A web-based educational software for structural

dynamics⋆

Christos G. Panagiotopoulos1 and George D. Manolis2∗

Abstract

In this work, we present a web-based educational package for teaching structural dynam-ics, as well as for providing an easily accessible on-line tool for interactive experimentation,useful for both undergraduate and graduate training. Current version of the package com-prises five software modules, namely (a) time history analysis of a single degree-of-freedomsystem, (b) modal analysis and time history analysis of a single storey space frame, (c)modal analysis and time history analysis of planar frames up to three stories in height,(d) construction of response spectra from ground acceleration signals, and (e) generationof synthetic ground motions for layered soil formations. These second-generation softwaremodulus are now freely available over the internet and derive from a previously developededucational package that has been used over the past five years as a virtual laboratory forteaching structural dynamics fundamentals to civil engineering undergraduates.

Keywords: structural dynamics, multiple-degree-of-freedom systems, earthquake spectra,modal analysis, time-stepping, synthetic record generation

1 Introduction

The most earthquake-prone country in the European Union is Greece, and any type of infrastruc-ture built must conform in terms of structural design to the European design code regulations(CEN, 2004). In addition, continuous effort is expended at gradually retrofitting and upgrad-ing the existing national building stock (KANEPE, 2007), starting from public buildings suchas schools and hospitals and moving on to residential and industrial structures, to what aredeemed ‘acceptable’ standards for survival following an earthquake. Thus, Civil Engineering de-partments in all five major Greek Universities and four Polytechnics have been offering coursesin structural dynamics and earthquake engineering since the 1980’s and 1990’s, respectively, atrend that has since accelerated with the last substantial upgrading of the national Greek seis-mic code (EPPO, 2000), now superseded by the Eurocode (CEN, 2004). These types of coursesturned out to be difficult to teach for two reasons: (a) they require students that have a goodstructural analysis background from past semesters, which is often shaky and (b) any mean-ingful examples that must be worked out carry a relatively large volume of computation, oftennecessitating the use of finite element method (FEM) packages that turn out to be cumbersome,since numerical modelling techniques must be taught in parallel with structural dynamics inorder to assure their rational use.

1.1 Structural dynamics analysis overview

In Civil Engineering, structural dynamics are encountered any time the external loads involvedare of transient nature, with time-harmonic cases being a special class. Thus, buildings placedin earthquake and/or high wind zones, offshore structures, bridges, industrial structures, etc.,are typical cases that require a dynamic analysis in parallel with the conventional static analysis(ASCE, 2010). In here, we focus on earthquake-induced loads, which pose a special type of

⋆Preprint of the paper submitted in Computer Applications in Engineering Education1Institute of Applied and Computational Mathematics, Foundation for Research and Technology - Hellas,

Nikolaou Plastira 100, Vassilika Vouton, GR-700 13 Heraklion, Crete, Greece, pchr@iacm.forth.gr2Laboratory of Statics and Dynamics, Department of Civil Engineering, Aristotle University, GR-541 24 Thes-

saloniki, Greece, gdm@civil.auth.gr

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problem given the uncertainty as to the magnitude, direction, duration and timing of an antic-ipated major seismic event. In general, seismic events are classified according to the damagethey cause, which ranges from minor to catastrophic. The current European design philoso-phy (CEN, 2004) is rather straightforward in that structures should suffer minor damages forsmall seismic events, repairable damage for events comparable to the design earthquake, whilestructural collapse should be avoided for large magnitude earthquakes.

More specifically, structural systems exhibit linear elastic response under operational loadsand nonlinear response when experiencing severe excitations generated by natural hazards. Inthe latter case, the restoring forces which develop within the structural members comprisingthe structure trace hysteresis loops over time when plotted against the kinematic variables.The hereditary nature of these nonlinear restoring forces indicates that they can no longer bedescribed as functions of the instantaneous displacement and velocity, so that the conventionalconstitutive models must be augmented to include the time-dependent nature of the hysteresisphenomenon. To this end, modern dynamic analysis should include the nonlinear response of astructure in the time domain (Chopra, 1995). Nowadays, this is computationally feasible evenfor complex structural systems requiring the use of thousands of degrees-of-freedom (DOF) fortheir description in a FEM environment, given the high increase in computational power overtime and the parallel evolution of engineering software. As such, nonlinear analysis has beenincorporated, either directly or indirectly through the use of ductility factors, in the majority ofmodern seismic design codes as an option for all categories of structures. Of course, the abilityto handle such models presupposes the engineer has mastered both modal analysis, responsespectrum analysis and elastic transient analysis, namely the targets of the present educationalsoftware development.

Furthermore, seismic isolation systems (Naeim and Kelly, 1999) placed within a structuralsystem have generated much interest among practising engineers in recent years, as they ac-complish two basic functions: (a) They absorb much of the energy imparted to the structureby seismic actions, wind loads, ambient vibrations, etc., and (b) they alter the dynamic char-acteristics of the combined superstructure-base isolation system, which limits the amount ofvibratory energy imparted to the structure from the surrounding soil. At the same time, struc-tural integrity is maintained as the level of forces which develop within the structural membersis reduced. Base isolation systems can also be placed locally within a structure at the vari-ous floor levels so as to protect secondary systems and equipment from damage. In general,seismic isolation systems can be classified in three basic groups: (a) Lead-rubber bearings forenergy absorption; (b) plain friction and friction pendulum systems for sliding of the base and(c) a combination of both. In our present educational software package, the option for limitednonlinearity at the base of a frame allows for a first approach to understanding base isolationprincipals from the part of the student.

