Revista Intersectiii No1_eng

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Volume 2, No.1, 2005

ISSN 1582 - 3024

St uc ura Me h nicSSttrrruucctttuurraalllMMeeccchhaaanniiccsss

Societatea Academica

"Matei - Teiu Botez", Iasi

Publishing House of the


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http://www.ce.tuiasi.ro/intersections

Editorial Team

Constantin AMARIEI

Department of Structural MechanicsFaculty of Civil Engineering

Gh. Asachi Technical University of Iai, [email protected]

Mihai BUDESCUTeam LeaderDepartment of Structural MechanicsFaculty of Civil EngineeringGh. Asachi Technical University of Iai, [email protected]

Ioan CIONGRADI

Department of Structural MechanicsFaculty of Civil EngineeringGh. Asachi Technical University of Iai, [email protected]

Gabriela COVATARIUSecretaryDepartment of Structural MechanicsFaculty of Civil EngineeringGh. Asachi Technical University of Iai, [email protected]

Dan PRECUPANUDepartment of Structural MechanicsFaculty of Civil EngineeringGh. Asachi Technical University of Iai, [email protected]

Doina TEFAN


Mihai VRABIE


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Intersections/Intersecii, Vol.2, No.1, 2005 1


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Content

TAssessment of the Systems Dynamic Characteristics UsingIdentification Technique

by Doina Stefan, Violeta-Elena Chitan

TTheoretical and Experimental Studies of Steel Profilesby Mihai Budescu, Ioan P. Ciongradi and Octavian V. Roca

TEffect of thickness variation upon plates subjected to bendingby Mihai Vrabie, Cornel Marincu, Nicolae Ungureanu and Mihaela Ibnescu

THomogeneity on Designing the Structural Systems

by Florentina Luca and Septimiu-George Luca

TFormal Similarities Between Unidimensional Finite ElementsUsed in Deformability and Diathermacy Anaysis

by Dan Diaconu-otropa and Mihaela Ibnescu

TSynthesis of the Problematics Approach of the WorshipSpaces Design

by Drago Ciolacu and Lucian Strat

TProcedure for Checking the Requirement Referring to FireInsulation Capacity of a NonStructural Wall from the FireCompartment

by Dan Diaconu-otropa and Mihaela Ibnescu

TExperimental Tests of a New Class of Steel Jointsby Mihai Budescu, Ioan P. Ciongradi and Octavian V. Roca

SSN 1582-3024 Intersections/Intersecii, Vol.I1, No1, 2005 2


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Intersections/Intersecii, Vol.I1, No1, 2005 3

Content

TFinite Elements in the Analysis of Open Thin-Walled BarsSubjected to Torsion

by Mihaela Ibnescu and Dan Diaconu-otropa

TDesign of the Spatial Structures Made of Elements withElastic Connections in the Nodes

by Constantin Amariei and Corneliu Ivac


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Structural Mechanics

Assessment of the Systems Dynamic

Characteristics Using Identification Technique

Doina tefan1, VioletaElena Chitan2

1Structural Mechanics Department, TU Gh. Asachi, Iai, 700050, Romania2Structural Mechanics Department, TU Gh. Asachi, Iai, 700050, Romania

Summary

This study, presents the assessment of the systems dynamic characteristics using

different identification techniques.

KEYWORDS: the Fourier transformation of the answer, transfer function, amplitude,

phase, frequency, mass matrix, damping matrix, rigidity matrix,

1. INTRODUCTION

The dynamic identification of structures can be considered a set of techniques

allowing the determination of the physical parameters occurring in equations, which

describe the behaviour of structure, subject of dynamic actions.

These techniques were developed due to the difficulties in the exact assessement of

rigidity, dampings and masses of real structures.

In most of the identification methods it is acted from the outside upon the system with

an iposed, known excitation, making easier the interpretation of the measurements. It

is necesary that during the experimentation, the influence of other disturbing sourcesshould be reduced to the minimum and the equipment used in the excitation of the

structure, as well as the one used in the measurement of the answer should not

considerably change its parameters.

2. TYPES OF EXCITATIONS USED IN THE STRUCTURES

IDENTIFICATION

The dynamic characteristics of a system where are considered a single input and a

single output are described in the time domain by the weight function h(t), and in the

frequencies domain by the answer function in frequency (FRF, transfer function) and

H(j) with which a Fourier transformation couple is formed:

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Assessement of the System Dynamic Characteristics Using Identification Technique


)U(

)(=)H(

(1)

Ussing the following relations, the FRF will have a value:

U ( ) = u(t) e dt--j t

( )= y(t)e dt- -j t (2)

where, Y() is Fourier transformation of the answer, and U() is Fouriertransformation of the excitation.

2.1 The Harmonic Excitation

In this case the structure is excited by means of an electrodynamic generator,

execising upon the structure a sinusoidal punctual force whose amplitude, phase and

frequency are ruled.

The identification of the dynamic characteristics is performed in the frequency

domain.

In the linear vibration they correspond to the peaks oscillation from the curve of the

transfer function.

The modal damping ratios are determined through the classic method of the

semipower. The modal forms are determinated by the ordinate function of these peaks

and their sign is obtained by the phase function of the transfer function.

The imput and the output are:

eu=u(t)tj

0

y( t ) =y e0j( t+ )

(3)

and the transfer function is:

H( )= y

ue

0

0

j

(4)

The modulus of the transfer function is obtined from the amplitude-pulsation

characteristic (y0/u0-) and the argument of this () function from the phase-pulsationcharacteristic (). These two characteristics can be traced either point by point,

performing measurements on discrete frequencies in fixed regime, or continuously in

quasistationary regime, using a frequent scavenging slow enough to allow the

establishment of the answer on every frequency especially in the area of weak

damping resonance.

SSN 1582-3024Article no.20, Intersections/Intersecii, Vol.2, 2005, No.1, Structural Mechanics 5

Now, the identification methods with harmonic excitation are the most used, but in

case of model coupling or of structure nonlinear behavior, the result can be oftenwrong.


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D. tefan, V. E. Chitan


2.2 The Excitaton With Transitory Signals Or Impulses

The simplest shape is represented by rectangular, triangular, trapezoidal, sinusoidal

form, a.s.o.

