\documentclass[12pt]{elsarticle}
%\documentclass[12pt]{article}
% % %BJ
% %BJ %\usepackage[nomarkers,figuresonly]{endfloat}
% \usepackage{endfloat}
% %BJ
%\usepackage{lineno,hyperref}
\usepackage{subfig}
\usepackage{multirow}
\usepackage{graphicx}
%\usepackage[autostyle]{csquotes}
\usepackage{csquotes}
\usepackage{leftidx}
\usepackage{amsmath}
%%%% HYPERREF HYPERREF HYPERREF HYPERREF HYPERREF
\usepackage[pdfauthor={Boris Jeremic},
colorlinks=true,
linkcolor=blue,
citecolor=blue,
urlcolor=blue]
{hyperref}
% ----------------------------------------------------------------------
%
% TIME OF DAY
%
%
\newcount\hh
\newcount\mm
\mm=\time
\hh=\time
\divide\hh by 60
\divide\mm by 60
\multiply\mm by 60
\mm=-\mm
\advance\mm by \time
\def\hhmm{\number\hh:\ifnum\mm<10{}0\fi\number\mm}
%BJ \journal{Comput. Methods Appl. Mech. Engrg.}
%\bibliographystyle{elsarticle-num}
%\usepackage[numbers,round,colon,authoryear]{natbib}
\usepackage{natbib}
%% Water mark
%\usepackage{type1cm}
%\usepackage{eso-pic}
%\usepackage{color}
% \makeatletter
% \AddToShipoutPicture{%
% \setlength{\@tempdimb}{.05\paperwidth}% X coord placement
% \setlength{\@tempdimc}{.25\paperheight}% Y coord placement
% \setlength{\unitlength}{1pt}
% \put(\strip@pt\@tempdimb,\strip@pt\@tempdimc){
% \makebox(-5,5){\rotatebox{90}{\textcolor[gray]{0.7}
% {\fontsize{0.6cm}{0.5cm}\selectfont{DRAFT of Han et al, work in progress!}}}}
% }
% }
%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% spacing REMOVE FOR official version
%
%
%baselinestretch
%
\renewcommand{\baselinestretch}{1.40}
\small\normalsize % trick from Kopka and Daly p47.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\setcounter{tocdepth}{4}
%\tableofcontents
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theoretical and Computational Formulations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Acknowledgments}
This work was supported in part by the US-DOE.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{References}
\bibliography{refmech,refcomp}
\bibliographystyle{abbrvnat}
\end{document}