Finally, current research work on structural risk analysis (Bazzurro and Cornell, 1994) hasdemonstrated that among all possible sources of uncertainty stemming from the material prop-erties, from the design assumptions and from the earthquake-induced ground motions, the latterhas the highest effect on the variability observed in the structural response. Thus, the selectionof a reasonable set of earthquake motions for conducting dynamic analysis is an imperative andmany methods have been proposed for selection and scaling process of real records (Ambraseyset al., 2000). This fact is recognized in Eurocode (CEN, 2004) as well, which permits eitherselection of real records or generation of artificial ones that comply with some general criteriarelated to matching a target response spectrum. To this end, there is a dedicated module in thepresent educational package that can be used in constructing response spectra from time historyrecords. This is an important addition, since it allows students to compare real earthquakespectra with the design spectrum. Furthermore, there is an additional module that allows forconstruction of artificial seismic signals in the form of displacement, velocity and accelerationtime histories that can be inserted in the aforementioned module to produce response spectra.

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1.2 Structural dynamics web-based software development

The development of professional and educational packages on various types of electronic plat-forms is an ongoing activity starting from the early days of expert systems (Maher, 1987) on-wards, but the introduction of web-based educational tools is more recent (Jackson et al., 1993;Reed and Afjeh, 1998). As examples, we mention diverse work ranging from that of (Katsanoset al., 2014) on seismic design of reinforced concrete buildings based on the Matlab platformfor professional engineers, that of (Sobhaninejad et al., 2011) on integrated earthquake scenariosimulations comprising hazard quantification, disaster quantification and post-earthquake recov-ery for use by government agencies and planners, and that of (Zea et al., 2009) on educationalvideogames for children. Closer to the field of structural mechanics we mention the work of(Almeida Barretto et al., 2003) on general static analysis of typical structures and structuralsystems and that of (Romeo and Padoan, 2008) on the static analysis of trusses. Furthermore,an early attempt of simulating case specific problems, representing a selection of typical kine-matics and kinetics problems for both particles and rigid bodies, for educational purposes hasbeen presented in the past (Flori et al., 1996). Furthermore, (Jacquot et al., 1995) developeda computer animation package for complementing the teaching of structural dynamics, (Gaoet al., 2005) focused on the development of a series of virtual laboratories for structural dynam-ics experiments and developed java code, while some extension of that was presented later (Simet al., 2009) including a multi-storey shear building model. Finally (Clarke, 2011) presented asoftware package for dynamic analysis in earthquake engineering with nonlinear capabilities.

Experience gained from past teaching practice leads to the conclusion that versatile andeasy to access educational tools must be continuously developed, in order to provide studentswith a user-friendly environment for solving standard types of problems in structural dynam-ics and structural mechanics. The world-wide-web turns out to be the preferred way for easyaccess. More specifically, we began in 2007 by setting up a dedicated electronic address ashttp://edusoft.civil.auth.gr in the civil engineering department’s server computers, withedusoft being an umbrella address that contains a sizeable number (about fifty) of applicationprograms associated with both required and elective courses offered by the Department of CivilEngineering at Aristotle University of Thessaloniki (AUTH). Access to edusoft is free, unlim-ited and applications are open-source under the GPL license. Two application modulus in theabove electronic address, with respective code numbers TE1800 and TE2400 for the Dynamics ofStructures I and II courses, contain four programs (jesdof, espec, jframe and jsisma). Laterin 2010, an extension of jframe, under the name jTframe was developed for the Dynamics ofStructures II, but was not uploaded on internet. Once accessed, the program environment isinteractive to allow students to build and correct the original problem at hand. Furthermore,good quality graphs, as well as real time animation of the response, are produced for a visualappreciation of what a dynamic response looks like, which is a valuable experience for students.We note that a (JAVA, 2011) runtime environment (JRE) has to be activated in the receivingcomputer for the programs to function and that the language of application is Greek.

These structural dynamics modules were fully overhauled in 2014, as part of an electronicbook (Manolis et al., 2015) that the authors prepared on that time, and made available throughthe http://dynasoft.civil.auth.gr site. This new package, labelled dynasoft, is freely avail-able for both immediate use and complete download by the external user, who can modify it forhis personal use. The package comprises five software modules, namely (a) transient analysis ofa single degree-of-freedom system (SDOF), modal and transient analysis for (b) a single storeyspace frame and (c) a three storey planar frame, (d) construction of response spectra and (e)generation of synthetic ground motions for layered soil formations. As previously mentioned,the first four modules were originally developed by the Statics and Dynamics Laboratory atAUTH under a European Union grant to the Civil Engineering department for upgrading un-dergraduate education by introducing web-based, interactive software that is accessible throughon-campus servers. More specifically, development of these modules was funded through a na-

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tional EPEAEK II research program entitled “Upgrading Civil Engineering Studies at AUTH”for the period 2007–09. These modules were then interfaced with the two structural dynamicscourses, one compulsory and one elective, taught at the eighth and ninth semesters, respectively,as previously mentioned. The fifth module was separately developed under a joint Bulgarian-Greek grant on “Elastic Wave Propagation in Inhomogeneous Media” in 2007–08. All thesemodulus were upgraded over the period 2010–12 with departmental funds directly earmarkedfor the development of a Virtual Applications Laboratory. Finally, in 2014–15 a second overhaulwas realized and the software was moved to the central servers of Aristotle University under thelabel dynasoft, thus making the programs accessible all outside users. In here, we outline thedevelopment of this software package and summarize the experience gained over the last fiveacademic years of continuous use by students.