The amplitudes spectrum of these impulsee are annulled on certain frequencies and

theoretically have an infinite domain of frequencies (fig. 1).

Fig.1 The amplitudes spectrum of the impulsee

In case of the system with nGLD there exists the posibility of certain resonaces failure

to be excited, as well as the possibility of excitation of resonances outside the interest

domain.

Using a cosntant amplitude sinusoidal excitation and a time variable frequency, these

shortcomings can be eliminated.

In case of studies on reduced scale models the excitation through striking with a

special hammer is enough in order to obtain u(t) and y(t) signals which, undertaken by

Fourier analyzer in real time give directly the answer function in frequency.

2.3

The Excitaton With Accidental SignsThe excitation force can be the wind or the earthquake. The behavior of the structure

is considered linear. In this case all the spectral components are simultaneously

excited and analyses in real time can be made.

Ussing the following relation, results the FRF:

H( ) =S ( )

S (

uy

uu

) (5)

where, Suu() is the power spectral density of the excitationSuy() is the power interspectral density of the excitation and the answer.



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If Suu() = S0= const. ("the white noise"), than, Suy() = S0H(), and theintercorrelation function between excitation and answer Ruy(

t) can be determinate with

the following relation:

uy - uyjR ( ) = S ( )e d

(6)

The trials in accidental regime have the advantage of simultaneously excite all the

spectral components of the interest domain, allowing an identification of the structures

with time variable parameters. Using an exponential mediation in the calculus of the

specters Suuand Suyan analyses in real time can be made. The procedure is useful in

the optimization of a structure answer through the modification of the masses, rigidity,

and damping distribution.

3. IDENTIFICATION PROCEDURES OF THE PROPER VIBRATION

FREQUENCIES AND FORMS

The various identification procederes propose to minimize a comparision criteriastarting from the different between the answering of the model and the one measured

in the real system. The experimenter is confronted with the problem of determine the

characteristics of the system represented by the masses (M), damping (C) and rigidity

(K) matrixes, which are not directly measurable out of such quantities as frequency,

modal forms.

The tests of the complex great structure is performed in two stages:

A it is roughly determited the number of the proper vibration mode and the

resonance frequency using a single vibrator;

B the modes are isolated through a distribution corresponding to a vibrators number

along the structure, matching the excitation forces from the vibrators, so that only themode subject of interest should be dominatly excited.

The most simple technique used for the determination of the resonance frequencies is

the method of the amplitude peak. The structure is acted by sinusoidal force from a

sole vibrator and the answer curves are registered under the form of the answer

function in frequency (FRF).

As the answer of the structure is due to the answer of all proper modes

simultaneuosly, the measured proper forms are seldom distorsion. This problem is

amplified in case of structures with close natural frequencies when the separation

of the modules becomes difficult.


As the precision of the modal parameters dipend by the precision which the FRF

can be measured, we must give a special importance to the excitation module of the

structure, to mentain the vibration amplitudes at the same level in every cases.


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This thing becomes easy to achieved in the frame of simple structures. In case of great

structures with numerous links and un-proportioned damping the nonlinear effects and

the proximity of the modes are frequent and not only that hinders the localization of

frequencies where the modules has to be identified but also the forms established

for a certain positions of the vibrator might not coincide with those established on

other positions of the vibrator.

In order to eliminate these difficulties the structure can be excited in several point

simultaniously.

3.1. The Identification Of The Frequencies And The Modal Forms Using The

Sole Excitation

A linear and elastic structure excited by a sinusoidal force will answer directly

proportional with the excitation force, having also the same frequency.

The measurement of the excitation force and the answer in a number of points from

the whole domain of frequencies is enough to describe the behavior of the

structure. The information can be represented through drawing the relation betweenanswer and excitation as a frequency function, the answer being in displacement,

velocity or accelerations.

It is considered a system where the action is after the GLDk direction:

k Kj t

F =F e

(7)

the answer being measured in the joint i, having the expression:

l l

j ty =y e

(8)

The complex admitance, which represents the ratio between the answer in

displacement and the action (force) can be also written:

Hj+H=eH=F

y=H

,,,jlk

k

l

lk (9)

where, H is the real part of the admittance

H" is the imaginary part of the admittance.

In the case of the system with proportional damping, the complex admittance is

graphically represented usually by one of the forms:

as amplitude pulsation diagram and phase pulastion diagram;

as vibration diagram with pulsation of the answer component in phase with the

force and the component in quadrature with the force;


as polar diagram representing the geometrical place of the extremity in complexplane, having the pulsation as parameters.


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The various analyses methods of the answer in frequency are different first of all,

through the hypothesis made upon the contributions of the nonresonant

vibrabration modes at the total answer in the proximity of resonance frequency and

separation procedure of the vibration modes when they close the pulsations.

No matter the adoptive method, the aim is the same, that is to determine thefollowing measures:

the resonance pulsation;

the proper forms {Yki} and modal matrix [Y];

the modal parametersi, mi, ki.so, the matrixes [M], [C], [K] can be know.

3.2 Identification Of The Linear Systems, Harmonically In Several

Joints

A complex of the structure vibrates simultaneously in several modes. For a correct

analyses of the results the undesired modes must be eliminated. This can be

achievd through the excitation of the structure using several vibrators forcing thestructure to vibrate in its main modes. When the structure is proportionally damped

it can be excited in any frequencies through a paricular set of forces which are in

phase or antiphase, one with the other, so that the answer measured in every point

should be all in phase or antiphase and the characteristic phase delay between the

force and the answer is unique. In the frequency there exists a number of

characteristic phase delays associeted to the distribution of independent linear

forces equal to the GLD number. The structure excited in this way for a particular

ratio of the forces shall vibrate in the main mode as a system with 1 GLD.

If the structure is nonproportionally damped, it can be excited in its main mode

only at the natural frequency corresponding through a set of monophasic forces.

We are actually interested in the conditions in which the exciting of propervibration can be achieved.

