)).

\item  No  energy is dissipated during unloading in both compressive and tensile
conditions.

\item  When  the  material  is  cyclically    loaded  under  compression, energy
dissipation only happens when the stress reaches the upper bound $\sigma_{max}$.

\item  No  energy is dissipated during cyclic loading when the material is under
tension.

\end{itemize}
%

%
For  a  single  loading  step  from  stress  state $a$ to $b$ in each subplot of
Figure~\ref{figure_concrete_fiber}, the energy dissipation is represented by the
shaded area.
%
\begin{figure}[!htbp]
 \centering
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Concrete_Fiber_1.pdf}}
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Concrete_Fiber_2.pdf}}
 \\
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Concrete_Fiber_3.pdf}}
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Concrete_Fiber_4.pdf}}
 \caption{\label{figure_concrete_fiber}
 Energy computation of uniaxial concrete fiber: 
 (a) First compression;
 (b) Single compressive unloading-reloading cycle; 
 (c) Unloading-reloading cycles within compression envelope; 
 (d) First tension.}
\end{figure}
%


%
If  the  material  is under compression (Figures~\ref{figure_concrete_fiber}(a),
\ref{figure_concrete_fiber}(b),  and \ref{figure_concrete_fiber}(c)), the amount
of  energy  dissipated  in  the concrete fiber $D_P$ is calculated by using the
area of a polygon $abcdef$.
%
This polygon is formed by the two unloading paths originating from stress states
$a$ and $b$:
%
\begin{equation}
d {D}_{P} = 
\frac{1}{2} \left[ 
(\sigma - \sigma_{c}) d \epsilon + (\epsilon_{c} - \epsilon) 
d \sigma + (\epsilon_{c} - \epsilon_{f}) \sigma + (\sigma_{f} - \sigma_{c}) 
(\epsilon - \epsilon_{t}) + \sigma_{c} d \epsilon_{t} 
\right]
\label{equation_concrete_dDP}
\end{equation}
%
where    the   stress and strain at point $f$ can be computed based on equations
that define respective unloading paths, using the following expression:
%
\begin{equation}
\epsilon_{f} = \frac{\sigma + 0.5 E_{r} \epsilon_{t} - E_{c} \epsilon}{0.5 E_{r} - E_{c}}
\quad \quad \quad
\sigma_{f} = 0.5 E_{r} (\epsilon_{f} - \epsilon_{t})
\label{equation_concrete_point_f}
\end{equation}
%
Point    $c$  can  be  calculated  using  the same approach, by using all stress
and strain variables evaluated at state $b$.
%


%
Note  that  the  polygon becomes  quadrilateral  in the cases of cyclic loading
within     the     monotonic     envelope,     as    can    be    observed    in
Figure~\ref{figure_concrete_fiber}~(b) and (c).
%
Nevertheless,           Equations~\ref{equation_concrete_dDP}                and
\ref{equation_concrete_point_f} remain valid.
%



%
Plastic  free energy $E_P$ of concrete material is calculated by using the
triangular area $fge$ at state $a$:
%
\begin{equation}
{E}_{P} = \frac{1}{2} \left[ \left(\epsilon - \frac{\sigma}{E_c} - \epsilon_{t} \right) \sigma_{f} \right]
\label{equation_concrete_PF}
\end{equation}
%
The  incremental  form  of  Equation~\ref{equation_concrete_PF}  is  obtained by
taking the difference between the plastic free energy at states $a$ and $b$:
%
\begin{equation}
d {E}_{P} = \frac{1}{2} \left[ \left( \sigma_{c} - \sigma_{f} - \frac{1}{E_{c}} \sigma \right) (\epsilon - \epsilon_{t}) - (d \epsilon - d \epsilon_{t}) \sigma_{c} - \frac{1}{E_c} \sigma{c} d \sigma \right]
\label{equation_concrete_dPF}
\end{equation}
%