2 The structural dynamics software package dynasoft

Application name dynasoft comes from the merger of two words, namely (structural) dynamicsand software, with the former indicating the subject matter and the latter targeting the means bywhich to solve basic problems encountered in the teaching process. It is not our intention here todevelop of a comprehensive finite element method program that covers large scale mathematicalmodeling in structural dynamics, as this is the domain of scientific computing with applicationto engineering problems, and involves collaboration with software developers. More specifically,use of dynasoft involves five major sub-entities, which together form the framework of a virtualcomputational laboratory to be used within the context of structural dynamics teaching, placingthe emphasis in earthquake engineering.

The programming language chosen for dynasoft development was JAVA (JAVA, 2011),and that was decided mainly for the following reasons. First of all, (a) java is freely availablethrough the internet; (b) java use is widespread, and it has been installed in a variety of electronicdevices, which implies cross-platform operation; and (c) because of object-oriented paradigm,java applications can be easily extended and reproduced, making it a particularly useful toolfor educational as well as more specialized scientific purposes. Formerly, through java appletsand later using the java web start, java language has became a valueable tool for web-basedapplications. Finally, java internationalization (I18N) is a mechanism to create applicationSthat can be adapted to different languages and regions. Application dynasoft, adopting thejava web start technology, is directly accessible online without passwords, and can be downloadedand operated directly on personal computers or other devices. It is available, along with codelicensed under the GPL, at site http://dynasoft.civil.auth.gr/.

Briefly, dynasoft features (a) the analysis of standard structural systems, referred to asdynasoft-construction, (b) calculation of dynamic and modal characteristics of these structures,(c) loading functions can be described either by mathematical formulas or in discrete form usingdata stored in files, (d) dynamic analysis to the input provided including calculation of timehistories, (e) construction of spectra and calculation of spectral parameters, and (f) handling ofbase motion induced by earthquakes.

In what follows, after presenting the graphical unit for interaction with the user, we introducethe modules currently included in dynasoft the assumptions made and capabilities supplied,and we finally present some practical example.

2.1 The graphical user interface (GUI)

The GUI implemented in dynasoft consists of four parts, each with distinct function and pur-pose. As shown in Fig. 1, we first have the graphic display of the dynamic system in questionand its kinematic state (deformations, eigenvectors, transient response). Next is the informa-tion text area that stores both input data and the computed response. Then, the diagramdisplay area over which time-dependent graphs, as well as spectral quantities that depend on

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a particular eigenmode, are produced. Finally, there is a drop bar for the main menu witha number of suboptions, all relating to data management and selection of program features.The software has been designed with the principles of object oriented programming in mind, sothat changes/additions may be adaptive and easily implementable. For instance, the solutionprocedure in the Fourier transformed domain could be such an addition.

Main menu bar

Information text area

Chart display areaGraphic display area

Figure 1: Graphical user interface.

A more detailed description of the four GUI parts starts with the main menu bar. TheFile button contains two sub-menus, New and Define. The latter selection defines the sixprogram sub-entities. More specifically, the SDOF oscillator is entity sdof, the 3D single-storeybuilding is entity storey, the three storey planar frame is entity frame, the response spectra areentity specalc and the synthetic ground motions are entity ground. Obviously, entities sdof,storey and frame share some common features, as will be discussed later on. The Define

menu is dormant until dynasoft construction is initiated and then specifies loads (loading),initial conditions (initial) and a spectrum from which spectral values can be extracted. TheEdit button allows selection of parameters for the Newmark-beta time integration method (b-Newmark parameters), with default values γ = 1/2 and β = 1/4 corresponding to the constantaverage acceleration algorithm (Newmark, 1959). The integration time step can also be changedfrom some default value, that of 0.001sec. This button also allows for the initiation of thetime-stepping analysis (transient analysis), which is originally inactive. Next, from the View

button one may handle plot settings, choosing between either time plots and charts in sectionimaging or graphs from the calculation of spectra, as well as language settings with plus thevisualization of an eigenvector in the graphic display segment. Finally, it accesses parametersrelated to rudimentary animation possibilities (animation). The last button on the main menuis Help, which gives general information about dynasoft .

2.2 Entity sdof for single degree of freedom system

Perhaps the simplest, yet basic, entity in structural dynamics is that of a single degree of freedomlinear oscillator. The equation of motion for such system is given as:

mu(t) + cu+ ku(t) = f(t), (2.1)

where m is the mass, k the stiffness and c the damping coefficient, while u(t) the displacementand f(t) the external applied loading function in time. In order for this equation to be well

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defined initial conditions (say at t=0) must be given, i.e.,

u(0) = u0 (2.2a)

u(0) = v0 (2.2b)

for displacement and velocity, respectively.Students can benefit greatly by having the opportunity to solve and direclty see the plots of

results for several combinations of mechanical properties (mass m, stifness k and damping c),initial condtions u0, v0 and external loading f(t). The first entity sdof of dynasoft deals withthat problem.