It considered the system with nGLD with a viscous damping, whose movement

equation is:

[ ]{&& ( )} [ ]{ & ( )} [ ]{ ( )} { ( )}y t C y t K y t F tk k k k + + = (10)

or in generalized coordonate:

[ ] {&& ( )} [ ]{& ( )} [ ]{ ( )} { ( )}q t C q t K q t F t gi i i gi i gi i gi+ + = (11)

We have to determine the pulsation harmonic excitation , that makes the system tovibrate in the main mode at the pulsation i, that is generating an answer:


{ y( t )} =A {Y}e0 i j( t- ) (12)


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where, is trail behind of the answer confronted by the excitation;A amplitude factor.ois the

If Ao=1 we can know the excitation force:

{ F } = { F }ej( t+ )

(13)

which will generate an answer:

{ y( t )} = { Y}eij t

(14)

Knowing the folowing relation:

{ y( t )} = { Y}qi=1

N

i i

m

(15)

the relations (14) i (15) must be equal:

{ y( t )} = { Y}q = {Y}ei=1

N

i i ij t

m

(16)

So, for isolate of the mode i, it is necessary that in the equation (11) to replace:

{q ( t ) } = { I } eij t (17)

i

where, {I}iis the "i" column in the matrix [I]:

{F ( t ) } = { F }egij( t+ )

(18)

Replacing (17) and (18) in (11), simplifying with ej0t

and spareting the real part from

the imaginary one, we get:

( [K ] - [M ] ){ I} = { F }gi 2 gi i cos (19)

wher, {C}iis the "i" column in the matrix [Cgi].

It comes out that in order to excite a structure in one of the main vibration modes

the force distribution must have the following form:

{ C} = { F }i

sin (20)

The relations (19) and (20) can be written in another way:

( - )[ Y ] [ M ]{ Y} = [ Y] {F}i2 2 T

iT ,

(21)

[ Y] [C]{ Y

} = [ Y }] {F

}T

iT ,, (22)


where, are evidencing the two vectorial components of the excitation:


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{ F } = { F+ jF}e, ,, j t (23)

Simplifying with [Y]T, we get:

{ } ( )[ ]{ } { } [ ]{ } = =F M Y Fi i C Yi 2 2

(24)

It results that for excitate a structure in one of the main vibration mode, thedistribution of the forces must be:

{ } ( )[ ]{ } [ ]{ }F M Yi i= C Yi + 2 2

(25)

The component {F'}in phase with the displacement is necessary for the balancing of

the harmonization elastic and inertial forces and the component {F"} is in quadrature

before the displacement and it is necessary for the balancing the damping forces.

Between the two components, there is a relation:

{F} =-

[ C ] [ M ] {F},,

i2 2

-1 ,

(26)

For the excitation of the pure modes of the system distribution of forces in phase, isused:

{ F } = { F }ejt (27)

where, {F} is a vector with real elements.

4. CONCLUSIONS

In case of great structures with numerous links and non-proportional damping, the

effects of nonlinearity and the proximity of the modes are the frequencies, a fact

implying the positioning of the frequencies where the modes have to be identified.Out of this reason there were determined the identification techniques of the

frequencies and modal forms specific to the structure using the unique excitation or

excitations from several points.

The excitations for the identification of the system can be hamonic, transitory,

accidental type or excitations of the normal functioning.

According to the goal of the identification the results can be used for the

verification and validation of the analytical models, the determination of some

measures, which are not directly measurable the foreseeing of the structural

modification effects, determination through calculus of the answer at other

excitations or several simultaneous excitations based on the answer measured for a

certain type of action.



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Article no.20, Intersections/Intersecii, Vol.2, 2005, No.1, Structural Mechanics 12


References

1. Argoul P., - Identification des structures vibrantes.These de doctorat de lENPC, Paris, 1990

2. Genatios C., - Sur la technique de vibration harmonique pour levaluation dynamiqueexperimentale des structures, Annales des Ponts et chaussees, trim.2, 1993

3. tefan D. , - Elemente de dinamic i identificarea dinamic a structurilor de construcii, Ed.

Vesper, 20014. Zaverin K., Phil M., - Modal analysis of large structures multiple exciter systems (Bruel &

Kjaer), 1984


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Theoretical and Experimental Studies of

Steel Profiles

Mihai Budescu1, Ioan P. Ciongradi2, Octavian V. Roca31Structural Mechanics Department, TU Gh. Asachi, Iai, 700050, Romania2Structural Mechanics Department, TU Gh. Asachi, Iai, 700050, Romania3Structural Mechanics Department, TU Gh. Asachi, Iai, 700050, Romania

Abstract

The main purpose of the tests is the checking of the stiffness characteristics of steel

sheets at several load levels.

The tests were carried out in collaboration with INCERC, Iai Branch.The testing

stand was built inside the Laboratory of the Structural Mechanics Department

from the Faculty of Civil Engineering and Architecture, Technical University Gh.

Asachi of Iai.

The testing of the elements was carried out according to EC3, chap.9, Testing

Procedure because the elements are classified as cold-rolled thin-gauge profiles

as stated in Romanian Norm NP 012-92 (EC 3 parts 1-3).

The testing procedure consisted of several repeated loading-unloading cycles.

Finally, one specimen from each class was loaded until collapsed. The local

buckling of the edge ribs caused the collapse of the profiles (in reality this is

impossible because the steel sheets are coupled).

The ultimate deflections are limited according to several Norms between L/100 and

L/200. The loading-unloading cycles pointed out the lack of permanent strains for

maximum displacements below the L/200 limit. Out of this limit the permanentstrains appear i.e. the rib folding in the support areas.

KEYWORDS: Thin-walled steel profiles, Local buckling, Quasi-static testing



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M. Budescu, I.P. Ciongradi, O.V. Roca


1. INTRODUCTION

The main purposes of the research (as stipulated in contract) are the study of the

behavior under gravitational loads of the following types of NERGAL roofprofiles: (a) 0.5mm, (b) 0.75mm and (c) 1.00mm. The strip of sheets is made of

DX51DG steel according to the EN 10142 Euronorm and the EN 10027 parts 1 and

2. According to the EN 10142 this steel is denominated as 1.0226 and the ultimate

strength is Rm= 500 N/mm2.

It was intended to establish the element behavior when are subjected to

gravitational loads, according to EC3, chap.9, Testing Procedure. This was

because the elements are classified as cold-rolled thin-gauge profiles as stated in

Romanian Norm NP 012-92 (EC 3 parts 1-3). Under these circumstances the

elements behave different as the usual rolled profiles due to the fact that local

buckling can occur, correlated with the profile shape and the sheet thickness.