%
Adding  Equation~\ref{equation_concrete_dSE},  \ref{equation_concrete_dDP},  and
\ref{equation_concrete_dPF} yields the incremental form of energy balance:
%
\begin{equation}
d E_S + d E_P + d D_P = \sigma d \epsilon
\label{equation_concrete_energy_balance}
\end{equation}
%
where the increment of three energy components add up to the increment of stress
power during any loading step.
%





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerical Studies}

Numerical    examples    presented   in  this paper are all simulated  using the
MS-ESSI  Simulator  \citep{Real_ESSI_Simulator}, and are available on the MS-ESSI
website \url{http://ms-essi.info/}.
%
Energy    dissipation  is  calculated  for  beam  finite  elements  made up from
inelastic concrete and steel fiber sections


We  begin  by  performing    numerical   simulation  of steel and plain concrete
columns under various
loading  conditions. This is done to study the energy behavior of uniaxial steel
and concrete material models.
%
Then, a model of reinforced concrete column, consisting of concrete and
steel  fibers, is constructed and simulated to illustrate the energy dissipation
in reinforced concrete structural elements.
%
Finally, a steel frame structure is modeled with fiber section elements and
loaded with dynamic, seismic motion.
%
Through  these  examples,  it  will be shown that the difference between plastic
work and plastic energy dissipation can be significant.



External loads are applied incrementally using displacement-control scheme.
%
System  of  equations  are  solved  using  Newton-Raphson  equilibrium  iteration
algorithm \citep{local-1} and UMFPACK
solver \citep{Davis2004b}, within MS-ESSI Simulator \citep{Real_ESSI_Simulator}.
%
Static,  displacement  control,    integration    algorithm  is  used  for  the 
column  loading  cases,  while Newmark integration is used for the dynamic steel
frame case.
%
Note that viscous and numerical damping are excluded from all cases, in order to
accurately  evaluate  energy  dissipation  due  to material elastoplasticity. In
other words, no viscous damping (Rayleigh or Caughey) is used, and for Newmark time
integration algorithm \citep{Newmark1959}, $\beta=0.25$ and $\gamma=0.5$ parameters are used.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Steel Column}
\label{Steel_Column}

In  order  to verify the proposed energy computation approach for uniaxial steel
material model, examples of steel columns are studied in this section.
%
As  shown in Figure~\ref{figure_steel_concrete_setup}, the one meter long column
model is fixed at the bottom, and loads are applied at the top.
%
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.8\columnwidth]{Figures/Steel_Concrete_Setup.pdf}
\caption{\label{figure_steel_concrete_setup}
Schematic of the steel/plain-concrete column modeled with fiber sections and uniaxial steel/concrete materials.}
\end{figure}
%
%
The size of the cross section is $100 {\rm mm} \times 100 {\rm mm}$.
%
The  parameters  for uniaxial steel material used in this section are summarized
in Table~\ref{table_steel}. Material model used for steel is based on
\citet{Menegotto1973}    and   \citet{Filippou1983},   as   noted   in   section
\ref{Thermomechanical_Framework}.
%
%
\begin{table}[htb]
\centering
\caption{Material model parameters used in steel column examples.}
\label{table_steel}
\resizebox{0.75\textwidth}{!}{
\begin{tabular}{cccccccccc}
\hline
$\sigma_{y}$ [MPa] & $E$ [GPa] & $b$ & $R_{0}$ & $c_{R_{1}}$ & $c_{R_{2}}$ & $a_{1}$ & $a_{2}$ & $a_{3}$ & $a_{4}$ \\
\hline
413.8 & 200.0 & 0.01 & 18.0 & 0.925 & 0.15 & 0.0 & 55.0 & 0.0 & 55.0 \\
\hline
\end{tabular}
}
\end{table}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Cyclic Axial Loading}

%Since  the  fiber  material  model  is uniaxial in nature, axial loading case is
%being investigated first.
%
The  evolution  of  energy  parameters  for uniaxial steel material are computed
using     Equations~\ref{equation_steel_dSE},    \ref{equation_steel_dDP},    and
\ref{equation_steel_dPF}.
%
Figure~\ref{figure_steel_axial} shows the stress--strain response as well as the
energy calculation results of the steel column under cyclic axial loading.