Entity sdof example

As example, we define a SDOF oscillator with stiffness k=2000 kN/m2 and mass m=10 tn,initially at rest and then subjected to an external force given as fext=sin(1.1ω0 t) kN. Wespecify a time interval from t=0 to t=10 sec, with ω0 the natural period of the oscillator. Wenow wish to compute the time response of the oscillator in terms of displacement, velocity andacceleration. From the menu we select sdof and the input window shown in Fig. 2 appears.Three parameters are needed to describe the oscillator, namely stiffness, mass and damping(as a percentage ζ). Output from the graphic display section yields the natural period =0.444sec and frequency ω=14.142 rad/sec, plus a back-calculated damping coefficient c. Figure 3shows the evolution of this input-output process. Next, load input is done from the menu usingFile→Define→loading, see Fig. 4. In addition, the loading time step and duration need beto be specified as dt=0.01sec and Td=10sec, where the former quantity can be differnt form thetime step used in the time integration method of Newmark (i.e., Edit→b-Nemark parameter).As previously mentioned, loads are specified either through a mathematical expression, as wasthe case here, or from an external ASCII file in discrete form. For the second option we activatethe record button, and read a column vector with 1000 entries since the load time step is stilldt. Any additional load values are ignored, while missing values are assumed to be zero. Finally,by pressing for an update, the information block gives the message Prescribed external loadingin time.

Figure 2: Dialog for sdof input. Figure 3: GUI after defining an sdof.

Analytical specified loading functions in time by mathematical expressions are defined by us-ing, the Jep Java - Math Expression Parser, under the GNU General Public License (GPLv3),retrieved from http://sourceforge.net/projects/jep/. Common used functions (e.g., trigono-metric, exponential, etc.) as well as the if function defined as if(condition, function for conditionbeing false, function for condition being false). Furthermore, since most loads have a finite timeduration, to reproduce this mathematically, we have defined and is possible to use the Heavisidefunction defined as,

h(t− t1) =

1, if t ≥ t1,

0, elsewhere.(2.3)

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The Heaviside function combination given below assures a finite time duration in [t1,t2] for someexternal load, i.e.,

f(t) = f(t) (h(t− t1)− h(t− t2)) .

Activation of the Edit button will now allow for the time history (or transient) analysis ofthe SDOF oscillator. Following the analysis, the GUI will produce output in the form givenby Fig. 3. The information of text section contains the SDOF displacement u(t), velocity v(t)and acceleration a(t), while the chart display section can be already used to plot. In order tomanage for which variables the plot will be currently given on the chart display area, we mayaccess the repsective controller through Tools→plot parameter. A “right click” on the textinformation display the cleaning option to erase results and the copy option to store information.Furthermore, a “right click” on the chart display section yields the options of, saving current’splot data in some file, saving the current chart as a png file or, finally, copying current’s plotdata in memory.

Figure 4: Dialog for loading input. Figure 5: GUI after transient analysis for ansdof.

2.3 Entity storey for the dynamic analysis of a single-storey 3D building

The single-storey is a structural system which due to its relative simplicity it is very instructivefor the understanding of several concepts of dynamics of structures (Manolis et al., 2015). Staticand dynamica analysis of the three-dimensional single-storey structure, in the past, was the basefor antiseismic design of structures, especially in Greece (Anastasiadis, 1999).

The single-storey building considered here comprises an arbitrary number of columns thatremain axially undeformed and respond only in bending. These columns are either hinged orfixed and support a roof slab that acts as a diaphragm (rigid surface). The motion of the roofslab is fully described by the two horizontal translations and a rotation about the vertical z-axis.The reference system will be that of origin the geometric center of the slab. It is obvious thatthis system has three degrees of freedom, the two translational displacements ux=u1 and uy=u2of the center and the angle of rotation θz=u3 about the vertical axis normal to the slap’s plane.The stiffness matrix K is a 3×3 symmetric one with elements that given as:

kxx =∑

i

kxi, kyy =∑

i

kyi, kxy = kyx = 0

kxz = kzx = −∑

i

yikxi, kyz = kzy =∑

i

xikyi (2.4)

kzz =∑

i

(kzi + y2i kxi + x2i kyi

)

where kxi, kyi and kzi stiffness coefficients of the ith column. The stiffness matrix could bediagonalized with in the principal reference system whose coordinates are given by,

(xk, yk) =

(kzykyy

,−kzxkxx

)

(2.5)

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and called elastic center. The three diagonal terms of the stiffness matrix would be given as:

kI = kII =kxx + kyy

2±

√(kxx − kyy

2

)2

kIII = kzz − x2kkyy − y2kkxx. (2.6)

For the simplified case we present here and adopted for the single-storey entity of dynasoft,the center of mass coincides with the geometric center of the slab. The mass matrix is diagonal3×3 with diagonal elements

mx = my = m, mz = Jm = mr2 (2.7)

where m the total mass of the slab, Jm the mass moment of inertia and r the radius of inertia(or radius of gyration). The mass moment of inertia for a rectangle Lx × Ly slab is given byJm = m(L2

x + L2y)/12. The equation of motion is given in matrix-vector form, as follows:

Mu(t) +Cu(t) +Ku(t) = f(t). (2.8)

For the seismic excitation case where an earthquake, given by ground motion u0(t) impinges atan angle β with respect to the x-axis, the force vector will have the following form:

f = −

mu0(t) cos βmu0(t) sin β

0

(2.9)

The damping matrix C in the numerical implementation of dynasoft has been defined in orderfor all three eigenmodes of the system to have a specific damping ratio ζ, given by the user.Finally, we notice that the decoupled eigenfrequencies, that are also computed by dynasoft,are those correspond to the system if the geometric center (also center of mass for the casesconsidered) coincided with the elastic center.