The main purpose of the tests is the checking of the stiffness characteristics of theNERGAL steel sheets at several load levels.

2. THE TESTING FACCILITIES

A special testing stand was designed in order to carry on the tests of the NERGAL steel

sheets. The two KB600-5 profiles of the stand are assembled with four bolts, the span

between the supports is 1500mm. The supporting elements of the displacement

inductive transducers are attached to the KB profiles. In the figure No. 1 it is presented

the testing stand built inside the Laboratory of the Structural Mechanics Department

from the Faculty of Civil Engineering and Architecture, Technical University Gh.

Asachi of Iai. The tests were carried out in collaboration with INCERC, Iai Branch.

The load transfer is performed according to the EC3 provisions, Chap.9, Testing

Procedure, by the means of an air mattress that assures the uniform load repartition

and the keeping unaltered the stiffness characteristics of the specimen. The direct

placement of the ballast on the steel sheet may alter the stiffness characteristics by

friction and vault effect.

The displacements were measured in three points, at the midspan and the quarter of

span in every space between the ribs. Inductive transducers were used to measure the

transverse deflections; their positions are presented in Fig No. 2.

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Theoretical and Experimental Studies of Steel Profiles

Article no. 21, Intersections/Intersecii, Vol.2, 2005, No.1, Structural Mechanics 15


Fig. 1 View of the testing stand


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Article no. 21, Intersections/Intersecii, Vol.2, 2005, No 1, Structural Mechanics 16


Fig.2. The testing stand with the inductive transducers

The displacements were recorded by the means of inductive transducers in the

format of analogical electric signal. The loading was performed by ballast with

successive layers with 50N gravel filled sacks.

In the case of the 0.75 and 1mm NERGAL steel sheets the air mattress couldnt be

used because the load capacity was lower then the loading level corresponding to

collapse. In this case it was performed the direct ballasting with sacks only on the sheet

ribs.

3. THE TESTING PROCEDURE

The testing procedure consisted of two steps:

(i) The ballasting was performed in loading-unloading cycles for checking the stiffness

characteristics;

(ii) The ballasting was performed up to a level corresponding to the specimens

collapse by local buckling.

In the Table No. 1 there are presented the theoretical values of the geometrical

characteristics and the tested sections, in order to be compared to the experimental

stiffness characteristics of the NERGAL profiles.

The length of the specimen is 1600mm and the span between the supports (bolted

connections were used) is 1500mm.


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Table 1 The geometric characteristics of the NERGAL profiles

Position ofPos.

No.

Thickness

(mm)Area

(cm2)

Weight

(kg/m2)XG(cm) YG(cm)

Moment of

(cm4)

0 1 2 3 4 5 6

1 0.40 4.327 3.397 -0.291 -0.146 0.126

2 0.45 4.868 3.821 -0.291 -0.148 0.142

3 0.50 5.409 4.246 -0.291 -0.151 0.158

4 0.60 6.491 5.095 -0.291 -0.156 0.191

5 0.75 8.114 6.369 -0.291 -0.163 0.240

6 1.00 10.818 8.492 -0.291 -0.176 0.324

7 1.25 13.523 10.616 -0.291 -0.188 0.411

centroid ineria

3. THE 0.5mm PROFILE TEST

The tests for all three specimens were carried out in increasing loading-unloading

cycles. The maximum loading level for every cycle was of 100, 150 and 200

daN/mp.

The force-displacement relationship for all the three specimens is sinuous, the steel

sheet acting relatively unstable. For example, in the Figure No. 3 there is presented

the average displacement of the T1, T4, T7 and T10 transducers, mounted at themidspan of the E2-0.50 specimen.

The displacements measured at midspan are greater in average by 30% up to 40%

than the computed values. These increases are explained by the sheet deformation

in the support areas. Because of that, the stiffness characteristic is determined by

taking into account the relative displacement at the middle and quarter of span.

SSN 1582-3024 Article no. 21, Intersections/Intersecii, Vol.2, 2005, No.1, Structural Mechanics 17

By analyzing the results one notice the fact that stiffness differs as a function of the

loading level, i.e. it decreases as the load increases. Thus, in the case of 100

daN/mp loading step the stiffness reduction is only 10.2% while in case of 200

daN/mp loading step the reduction reaches 22.75%. The explanation of this

phenomenon is given by the sheet folding when subjected to load. Folding

diminishes the rib height, thus the stiffness characteristic is significantly decreased(the moment of inertia).


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Deplasarea medie la L/2

0

50

100

150

200

250

0 2 4 6 8 10

v [mm]

q[daN/mp]

Fig. 3 The measured displacement at the midspan of the E2-05 specimen

The E3-050 specimen was loaded up to the occurrence of the local buckling

phenomenon. The collapse occurred suddenly at a loading level of 270 daN/mp.

The buckling was local and occurred simultaneously at the midspan of the two

edge ribs. In the Figure No. 4 it is presented the collapse of the specimen.


Fig. 4 The collapse of the E3-050 specimen (buckling)


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4. THE 0.75mm PROFILE TEST

In the same way that in case of the NERGAL 0.5mm profile the tests for all three

0.75mm specimens were carried out in increasing loading-unloading cycles. The

maximum loading level for every cycle was: 100, 150, 200, 250, 300, 350 and 400

daN/mp.

For all kinds of specimens the loading-unloading cycles were carried out in order

to obtain the residual strains. The results prove that no residual strains occur at low

levels of loads, the residual effects being nothing else but re-arrangements in the

support areas (these elements are very sensitive). At high levels of loading the

residual effects may be caused by the change of the profile cross-section.

In the Figure No. 5 there is presented the load-average displacement relationship

(transducers T1, T4, T7 and T10, mounted at the midspan) during a loading-

unloading cycle up to 100 daN/mp. One can notice a linear shape of this variation,

straighter that in case of 0.5mm profiles. Some non-linearities are caused by the

different stiffness of the edge ribs.Deplasarea medie la L/2

0

20

40

60

80

100

120

0 0.5 1 1.5 2 2.5

v [mm]

q[daN/mp]

Fig. 5 The behavior of E1-075 specimen during two consecutive cycles

After reloading of the specimen for next cycle one notice a path almost identical to

the last curve, thus meaning the residual deflections were consumed after the first

loading cycle.