\begin{figure}[!htbp]
 \centering
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Steel_Axial_Stress_Strain.pdf}} 
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Steel_Axial_Energy.pdf}}
 \caption{\label{figure_steel_axial}
 Energy    analysis  of  steel  column  under  cyclic axial loading: (a) Cyclic
 stress--strain response;
 (b)    Evolution  of different forms of energy with cycles: Input work, plastic
 dissipation, plastic work, plastic free energy, and strain energy.}
\end{figure}


As       expected,      the      stress--strain      response      shown      in
Figure~\ref{figure_steel_axial}  follows  the  constitutive  model  presented in
Figure~\ref{figure_steel_model}.
%
Due  to  the  choice  of  hardening  parameters  ($a_{1}$, $a_{2}$, $a_{3}$, and
$a_{4}$),  isotropic  hardening after first loading reversal is relatively small.
%
The evolution of plastic free energy, which is related to the hardening behavior
of  the constitutive model, is also observed to be insignificant after the first
loading reversal.
%
Energy          balance          in          the          steel         material
(Equation~\ref{equation_steel_energy_balance})   is   maintained  during  entire
simulation.


In  this particular case, the difference between plastic dissipation and plastic
work  is  significant  during  initial  loading (or monotonic loading), but then
becomes  less  significant during  cyclic  loading.
% which is probably the reason of
%ignorance of plastic free energy in many studies.
%
It  is  important  to  point  out  that  such difference could be significant if
different  hardening  parameters  are  chosen or complex loading conditions (for
example seismic loading) are applied.



Another   observation is that the ratio between plastic dissipation  and plastic
work, the  Quinney--Taylor  coefficient \citep{Taylor1934}, changes from 0.5 to
0.9 in just a few loading cycles.
%
Based  on this, it is recommended that Quinney--Taylor  coefficient be variable,
calculated directly, as was done here, and not prescribed as a fixed number.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Cyclic Bending Loading}

%It  has  been  proven  that fiber section elements perform well under axial- and
%bending-dominant loading conditions.
%
The same column used in section \ref{Steel_Column} is loaded with cyclic bending
moment on the top.
%
Figure~\ref{figure_steel_bending} shows the moment--rotation response as well as
the  energy  calculation  results  for  the  steel  column  under cyclic bending
loading.

\begin{figure}[!htbp]
 \centering
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Steel_Bending_Stress_Strain.pdf}} 
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Steel_Bending_Energy.pdf}}
 \caption{\label{figure_steel_bending}
 Energy analysis of steel column under cyclic bending loading: (a) Moment--rotation response;
 (b) Evolution  of different forms of energy with cycles: Input work, plastic
 dissipation, plastic work, plastic free energy, and strain energy.}
\end{figure}


% Clearly,  the moment--rotation response and energy results in this case are very
% similar to those in the axial loading case.
%
When  a  beam element is loaded in  pure bending, half of the fibers will be in
tension   while  the  other  half in compression.
The  normal  stress and strain distribution on any cross section is symmetric.
%
Since  the  fiber  material  model  used  in  this  case  has   the  same
stress--strain  response  under  tension  and compression, the energy results in
for this bending case share the similar  pattern with those in the axial
loading case.


Note  that in both axial and bending cases, the strain energy accumulated in the
material body is much smaller than the plastic dissipation.
%
This  means  that  most  of the input work results in plastic deformation of the
material,  and indicates  possibility of large deformation and material
damage.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Plain Concrete Column}

In  order  to  verify  the  proposed  energy  computation  approach for uniaxial
concrete  material model, examples of plain concrete columns are studied in this
section.
%
The size and setup of the model are the same as those of the steel column, which
has been shown in Figure~\ref{figure_steel_concrete_setup}.
%
The  parameters  for  uniaxial  concrete  material  used  in  this  section  are
summarized in Table~\ref{table_concrete}.