Entity storey example

As an example, consider the single-storey building of of Fig. 6, where E=2.97 kN/m2 is themodulus of elasticity, m=1.0 tn/m2 is the mass per area (Lx times Ly) and the building heightis h=4.0 m. Also, the damping ratio is taken as equal to ζ=5%. Data input is through fileFile→New→storey, and the dialog box of Fig. 7 appears. This requires filling out values forLx, Ly, m, E, h and ζ, plus defining a rigid plate and continuing with the introduction of thevertical elements. The input keys are as follows:

Once all the data for this example is added, the program automatically calculates and recordsin the text information section the dynamic properties of the building (i.e., stiffness, mass anddamping matrices, elastic center, radius of inertia, coupled as well as decoupled eigenfrequenciesand their associated eigenvectors). In the section on graphic display, we can view the first eigen-mode (see Fig. 8), while the remaining ones can be selected for viewing using the Tools→plot

parameter option (see Figs. 9-10).Next we continue with building response to time-dependent external loads. We will select as

input an accelerogram stored in an external file and recorded at 0.1sec time intervals, impingingfrom an angle of 30 degrees with respect to the horizontal x-axis.

We activate the File→Define→loading option to import data, and we also enable the “basemotion” key so that the external force file is disabled. We note here that it is possible to havebase motion and external forces applied simultaneously to the building, but for simplicity theseare kept separate. Once the above information is added for the ground motions, it is necessary tospecify the total time, which may be greater than the duration of the earthquake record itself.More specifically,we define a total time of 25sec, while the accelerogram duration is 19.9sec.

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Ly=7m

Lx=10m

(x1, y1)=(-5.0,0.0)

30/150 cm

(x2, y2)=(0.0,-3.5)

200/30 cm

(x3, y3)=(5.0,3.5)

30/40 cm

x

y

Figure 6: Plan of the single-storey building used in example.

Figure 7: Dialog for single-storey input. Figure 8: After importing the single-storey.

Figure 9: Second eigen-mode of single-storey. Figure 10: Third eigen-mode of single-storey.

We also note that acceleration values must be consistent with those used in describing thebuilding, and here we specify 0.15g in m/sec2 for the maximum acceleration. After filling in allrequired fields and pressing update , we can now move on to the analysis using Edit→transient

analysis. This option will calculate the transient response of the building. We either definethe parameters of the Newmark-beta algorithm or use default values. Once the time-steppinganalysis is finished, the calculated system response is written in the information text block forevery degree of freedom (DOF) in sequence. Through the Tools→plot parameters option wecan now choose the combination of variables for plotting (t, u(t), v(t), a(t), f(t)). respective toany degree of freedom, to plot in chart display section. The plots can be saved and a “right-click” on the screen yields the save chart option. This way we have generated phase diagramssuch as that of Fig. 11 and response history as in Fig. 12.

Another possibility is the dynamic spectrum analysis of the building. To this end, we activate

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the sequence File→Define→Spec, which opens the dialog box. We now see a window dividedinto three sections, namely the graphic display (above right), the spectrum table (top left) andthe administration bar (bottom). When opened for the first time, the window gives the Greekdesign code spectrum (EPPO, 2000) that also appears in the Eurocode provisions for aseismicdesign (CEN, 2004) ,which is calculated at six characteristic points. In this context, the spectrummight be seen as some kind of external loading equivalent for the structure. The text informationblock, is then updated and incorporates a table with a list of spectral accelerations, spectralvelocities and spectral displacement values corresponding the three DOF of the single-storeybuilding example.

Figure 11: Displacement with velocity plot forthe second dof of the single-storey building un-der base motion excitation.

Figure 12: Response in time of y-displacement(second dof) of single-storey under base motionexcitation.

2.4 Entity frame for the dynamic analysis of a single-storey 3D building

Since the basic structural entity by which modern multi-story buildings are constructed is theplanar frame comprising a lattice of beams and columns, it is essential that this should be mod-eled here in terms of its matrix representation (Przemieniecki, 1968). Since the dynamic analysisof even the simplest frame requires modal analysis as a first step, which is a computationallyintensive process, the present module allows for a completely automated solution. Transientanalysis in time domain and spectrum analysis can be performed similary to what has beenpresented in the previous section 2.3.

There are two main option regarding the junctions of horizontal structural elements of theframe, these are the restrained and the released joints given in dynasoft as fixed or hingedcolumns selection, respectively. For a single horizontal element of rectangle cross section withmoment of inertia I, height h, elasticity modulus E and considered fixed on the ground levelthe stiffness coeficient will be

k = µEI

h3, (2.10)

where coefficient µ=3 for hinged and µ=12 for fixed column, respectively. Using the stiffnessor flexibility coefficients we easily can construct the stiffness matrix M of the structure. Thefloor lever assumed always to act as a rigid body diaphragm. While the masses of the frameare assumed to be concentrated on the floor levels resulting a diagonal mass matrix M. Forthe damping matrix same assumption of constant damping ratio, defiend by the user, for allthe eigenmodes, is adopted as in section 2.3. For the case of base isolation, one additionaldegree of freedom is considered for the ground level, with corresponding stiffness, mass anddamping coefficients, Kb, Mb and ζb respectively, defined by the user for a simplified baseisolation representation.