By analyzing the stiffness characteristics on the basis of the recorded deflections at

midspans there are noticed differences up to 20-30% when compared to thetheoretical values. Under these circumstances it is noticed a better behavior of the


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0.75mm NERGAL sheet than the 0.5mm profile. Even though, the stiffness

characteristics were also obtained from relative deflections, to avoid distortions.

In the same way like 0.5mm NERGAL sheet, the 0.75mm profile provides a

stiffness depending on the loading step. In the Figure No. 6 it is presented this

correspondence after processing the results from the three specimens.

91.664

90.552

89.100

87.970

87.110

86

87

88

89

90

91

92

0 100 200 300 400

Nivelul incarcarii [daN/mp]

Ir/Ic[%]

Fig. 6 The variation of stiffness characteristic vs. load level (0.75mm steel sheet)

Thus, at the 100daN/mp loading step, the stiffness decay is only of 8.336%, at the

200daN/mp loading step it attains 10.900% and when the 300daN/mp step is

applied, the reduction is of 12.890%.

The folding effect that leads to the reduction of the moment of inertia is less

significant that in case of 0.50mm steel sheet. Moreover, during a significantincrease of the load it is not observed an important stiffness decrease, as it was

expected, thus the shape of the graph from Fig. No. 6 is approximately linear.

In the end the E3-075 specimen was ballasted in order to obtain the ultimate load.

Thus it was attained a 500daN/mp load, when the first signals of damage occurred,

i.e. noises that forecast the stability loss. In order to avoid the damage of the

equipment, the experiment was interrupted because the specimen loading was very

large.


The maximum average displacement recorded at this loading level was of

11.391mm. The load applied directly changes dramatically the specimen behavior,

i.e. the loading-unloading relationship. Thus, in the Figure No. 7 it is presented the

situation of the last two loading steps of the E3-075 specimen, first with airmattress, and the second without. In the first loading step without air bed one


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notice a stiffness increase, after that the slope becomes similar to the situation

when the load is transmitted though the airbed.


0

100

200

300

400

500

600

0 5 10 15

v [mm]

q[daN/mp]

fara perna de aer

cu perna de aer

Fig. 7 The behavior of the E3-075 specimen (with / without airbed)

5. THE 1.0 mm PROFILE TEST

The tests of the 1.0mm NERGAL profile were carried out in the same way like the

previous two profiles, i.e. the loading and measurements. The behavior of these

specimens looks more stable than those of the 0.5 and 0.75.

The shape of the F- relationship is almost linear, the sinuosity is due to the

averaging, the measured stiffness characteristic is computed from the relativedeflections at L/2 and L/4 and for the first two cycles it represents 95.186% from

the theoretical value.

In the case of the 1mm NERGAL profile the folding effect that leads to the reduction

of the moment of inertia is less significant than in the other cases, see Fig. No. 8.

Finally, the E3-1 specimen was ballasted in order to find out the ultimate load. The first

signs of collapse were similar to those of the 0.75mm sheet, i.e. specific noises. The

maximum loading level was of 700daN/mp, which corresponds to a maximum mean

deflection at the midspan of 11.049mm.

In the case of this test it was noticed the lack of the stiffness difference caused by the

direct placement of the load, thus meaning that for bigger thickness the stiffness

increase due to the loading fashion (independent poliplan sacks) is insignificant.



23/96




The Fig. No. 9 presents the force-deflection relationship in case of the E3-1 specimen

at the last test with the airbed and with direct placement of load over the sheet.

96.485

95.722

95.232

94.622

94.22294

94.5

95

95.5

96

96.5

97

0 100 200 300 400

Nivelul incarcarii [daN/mp]

Ir/Ic[%]

Fig. 8 The stiffness characteristic-load level relationship (1mm thickness sheet)


0

100

200

300

400

500

600

700

800

0 5 10 15

v [mm]

q[d

aN/mp

cu perna de aer

fara perna de aer

Fig. 9 The behavior of the E3-1 specimen (with / without airbed)



24/96




6. CONCLUSIONS

This paperwork deals with the results of the experimental analysis of the 0.50, 0.75

and 1.00mm NERGAL profiles when subjected to gravitational loads. The purpose

of the tests was the checking of stiffness characteristics for the NERGAL sheets at

several loading steps.

The NERGAL profile is made of DX51DG steel sheet according to the EN 10142

and EN 10027 parts 1 and 2 (Euronorms), the steel is denominated as 1.0226

according to EN10142; the ultimate strength is Rm= 500 N/mm2.

The testing of the elements was carried out according to EC3, chap.9, Testing

Procedure because the elements are classified as cold-rolled thin-gauge profiles as

stated in Romanian Norm NP 012-92 (EC 3 parts 1-3). The tests were performed

on a special stand. The loading was performed by ballasting with 50N sacks,

distributed over an air mattress that provides the uniform load distribution and

doesnt affect the stiffness characteristics of the specimen.

The deflections were measured at every three points at midspan and quarter span,on every space between the ribs, thus using 12 measurement points.

The testing procedure consisted of several repeated loading-unloading cycles.

Finally, one specimen from each class was loaded until collapsed.

The loading-unloading cycles pointed out the lack of permanent strains for

maximum displacements below the L/200 limit. Out of this limit the permanent

strains appear i.e. the rib folding in the support areas.

The local buckling of the edge ribs caused the collapse of the NERGAL profiles (in

reality this is impossible because the steel sheets are coupled). As a consequence,

the assembly technique of the steel sheet edges becomes very important.

The stiffness reduction of the NERGAL tested profiles in case of a limited displacement(L/200):

For the 0.50mm profiles it reaches about 20%;

For the 0.75mm profiles it reaches about 10%;

For the 1.00mm profiles it reaches about 5%.

One may notice that the ultimate deflections are limited according to several

Norms between L/100 and L/200.