\begin{table}[htb]
\centering
\caption{Material model parameters used in plain concrete column examples.}
\label{table_concrete}
\resizebox{0.75\textwidth}{!}{
\begin{tabular}{ccccccc}
\hline
$f_{cs}$ [MPa] & $\epsilon_{cs}$ & $f_{cu}$ [MPa] & $\epsilon_{cu}$ & $\lambda$ & $f_{ts}$ [MPa] & $E_{t}$ [GPa] \\
\hline
-30.2 & -0.00219 & -6.0 & -0.00696 & 0.5 & 3.02 & 5.0 \\
\hline
\end{tabular}
}
\end{table}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Monotonic Axial Loading}

As  stated  in  the  assumptions for energy dissipation in the uniaxial concrete
model,  the  amount of energy dissipated during monotonic loading is much larger
than that during subsequent unloading/reloading.
%
Such  assumption  is  made based on the brittle nature of concrete materials, in
which damage caused by fracture is the main source of energy dissipation.
%
In  this  case, the stress--strain response as well as the energy results of the
plain  concrete  column  model under monotonic axial compression is investigated
and presented in Figure~\ref{figure_concrete_monotonic}.


\begin{figure}[!htbp]
 \centering
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Concrete_Monotonic_Stress_Strain.pdf}} 
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Concrete_Monotonic_Energy.pdf}}
 \caption{\label{figure_concrete_monotonic}
 Energy  analysis  of  plain  concrete column under monotonic axial loading: 
 (a) Stress--strain response;
 (b) Evolution  of different forms of energy with cycles: Input work, plastic
 dissipation, plastic work, plastic free energy, and strain energy.}
\end{figure}

The  stress--strain  response  shown  in  Figure~\ref{figure_concrete_monotonic}
follows    the    compressive    constitutive    response    as   presented   in
Figure~\ref{figure_concrete_model}.
%
Energy          balance          of        the          model,  expressed     by
Equation~\ref{equation_concrete_energy_balance},  is  maintained  during  entire
simulation.


As observed in Figure~\ref{figure_concrete_monotonic}, large amount of the input
work is dissipated during monotonic compression.
%
It is important to point out that the difference between plastic dissipation and
plastic work is significant.
%
Plastic  free  energy starts to accumulate after maximum compressive strength is
reached and continue to increase even after crushing.
%
Such  behavior  can  be  explained  by  considering  that the micro-structure of
concrete  continues  to  evolve as external loads continues to be applied on the
solid/structure.


The  strain  energy  starts  to  decrease  after  maximum  compressive  strength
is reached and gradually decreases to almost zero after crushing.
%
This  observation  is  consistent  with the fact that the micro-fractures expand
rapidly after maximum strength is reached, which leads to the release of elastic
strain energy and energy dissipation caused by fracture and crushing.









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Cyclic Axial Loading}

Due  to the complex unloading--reloading rules of the model, the cyclic behavior
of  the  uniaxial  concrete  material  is much more complicated than that of the
steel model.
%
Figure~\ref{figure_concrete_cyclic} presents the stress--strain response as well
as  the  energy  calculation  results for the plain concrete column under cyclic
axial loading.


\begin{figure}[!htbp]
 \centering
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Concrete_Cyclic_Stress_Strain.pdf}} 
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Concrete_Cyclic_Energy.pdf}}
 \caption{\label{figure_concrete_cyclic}
 Energy analysis of plain concrete column under cyclic axial loading: (a) Stress--strain response;
 (b) Evolution  of different forms of energy with cycles: Input work, plastic
 dissipation, plastic work, plastic free energy, and strain energy.}
\end{figure}


As   shown  in  Figure~\ref{figure_concrete_cyclic},  the  majority  of  plastic
dissipation occurs during initial, monotonic loading branch.
%
It  is  important  to  note    that    there    are    negative  increments  in 
plastic work during unloading, for example at time $t=5$s,
however  plastic  dissipation  never  shows  any  negative  increments.  This is
consistent   with   the   requirements  of  the  second  law  of  thermodynamics
(Equation~\ref{equation_internal_dissipation_final}).


It   should be mentioned that there is a small amount of energy dissipation when
the material is in tension, for example between $t=6-9$s.
%
However,  this   energy dissipation is  much smaller than that when the material
is in compression.
%
This  can  be  explained  by  the  low  tensile strength of concrete material in
general.