Entity frame example

We now consider a three-story, two-column planar frame, with =2.97 kN/m2, floor masses of20tn and damping ratio ζ=5%. The floor heights are h1=4, h2=3, h3=3m, and the column cross

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sections are 40/30 (cm). Also, the floor beams of the floors have infinite stiffness, which impliesdiaphragm action at the floor level.

Figure 13: Dialog container for introduction offrame entity.

Figure 14: After introduction of frame the firsteigen-mode is drawn.

In order to input this information, the corresponding dialog box File→New→frame is ac-tivated, and Fig. 13 appears. We note here that the span length L has no influence, since thefloor beams are assumed rigid. In the Building geometry option we may specify the numberof floors from one-storey up to three-stories. It is also possible to consider base isolation byactivating the repsective fields and specifying the appropriate elastic, inertial and viscoelasticproperties at the base supports.

Once the frame input information is complete, then we press the update button for the pro-gram to build the mathematical model of this structure, as entity frame. This implies formationof the stiffness, damping and mass matrices, plus calculation of eigenfrequencies and eigen-vectors. This information can be viewed in the text information area. Simultaneously, theframe’s first eigenmode appears in the graphic display section (see Fig. 14), while the remainingtwo eigenmodes can be viewed by accessing the dialog box from the Tools→plot parameters

option (see Fig. 15-16).

Figure 15: Second eigen-mode computed,drawn and saved as figure using dynasoft .

Figure 16: Third eigen-mode computed, drawnand saved as figure using dynasoft .

Next, we calculate the transient response of the frame under initial floor displacements ofu1(0)=0.01m, u2(0)=-0.005m and u3(0)=-0.02m, along with zero initial velocities. At the sametime, we impose external point loads at the three floors equal to f1=100kN, f2=-20kN and f3=-0. The load time variation is a rectangular step function with start and end times of t1=3.5secand t2=4sec, respectively, yielding a total time load duration of 10.5sec. This information isprescribed by activating the File→Define→initial conditions options, followed by activa-tion of the File→Define→loading option. In order to use a mathematical function for theload duration, we build the expressions 100(h(t−3.5)−h(t−4.0)) and −20(h(t−3.5)−h(t−4.0))for the first and second floors, respectively. The time step is selected as equal 0.01 sec, which

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Table 1: Reproduction of output for spectral quantities

Spectral acceleration (m/sec2) Spectral displacement (m) Spectral Velocity (m/sec)

Sa,1=6.658 Sd,1=2.90e-2 Sv,1=4.39e-1

Sa,2=2.347 Sd,2=9.27e-2 Sv,2=4.66e-1

Sa,3=1.759 Sd,3=2.77e-2 Sv,3=2.21e-1

coincides with the numerical integration time step, and the total time is set to 10.5 sec. We nowpress the update button and the form of GUI that appears is Fig. 17-19.

Figure 17: GUI after introduction and analysis of the frame entity.

Figure 18: Second floor velocity in time. Figure 19: Elastic force in the third floor intime.

We complete this example by performing a dynamic spectrum analysis of the frame. Inparticular, we will access response spectra that are calculated in the following entity labeledspecalc. To this purpose, we retrieve the acceleration spectrum stored in a file and arrangedin column form, with the first column entry containing natural period values and the secondcolumn acceleration values, separated by a space. We open the input dialog box by activatingthe File→Define→spec option, and then we press the load button in order to identify andselect in the computer file that contains the spectrum. Then, by pressing the update button,calculations are performed and spectral values for the accelerations, velocities and displacementsare stored in the information text, as shown in Table 1.

Finally, we introduce here some animation capabilities for dynasoft, accesible through theTools→plot parameter, where the respective controller will appeared similar to Fig. 20. Thiscontroler consist of the two plot’s axes variables selectors, the forward and backward buttonsfor eigen-shape browsing, the play and stop buttons for triggering and stop animation, and

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finally a slider to control speed of animation. When animation is active, the deformed shapeis shown in the graphic display area, while in the chart display segment a moving green dotmoves on the currently active plot and indicates the current position in correspondence to theanimated deformed shape (see Fig. 21).

Figure 20: Plot and animation controller. Figure 21: Snapshot while animation is active.

2.5 Entity specalc for calculating response spectra

Modern earthquake engineering analysis and design is for the most part conducted by spectralanalysis, which builds on both modal analysis results and the availability of ground motion timehistories. The latter are converted into either analysis or design spectra and their computationfollows standard procedure by which motion is filtered through a collection of sdof systems andthe maximum response recorded as a function of frequency. Thus, it is important, not just foreducational purposes, to automate the construction of spectra. In this section, we will focuson the construction of response spectra from recorded seismically-induced ground accelerations,although any type of transient motion can be considered.