25/96

SSN 1582-3024



Article no. 21, Intersections/Intersecii, Vol.2, 2005, No 1, Structural Mechanics 24


References

1. Budescu, M., and all. Incercari experimentale Panouri de tabla tip NERGAL, Contract:K2603/2003.

2. Wei-Wen Yu Cold-formed steel design, 3rd Ed., Wiley & Sons, Inc., NY, 2000.3. American Iron and Steel Institute, Cold-formed steel design manual, 50th Commemorative

Issue, 1996.4. American Iron and Steel Institute, Load and Resistance Factor Design Specification forCold-

formed steel Structural Members, 1991.5. European Committee for Standardisation,Eurocode 3: Design of Steel Structures, ENV 1993.6. STAS 10108/2-83,Romanian Specification for Calculation of Thin-Walled Cold-Formed Steel

Members.7. CEN/TC250 EUROCODE 1:Basis of design and actions on structures, ENV1991-1.8. NP 012-1997,Normativ pentru calculul elementelor din oel cu perei subiri formate la rece.9. STATS 10101-0A/77,Aciuni n construcii.10. STATS 10108/0-78, Calculul elementelor din oel.


26/96



Effect of thickness variation upon plates subjected to bending

Mihai Vrabie1, Cornel Marincu2, Nicolae Ungureanu1and Mihaela

Ibnescu1

1Faculty of Civil Engineering, Technical University Gh. Asaki, Iasy, 700050, Romania2S.C. IPTANA S.A., Bucharest, Romania

Summary

The variation of the plate thickness is an element that introduces a major difficulty

in solving the differential equations of the deformed middle surface, because the

plate rigidities are also variable.

In the paper, the effect of the thickness variation upon shear forces is presented.

These shear forces must be corrected by adding the shear stresses caused by

bending and twisting moments.

A physical interpretation is given to these corrections and the separation of effects

is also pointed out. There are also discussed the additional energy effects caused

by thickness variation.

KEYWORDS: Bending plates, variable thickness, effect of transverse sheardeformation, strain energy.

SSN 1582-3024 Article No.22, Intersections/Intersecii, Vol.2, 2005, No.1, Structural Mechanics 25


27/96


28/96

( )yx

wDMMM

x

w

y

wDM

y

w

x

wDM

tyxxy

y

x

===

+

=

+

=

2

2

2

2

2

2

2

2

2

1

(1)

SSN 1582-3024



Article No.22, Intersections/Intersecii, Vol.2, 2005, No. 1, Structural Mechanics 27


where the bending rigidity of the plate is variable

( )( )2

3

112

,

=

yxEhD (2)

x

wx

=)(

p(x,y)

h(x,y)

w(x,y)

z

Fig. 1. A deformed segment of the bending plate


29/96

( )

)

)

),

cQy

M

x

M

bQy

M

xM

ayxpy

Q

x

Q

y

yxy

xyxx

yx

=

+

=

+

=

+

(3)

SSN 1582-3024


M. Vrabie, C. Marincu, N. Ungureanu, M. Ibnescu

Article No.22, Intersections/Intersecii, Vol.2, 2005, No.1, Structural Mechanics 28


The differential equilibrium equations remain under the well-known form:

The equation (3a) is differentiate with respect to x and the equation (3c) isdifferentiate with respect to y. By adding the two obtained equations and takinginto account (3a), it results:

( )yxpy

M

yx

M

x

M ytx ,22

22

2

2

=

+

+

(4)

The moments Mx, My, Mtare substituted by their expressions given by (1) [3]:

( )

( )yxpx

w

y

wD

y

yx

wD

yxy

w

x

wD

x

,

12

2

2

2

2

2

2

22

2

2

2

2

2

2

=

+

+

(5)

or

( )yxpx

wD

yyx

wD

yxy

wD

x

y

w

Dyyx

w

Dyxx

w

Dx

,2

2

2

2

2

222

2

2

2

2

2

2

2

222

2

2

2

2

=

+

+

+

+

+

(6)

The differential equation of the deformed middle surface for plates with variablethickness was obtained.

In equation (6) the differentiation operations are performed and the followingnotations are considered:

yx

DD

y

DD

x

DD

y

DD

x

DD

=

=

=

=

=

2

2

2

2

2

;;;; (7)

The deformed middle surface equation can be written as [2]:


30/96

( )

( ) ( )yxp

x

wD

yx

wD

y

wD

wDDwy

Dx

DwD

,21

2

2

22

2

2

=

+

+

++

+

+

(8)

SSN 1582-3024





When the thickness variation is only along one direction, as an exampley, equation(8) becomes:

( ) ( )yxpx

wDwDw

yDwD ,12

2

2

=

+

+ (9)

The few analytical solutions of equation (8) and (9), respectively have beenmentioned in introduction and they are referring to particular support and loadingcases.

The most frequent solutions are the approximate ones, based on numerical methods

(finite difference method, finite element method, boundary element method etc.)

3.THE EFFECT OF THICKNESS VARIATION ON SHEAR FORCES

The shear forces are expressed in terms of twisting and bending momentsderivatives, according to relations (3a) and (3b):

x

M

y

MQ

y

M

x

MQ t

y

ytx

x

+

=

+

= ; (10)

These relations include the thickness variation, so that, substituting the moments

Mx, My, Mtby relations (1), they must take into account the rigidity Dvariation. Inconsequence, the two shear forces will be:

( )

( )yx

wD

xx

w

y

wD

yQ

yx

wD

yy

w

x

wD

xQ

y

x

+

=

+

=

2

2

2

2

2

2

2

2

2

2

1

1

(11)

The shear force vector {QxQy}Tcan be expressed under the following shape:


31/96

+

=

t

y

x

y

x

M

M

M

x

D

y

D

y

D

x

D

Dw

w

y

xDQ

Q

0

01

0

0

2

2

(12)

SSN 1582-3024





and respectively:

+

=

t

y

x

y

x

M

M

M

x

h

y

h

y

h

x

h

hw

w

y

xDQ

Q

0

03

0

0

2

2

(13)

The second term represents the influence of thickness variation on the shear force.

In the particular case, when the derivatives xh / and are constant, the

shear stresses and have a linear variation over the plate thickness. In case

of a more refined analysis, the energy generated by the shear stresses oncorresponding shear strains can be introduced in the elastic strain energy stored bythe element.

yh /M

xzM

yz

4. PHYSICAL INTERPRETATION OF SHEAR FORCE CORRECTIONS

DUE TO THICKNESS VARIATION

A portion of a plate with continuous variable thickness and two sections parallel tothe coordinate planes x0z and y0z are considered. The plate thickness has thedirection of 0zaxis in the local coordinate system and intersects the intrados andextrados surfaces at point a and b, respectively. The bending moments, which

produce the tension of intrados fibers and compress the extrados ones, areconsidered as positive (positive twisting moments are correspondingly introducedaccording to stresses sign convention).