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Reinforced Concrete Column}

To    study    the    combined  influence  of  concrete and steel fibers, energy
calculations for a reinforced concrete column are presented.
%
The  schematic of the model is shown in Figure~\ref{figure_concrete_cyclic}, and
the material model parameters are summarized in Table~\ref{table_rc}.
%
The  cross  section  of the column is modeled with unconfined concrete, confined
concrete,  and  steel  fibers with uniaxial material models discussed in earlier
sections.


\begin{figure}[!htbp]
\centering
\includegraphics[width=0.8\columnwidth]{Figures/RC_Setup.pdf}
\caption{\label{figure_rc_setup}
Schematic of the reinforced concrete column modeled with fiber sections and uniaxial steel/concrete materials.}
\end{figure}

\begin{table}[htb]
\centering
\caption{Material model parameters used in reinforced concrete column examples.}
\label{table_rc}
\resizebox{0.6\textwidth}{!}{
\begin{tabular}{ccccc}
\hline
\multicolumn{2}{c}{\multirow{2}{*}{\textbf{Steel Fiber}}} & \multicolumn{3}{c}{\textbf{Concrete Fiber}} \\ \cline{3-5} 
\multicolumn{2}{c}{} & & \textbf{Confined} & \textbf{Unconfined} \\ \hline
$\sigma_{y}$ (MPa) & 413.8 & $f_{cs}$ (MPa) & -30.2  & -24.16 \\
$E$ (GPa)   & 200.0 & $\epsilon_{cs}$ & -0.00219 & -0.001752 \\
$b$     & 0.01 & $f_{cu}$ (MPa) & -6.0  & 0.0  \\
$R_{0}$    & 18.0 & $\epsilon_{cu}$ & -0.00696 & -0.005568 \\
$c_{R_{1}}$   & 0.925 & $\lambda$   & 0.5  & 0.5  \\
$c_{R_{2}}$   & 0.15 & $f_{ts}$ (MPa) & 3.02  & 0.0  \\
$a_{1}$, $a_{3}$ & 0.0 & $E_{t}$ (GPa)  & 5.0  & 0.0  \\
$a_{2}$, $a_{4}$ & 55.0 &     &   &   \\ \hline
\end{tabular}
}
\end{table}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Cyclic Axial Loading}

Figure~\ref{figure_rc_axial}  shows  the force--displacement response as well as
the  energy  calculation results for the reinforced concrete column under cyclic
axial loading.

\begin{figure}[!htbp]
 \centering
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/RC_Axial_Force_Displacement.pdf}} 
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/RC_Axial_Energy.pdf}}
 \caption{\label{figure_rc_axial}
 Energy analysis of reinforced concrete column under cyclic axial loading: (a) Force--displacement response;
 (b) Evolution  of different forms of energy with cycles: Input work, plastic
 dissipation, plastic work, plastic free energy, and strain energy.}
\end{figure}


Since  concrete  fibers  have  much  higher  compressive  strength  than tensile
strength,  the  stress--strain  response  of  the  column  is  controlled by the
concrete  part  when  it  is under compression, and by the steel part when under
tension.
%
In  this  case,  the  initial loading curve clearly resembles the stress--strain
response of concrete fiber under monotonic compression.
%
Then the unloading--reloading cycles have the same pattern as those of the steel
fiber under cyclic axial loading.



By      comparing   the   energy   results  for  reinforced  concrete  shown  in
Figure~\ref{figure_rc_axial}     and     those     for     steel     shown    in
Figure~\ref{figure_steel_axial},  it  can  be  seen  that the energy dissipation
patterns  in  both  cases  are  similar after initial compression and tension of
concrete, after which steel takes over.
%
This indicates that the majority of input work is dissipated in the steel fibers
once the maximum strength of the concrete is exceeded.
%
Again,  it  can be observed that the difference between plastic work and plastic
dissipation is significant in this case.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsubsection{Cyclic Shear Loading}

% ???
% Maybe not shear, since fiber beam does not perform well for shear-dominant problems...