As a simply definition of response spectrum we retrieve from wikipedia (Wikipedia, 2015),“Aresponse spectrum is simply a plot of the peak or steady-state response (displacement, velocity oracceleration) of a series of oscillators of varying natural frequency, that are forced into motionby the same base vibration or shock. The resulting plot can then be used to pick off the responseof any linear system, given its natural frequency of oscillation. One such use is in assessing thepeak response of buildings to earthquakes.”

Entity specalc example

For this example, we will use a synthetic accelerogram stored as an ASCII file and has beenconstructed using the dynasoft entity ground that will be presented in the following subsection.More specifically, acceleration values are given for a time interval of 0.01sec. We activate therespective input dialog box from the File→New→specalc option and the display window givenin Fig. 22 appears on screen. We input all information starting with the spectrum’s lower T1 andupper T2 period values (in sec), the number of discrete spectrum values and the damping ratioζ. Then, the input file that contains the ground motion acceleration values must be selected,and the time increment for these discrete values must also be specified. As regards the time stepfor which discrete values of the accelerogram are given, there are two options. Having selectedthe checkbox below the file button (see Fig. 22), is asumed that an one-column file will beloaded which will contain single discrete values of the acceleration history per time step definedon the dialog. The other option, for not selected checkbox, is a two-columns acceleration file,where in each row the pair of time and acceleration values, seperated by a space, is given, whilea constant time step is expected. After updating the model in the chart section of the GUI theaccelerogram loaded will be shown (image will be very similar to Fig. 28 of section 2.6).

We note that the methodology followed for response spectrum construction involves solvingfor the response of a number of SDOF systems with fixed damping ratio and whose naturalperiod follow the spectrum of values specified a priori. In this case here, there are 400 SDOF

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Figure 22: Input dialog for specalc entity. Figure 23: Spectral displacements with (eigen-)period.

analyses, each done numerically using the Newmark-beta algorithm with default values for theintegration parameters (β,γ) and for the time step dt, although these values can be changedby accessing the Edit→-Newmark parameters. The maximum response of these 400 SDOFoscillators (i.e., the spectral displacements SD, the spectral pseudo-velocities SV and the spectralpseudo-accelerations SA) form the ordinates of the spectrum. By pressing the update buttonin the dialog box of specalc, these values are automatically calculated and stored in the textinformation section, while the plots are might be shown in the charts display section. In order toview the spectra, we access the Tools→plot parameters options to specify details concerningthe plots. For the present example, the three spectra are now shown in Fig. 23-25. Notice herethat it might be necessary to access additional options available for plotting from the displaysection in the View option by selecting either the Other Computed Values (Spectral, Synth.

Accelerations) Plots or the Structural Response Computed Values Plots buttons.

Figure 24: Spectral velocities with (eigen-) pe-riod.

Figure 25: Spectral accelerations with (eigen-)period.

2.6 Entity ground for synthetic earthquake ground motion

Given the importance of the availability of both analysis and design spectra for structuralanalysis and design purposes, respectively, it is essential that ground motion records be available.This can be done by searching seismic data bases, but it is most convenient to be able to producesynthetic seismic motions upon demand, which require as input information on the surroundinggeological profile and a measure of the magnitude of an expected earthquake. This type ofsoftware interfaces perfectly with entity specalc for continuing on with the construction ofearthquake spectra.

An elementary, yet basic model is that of horizontally polarized shear (SH) waves. As can beseen in Fig. 26, we start with a seismic source at large depth which emits a white-noise signal,i.e., one which remains constant over the frequency range of interest. Next, the signal passesthrough three filters, the High-Pass (H-P), the Kanaji-Tajimi (K-T) and the Low-Pass (L-P)filters, which respectively model a stiff halfspace and the overlying soil deposits, ranging fromfirm to loose (Manolis and Koliopoulos, 2001).

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f(t)u1(t)

u2(t)

u3(t)

m1

m2

m3

k1

k2

k3

c1

c2

c3

u(t)

Figure 26: Seismic motion simulation on ground considering a triple filter system.

The mathematical statement of the problem gives the power spectral density function (PSDF)for ground surface acceleration Sa as follows:

Sa(Ω) =(Ω/ω1)

4

(1− (Ω/ω1)2

)2+ 4ζ2

1(Ω/ω1)2

︸ ︷︷ ︸

high-passfilter

×1 + 4ζ2(Ω/ω2)

2

(1− (Ω/ω2)2

)2+ 4ζ2

2(Ω/ω2)2

︸ ︷︷ ︸

Kanai-Tajimifilter

×1

(1− (Ω/ω3)2

)2+ 4ζ2

3(Ω/ω3)2

︸ ︷︷ ︸

low-passfilter

× S0. (2.11)

The above equation represents a triple filter of the white noise power 2πS0 from the sourcethrough H-P (index 1), K-T (index-2) and L-P (index 3) filters. The corresponding equationsof dynamic equilibrium for the filters, assume the following normalized form:

u1(t) + 2ζ1ω1u1(t) + ω2

1u1(t) = f(t), (2.12a)

u2(t) + 2ζ2ω2 (u2(t)− u1(t)) + ω2

1 (u2(t)− u1(t)) = 0, (2.12b)

u3(t) + 2ζ3ω3 (u3(t)− u2(t)) + ω2

3 (u3(t)− u2(t)) = 0. (2.12c)