In the considered zone it is presumed that no surface forces exist, so that, thestresses at the extreme points will be directed along the tangents to the surface lines(Figure 2) [4].

The shear forces have two components, Qand Q, according to relation (13):

+

+=

yy

xx

y

x

QQ

QQ

Q

Q (14)

The shear forces xQ and yQ are given by the stresses xz , and yz parabolic

distributed over the thickness, and having the maximum values at the middle


32/96


33/96

)( txz M

)( xxz M

)( xxz Q

)( xxzb Mxb

xa)( xxza M

)( txzb M

)( txza M

xyb

xya

xz

SSN 1582-3024





bQ Mxxy x

a

a)

bMt

y Q xxa

b)

b

Qxx

a

c)

Figure 2. The shear stresses generated by xtx QMM ,,


34/96




5. ENERGY EFFECTS

The effect of thickness variation can be energetically pointed out, by including it inthe total strain energy of the shear forces Qxand Qy. For this reason, the vector ofdeformations in the local coordinate system, attached to a point of the middle

surface, is written as:

{ } { }TyzxzxyyxQ

M

n

t},{};,,{

=

=

= (20)

where { } { } { }TxyyxMt ,,==

{ } { }TyzxzQ ,=

are the deformations in the plane tangent to the

middle surface (deformations produced by Mx, My, Mt), and

are deformations caused by transverse internal forces, as

the shear forces Q

{ }n =

xand Qy(correcteddue to thickness variation).

Similarly the stress vector is divided, so that the constitutive equations in the local

coordinate system are:

{ } { } [ ]{ } [ ]

[ ]

===

=

Q

M

Q

MT

yzxzxyyx

Q

M

D

DD

0

0}{},{ (21)

where [D]is the material constitutive matrix.

For isotropic materials [DM]and [DQ]are:

[ ] [ ]

=

+

+

=Gk

GkD

G

G

G

DQ

Q

QM 0

0;

00

02

02

(22)

where21

=

E (23)

is the reduced Lames constant for the plane stress state;

E, G, the material elastic constants;

kQ correction coefficient due to the non-uniform distribution of transverse shearstresses inx0zandy0zplanes (for homogeneous materials kQ= 5/6).

In the local coordinate system, the total strain energy of the plate can be expressedas:

SSN 1582-3024 Article No.22, Intersections/Intersecii, Vol.2, 2005, No. 1, Structural Mechanics 33


35/96

{ } [ ]{ } { }[ ]{ } eQV

Q

T

QM

V

M

T

M UdVDdVD ++= 21

2

1 (24)

SSN 1582-3024





where U is the position energy of the applied loads.e

6. CONCLUSIONS

Sometimes the plate thickness can be continuous variable over the plate surface orover some of its areas. In a particular case, the thickness can be variable in steps, ordue to some stiffening elements. There are cases when equivalent systems (asexample, due to geometric orthotropy) become with variable thickness.

It is mentioned that the variable thickness does not modify the plate model, in thecase of the present paper, a thin plate. However, the thickness variation for platessubjected to bending produces some effects expressed in the internal forcesrelations and also in the energetic and variational approaches, frequently used in

such analyses.

Among these effects we can mention:

- the plate rigidities are variable because of thickness variation;- the shear forces, as resultants of shear stresses, are influenced by the

bending and twisting moments, when the thickness is continuously variable;- the thickness variation introduces corrections and additional effects,

respectively upon deformations and transverse stresses; in this paper, a geometricinterpretation has been given to this effect, also pointing out the effects separation;

- in the energy balance the effect of transverse deformation and the effect ofredundant stresses produced by the bending and twisting moments must beintroduced; the case studies show that for plates with variable thickness, havingslopes lying between 5%and 10%, the additional energetic effects and transversedeformations effects increase with 100% [4].

References

1. Timoshenko, St.P., Woinowski-Krieger, S. Teoria plcilor plane i curbe, Ed. Tehnic,Bucureti, 1968 (in Romanian), pp. 194-200.

2. Soare, M.V.Plci plane, Seciunea VI din Manual pentru calculul construciilor, Ed. Tehnic,Bucureti, 1977 (in Romanian), pp.1000-1001.

3. Ungureanu, N. Rezistena materialelor i teoria elasticitii, I.P. Iai, 1988 (in Romanian), pp.270-284.

4. Marincu, C. Modele structurale i metode de analiz static i dinamic a plcilor curbe de

grosime variabil, cu aplicaii la structuri hidrotehnice, tez de doctorat, U.T. Gh. Asachi Iai,

1998 (in Romanian).5. Hinton, E., Owen, D.R.J. Finite Element Software for Plates and Shells, Pineridge Press,

Swansea, U.K., 1984.


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SSN 1582-3024





6. Zhao, Z., Chen, W. New Finite Element Model for Analysis of Kirchhoff Plate, Int. Journ. For

Numerical Methods in Engng., 38, 1995, pp. 1201-1214.7. Karam, V.S., Telles, J.C.F. On Boundary Elements for Reissners Plate Theory, Engng.

Analysis, 5, 1988, pp. 21-27.


37/96



Homogeneity on Designing the Structural Systems

Florentina Luca1and Septimiu-George Luca

2

1National Institute for Building Research, Iai Branch, 700048, Romania2

Structural Mechanics Department, TU Gh. Asachi, Iai, 700050, Romania

Summary

The paperwork presents the analysis over the effects of the variation of constitutive

materials characteristics in the elastic designing of the structural systems. It has

been analyzed the transformation of the actions into sectional efforts using the

given techniques by the mechanic materials and static of structures. For the

structural analysis it has been used the finite elements technique and also the

stiffness matrix method. In this case it is found out that the stiffness matrix of an

elastic structure depends essentially on E, on the shape and geometrical

dimensions of finite elements. The case study from the last part of this paperwork

illustrates the random aspect of elasticity modulus.