% \begin{figure}[!htbp]
%  \centering
%  \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/RC_Shear_Force_Displacement.pdf}} 
%  \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/RC_Shear_Energy.pdf}}
%  \caption{\label{figure_rc_shear}
%  Energy analysis of reinforced concrete column under cyclic shear loading: (a) Force--displacement response;
%  (b) Plastic dissipation, plastic work, plastic free energy, strain energy, and input work.}
% \end{figure}








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Cyclic Bending Loading}

Figure~\ref{figure_rc_bending}  shows  the  moment--rotation response as well as
the  energy  calculation  results of the reinforced concrete column under cyclic
pure bending loading.

\begin{figure}[!htbp]
 \centering
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/RC_Bending_Force_Displacement.pdf}} 
 \subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/RC_Bending_Energy.pdf}}
 \caption{\label{figure_rc_bending}
 Energy analysis of reinforced concrete column under cyclic bending loading: (a)
 Moment--rotation  response; 
 (b) Evolution  of different forms of energy with cycles: Input work, plastic
 dissipation, plastic work, plastic free energy, and strain energy.}
\end{figure}


During initial loading, the concrete fibers on the compressive side of the cross
section take most of the compression, while during the first reverse loading, the
concrete  fibers  on  the  other  side  of  the cross section are compressed and
damaged.
%
This    process  is indicated in the moment--rotation curve where two bumps, for
positive and negative moments, are observed.
%
The  energy  computation  results  also  show that the concrete fibers dissipate
large amount of energy and get damaged during the first loading cycle.
%
After  that, the response of the reinforced concrete column is controlled by the
steel bars.


According  to  the two cases, axial and pure bending, of reinforced concrete column under cyclic loading,
the  concrete part of the column can dissipate the majority of the input work if
the loading is mainly monotonic compression.
%
For cyclic loading cases, if the loading does not exceed the maximum compressive
strength  of  the  concrete,  energy
dissipation is observed in both the concrete and steel.
%
However, if the cyclic loading does exceed the maximum strength of the concrete,
the  majority of energy dissipation is in the steel reinforcing bars after
the concrete is damaged.
%
This   conclusion   is   consistent   with   the   engineering  experience  that
reinforcement is crucial  to  the  performance  of concrete structure during
seismic events, when the beams and columns suffer from cyclic loadings.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Steel Frame}
%OVDE


Previous  examples  were  assumed to be static or quasi-static, cyclic. 
%
This    was  done  in  order  investigate the energy dissipation on material and
simple structure level without the influence of dynamics.
%
In other words, kinetic energy was not considered.
%
This  example features  a full dynamic  modeling of a steel frame  using fiber section elements
with uniaxial steel material, as shown in Figure~\ref{figure_steel_frame_setup}.
% 
Model is comprised of three levels, floors.
%
Each level of this frame is comprised of two vertical columns (beam-column elements) and
one horizontal beam (beam-column element) on top of these columns.
%
Steel frame model is loaded dynamically at the base using 1D seismic motion.
%
The peak acceleration of the input motion is 0.76 $g$.

\begin{figure}[!htbp]
\centering
\includegraphics[width=1.0\columnwidth]{Figures/Steel_Frame_Setup.pdf}
\caption{\label{figure_steel_frame_setup}
Schematic of the steel frame modeled with fiber section elements and uniaxial steel material.}
\end{figure}


The        energy       computation       results       are       shown       in
Figure~\ref{figure_steel_frame_energy}.
%
Input  work is computed from the input motion and reaction forces at the base of
model.
%
Kinetic    energy   is computed from velocities of nodes.
%
Strain energy, plastic free energy, and plastic dissipation at each  level 
are  computed using Equation~\ref{equation_steel_dSE}, \ref{equation_steel_dDP},
and \ref{equation_steel_dPF}.


\begin{figure}[!htbp]
\centering
\includegraphics[width=\columnwidth]{Figures/Steel_Frame_Energy.pdf}
\caption{\label{figure_steel_frame_energy}
Energy analysis of steel frame model under imposed seismic motion.}
\end{figure}