The above equations simply show that excitation f(t), which corresponds to white noise, sequen-tially moves through the H-P, K-T and L-P filters to give response u1(t) then u2(t) and after thatu3(t). In the numerical implementation of dynasoft it is possible for the user to choose whichof these filters will be active. Always, the last response in the previous sequence is with respectto the ground motion at the surface. The next step is to use the PSDF Sa from eq. (2.11) as thestartimg point for generation of ground accelerations under (pseudo) non-stationary conditions(Shinozuka and Jan, 1972) as,

u(t) =

N∑

n=1

An sin (Ωαt+ φn)ψ(t), (2.13)

where An=√2SaΩn∆Ω, Ωn=(n− 1/2)∆Ω, ∆Ω=(Ωβ−Ωα)/N and φn is the random phase angle

defined in the [0, 2π] interval. Furthemore, Ωβ and Ωα respectively are upper and lower boundsin the frequency spectrum. Also, index n goes from 1 to N the number of simulations, assumes

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values in the range 500-1000. Finally, the envelope function ψ(t), which gives non-stationaryproperties to the accelerogram, is given below as,

ψ(t) =

(tt1

)2

, 0 < t < t1

1, t1 ≤ t ≤ t2

exp (−b(t− t2)), t2 < t

(2.14)

where time markers t1 and t2 range from 10% to 90% of the total time (duration) of motion T ,and b=2/(t2−t1) is a constant (in 1/sec).

Entity ground example

As an application example, the above methodoly incorporated in dynasoft, will be use inconjunction with a geological medium whose properties, along withthe earthquake’s power, timeduration and frequency content, are given here. The H-P layer represented by ω1=0.8rad/sec,damping ratio ζ1=100%, the K-T layer by ω2=4.8rad/sec, damping ratio ζ2=75% and finally theL-P filter, by ω3=10.20rad/sec, damping ratio ζ3=140%. The earthquake’s power is assumedto be S0=0.135m2/sec2, total duration T=20sec, with t1=10%T and t2=60%T . The frequencyrange [Ωα,Ωβ]=[0.2,25]rad/sec. The dialog with which data for ground are passed in dynasoft

Figure 27: Dialog with which data for groundare declared.

Figure 28: Constructed (synthetic) acclelero-gram.

is shown in Fig. 27, while the repsective ground acceleration constructred can be seen in thechart soction of GUI depicted in Fig. 28.

Integration in time of the acceleration function gives sequentially velocity and displacementfunctions, which are shown in Fig. 29 and Fig. 30, respectively. Furthermore, if these dis-placements or velocities, were to be used they need further analysis (correction), however, thisprocedure is out of the scope of the current framework.

Figure 29: Ground velocity calculated by ac-celeration integration.

Figure 30: Ground displacement calculated byvelocity integration.

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3 Conclusions and discussion

Regarding actual use of the present educational package dynasoft, as well as of its predecessoredusoft, for structural dynamics analysis purposes, we conclude that the teaching experienceaccumulated during the last five years of use in actual classroom environment of a mandatorycourse on Structural Dynamics has been positive. Three modulus (i.e., single degree-of-freedomsystem, response spectra, plane frame analysis) were taught to 8th semester undergraduates afterthe first five weeks of instruction (roughly halfway through the semester), and to groups of 35 stu-dents for a one hour period in a computer laboratory. Three such groups were necessary to meetstudent demand, and laboratory use required three such periods. This particular type of sessionwas augmented by lectures summarizing basic structural dynamics concepts and originally acces-sible, in the past, through the electronic address https://blackboard.lib.auth.gr. This sitewas also moved in 2014 to a new address https://elearning.auth.gr where all such activitiesthroughout Aristotle University were centralized, utilizing the Moodle learning platform (Moo-dle, 2015). Also, the semester project assigned to all students required use of the aforementionedmodulus for its completion. The elective course on Advanced Structural Dynamics offered inthe 9th semester of study also benefitted from the analysis software, and primarily from the3D, single storey frame module, since emphasis was placed on practical problems encounteredwith actual buildings, where torsional effects appear because of structural asymmetry and thearbitrary directivity of the earthquake ground motion signal with respect to the principal axesof symmetry of the structure. Also, the improvement in the planar frame module that allowsfor limited nonlinearity at the base is also useful for rudimentary base isolation studies.

In sum, students responded favourably and felt they had learned to use a practical educa-tional tool. Of course, the final aim of both structural dynamics courses is to familiarize studentswith the Eurocode (CEN, 2004) and to allow them to perform professional dynamic analysesthat includes: (a) nonlinear effects (i.e., both lumped and distributed plasticity models), (b)earthquake record selection strategies, (c) structural performance of irregular building, (d) intro-duction of damage indices that correlate with observed damage, (e) base isolation techniques forearthquake mitigation and (f) inclusion of soil flexibility, soil damping, soil-foundation-structureinteraction effects and degradation of underlying soil layers for massive structures founded ofsoft soils.

The most obvious extension of the dynasoft application is in the realm of material nonlin-earities, such as elastoplasticity and damage mechanics concepts, so as to consider the effect ofintense loads on a structural system.

Acknowledgments

The authors wish to thank Professors Y. Krestenitis, PI for EPEAEK II project “Upgrading Civil Engi-neering Studies at AUTH 2007–2009”, C. Koutitas, project director for “Applications of Informatics tothe Civil Engineering Educational Process: The Virtual Laboratory 2010–2012” and A. Athanatopouloufor her cooperation.

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