It is noted that the energy balance if fully maintained at all times.
%
It   is  observed in  Figure~\ref{figure_steel_frame_energy}, that the sum of   
kinetic  energy,  strain energy, plastic free energy, and plastic dissipation of
the system equals to the total input work.
%
All   of the above energies are calculated independently, and then used to prove
energy balance of the system.
%


 
Close  inspection  of curve above plastic free energy (curve that represents sum
of  plastic  dissipation  for all three levels, and plastic free energy, reveals
small negative slope.
%
This curve represents plastic work and not plastic dissipation
hence negative slope is allowed.
%
On  the  other hand, curve representing sum of plastic dissipation for all three
levels, does not, and cannot have negative slope.
%
Negative slope of this plastic dissipation curve would mean energy production and that 
would violate second law of thermodynamics \citep{Yang2017a}.
%
At  the  end of simulation, more than 80\% of the total input work is dissipated
due  to material elasto-plasticity.
%
Approximately  13\%  of input work is transformed into plastic free energy that 
does not result in heating or material damage.
% %
% In  some  cases,  it  might be reasonable to use input work (or energy demand in
% some literature) as a parameter to evaluate structure safety.
% %
% However, as shown in this example, correctly computed energy dissipation is more
% appropriate  for  evaluation  of  material  damage  and structure performance in
% general.
% 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}

Presented  in  this  paper  was  a thermodynamic-based methodology for
computation  of energy dissipation in nonlinear structural elements, modeled using
fiber section and uniaxial material models.
%
Two  popular  material models for steel and concrete were examined, with focus on
their nonlinear cyclic behaviors.
%
Formulation for the energy storage and dissipation in these two material models
were  derived from the basic principles of thermodynamics, in combination with a
few reasonable assumptions.
%
The    proposed    methodology    was  illustrated  using  a series of numerical
simulations on beam column finite elements subjected to axial and bending loads.
%
In  addition,  energy calculations were performed for a three story steel frame,
excited with a 1D seismic motion at the base.




The    misconception  about plastic work and plastic dissipation, which leads to
the  violation of principles of thermodynamics, and that is found in a number of
papers on energy dissipation of structures, was addressed.
%
Theoretical  derivation  and  experimental  observation  have  both  proven that
plastic free energy is a basic form of energy that should not be neglected.
%
Taking   into account kinetic energy, strain  energy,  plastic  free energy, and
plastic  dissipation,  ensures  that  the  first  law  of thermodynamics, energy
balance, is maintained.


% Physically,  plastic  free  energy  is  related  to  the  evolution  of material
% micro-structure,  which  is  not  represented  by  specific parameters (like the
% internal variables in some elastoplasticity models).
%
Based     on  experimentally  observed  behavior  of  concrete  and  steel,  few
assumptions  were  made  within  concrete  and steel 1D fiber material models to
enforce thermomechanics.
%
Equations  for  energy  computation  were  derived and implemented in MS-ESSI 
\citep{Real_ESSI_Simulator}.
%
In  addition,  numerical  examples presented in this paper  are available on the
MS-ESSI website \url{http://ms-essi.info/}.




Presented   approach  was  illustrated  and tested using several concrete, steel
and  reinforced  concrete  beam-column  element  and a steel frame with different
loading conditions.
%
As  expected,  energy  balance  was  maintained  during entire simulation in all
tested cases.
%
It    was    shown   that  plastic  work  could  drop, have negative increments,
however  plastic dissipation was always non negative, as expressed by the second
law of thermodynamics.
%
It  was  also  observed  that  the  difference  between plastic work and plastic
dissipation could be significant.
%
The    ratio  between  plastic  work  and  plastic  dissipation, Quinney--Taylor
coefficient, did evolve in time.
%
It  is  thus  recommended  not  to  use  a  constant  value for Quinney--Taylor
coefficient, rather it should be calculated on a case by case basis.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Acknowledgments}

This work was supported in part by the US-DOE.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section*{References}

\bibliography{refmech,refcomp}

\bibliographystyle{abbrvnat}


\end{document}