KEYWORDS: elasticity modulus, homogeneity, structural system, stiffness matrix,variability, finite elements

1. INTRODUCTION

The structural designing is made in the elastic domain, the compulsory stage for allthe structural systems, and also in plastic or inelastic domain when the conditionsare imposing it. On designing the constructions are determined the displacements,

the efforts, the stress and deformation states and it is made an assessment of thesafety degree and when it is imposed it is made an assessment of the risk, such asthe designing at the action of earthquake. The calculation of the structures in theelastic domain covers a wide area of the designing and from the characteristics ofthe constitutive materials variation point of view will represent the object of certaininvestigations, essentially of the manner in which the homogeneity propertyintervenes.

On designing intervene:

the structure by its geometrical elements and constitutive material with itsproperties;


the actions that can be expressed through their idealization, as distribution andapplication manner on structure.


38/96




The designing in the elastic has certain particularities that allowed it a largeutilization. Designing the structures assumes first the transformation of the actionsinto sectional efforts, for example structures of beam, plates (structural walls) andcomplex structures case. The structural designing has more levels, most frequent

being the designing on the elastic limit and the designing on the ultimate limitstrength state.

SR (1)

where:

R represents the minimum capacity of strength, which is the elastic limit strengthor the ultimate strength in other cases;

S represents the maximum sectional effect of the loads. It is mentioned thepossibility of the existing of more loads hypothesis.

According to developments in this domain, the R and S values are given bystatistical and probabilistic distributions that have to be known. A probabilistic

designing, although possible in principle, in practice would become very complexand it is not used in it is most general form. In practice it is working with certainformulations that take into account the fact that a large number among the valuesthat intervene on designing have random character.

One first aspect which must be determined in relation (1) is that S represents thesectional effect of loads; thus it assumes the transformation of considered loadsinto efforts. In this transformation some characteristics of material from structurethat are random intervene. Besides, the sectional effect of S loads is affected by thevariability of some characteristics of structure, which are not constant values,although it does not present an ideal homogeneity.

In order to achieve the transformation of the loads into efforts are being used the

mechanical materials and structures theory lows that are determinist and precise, ifthe values which intervene are as well determinist. In reality statistical and

probabilistic uncertainties intervene.

2. ANALYSIS OF TRANSFORMATION OF THE ACTIONS INTO

EFFORTS

SSN 1582-3024Article No.23, Intersections/Intersecii, Vol.2, 2005, No.1, Structural Mechanics 37

In addition to it, we analyze the case of elastic designing namely the transformationof the actions into efforts using the given techniques by the mechanic materials andstatic of structures, but taking into account the variability of elastic characteristics

and the variability of the geometrical values: areas, thickness, moments of inertiaand lengths.


39/96


F. Luca, S.G. Luca


For a structural analysis is used the technique of finite elements and the stiffnessmatrix method. Using the stiffness matrix method, at static actions, we can reach toa system of linear equations as follows:

PKD= (2)

where:

K the stiffness matrix of discretization structure in finite elements;

D the vector of unknown nodal displacements;

P the vector of equivalent nodal forces.

The stiffness matrix is obtained through the assembling of the elemental stiffnessmatrix, of all finite component elements.

The stiffness matrix of finite elements can be determined using the relation:

= VT

e CBdVBK (3)

The matrix B (x, y) is the geometrical type and depends on the finite element type,the nodal degree of freedom and the shape function of finite element, respectively[1], [2].

C the constitutive matrix or elasticity matrix which characterizes the elasticproperties of material.

The matrix C for the plane finite elements is:

for the plane stress state:

=

2

100

01

01

1 2

EC (4)

for the plane deformation state:

+

=

)1(2

2100

011

01

1

)21)(1(

)1(

EC (5)


In case of the spatial finite elements matrix C can be expressed:


40/96

+=

2100000

0210000

0021000

0001

0001

0001

)21)(1(

EC (6)

SSN 1582-3024





It is noticed that in the general cases the constitutive matrix depends on the elasticcharacteristics E and . In relation (3) the elasticity modulus E consideredconstantly can be written:

'ECC= (7)

For a structure formed from the same material, the elasticity modulus E becomes

the factor for the stiffness matrix of structure K and can be written'EK .

Because the elasticity modulus E has large values for different constructionmaterials it represents the ponderosity in matrix C. It can be affirmed that thestiffness matrix of an elastic structure depends essentially on E, on the shape andgeometrical dimensions of finite elements. It is found out that these values have arandom character.

Accordingly, at an accepted load hypothesis expressed through vector P, the vectorof nodal displacements D is influenced by the variability of the elasticity modulus,the coefficient of Poisson, in a low extent, and the variability of geometricdimensions.

In case of bars structures,the stiffness of a bar finite element Kecan be written:

=

ejjeji

eijeiie kk

kkk (8)

where i and j are ends of bar.

It is found out that the elements of matrixes contain in their terms the elasticitymodulus E and geometric elements of bar: the areas of the sections, the central

principal inertia moments, the polar inertia moments and the lengths of bars.

Only these characteristics also appear in the assembled stiffness structure matrix.The sectional stiffness of bars can be considered random values and by adequatetesting, the obtaining of their statistical distribution could be possible.

3. RANDOM ASPECT OF ELASTICITY MODULUS


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F. Luca, S.G. Luca


The elasticity modulus as a random value has a normal distribution; the followingreasoning are assuming this theory. Relation (2) can be used to determine the nodaldisplacements of meshed structure and this relation can be written for differentsituations.

Thus, if the structure is totally realized by the same material such as: steel,reinforced-concrete, wood, then the elasticity modulus used in the deterministiccalculation is the same, the matrix K and the equation (2) can be written:

KEK = (9)

PDKEKD == (10)

The elasticity modulus with a normal distribution is situated in the domain:

maxmin EEE (11)

where Emaxand Eminare the maximum values and the minimum values respectively.

For the extremities of the variation interval the equation (19) is:

PDKEDK == maxmax (12)

PDKEDK == minmin (13)

Solving these equations, appropriate displacements can be obtained. For themaximum stiffness are obtained minimum displacements and for the minimumstiffness are obtained maximum displacements:

PKDE 1max= (14)

PKDE1

min= (15)

Results:

PKE

D 1

max

min

1 = (16)

PKE

D 1

min

max

1 = (17)

On designing and execution of the constructions in which norms are observed, theelasticity modulus of structure is variable and the most proximate value is themedium elasticity modulus.


PKDEmedium1

= and PKED medium

11 = (18)


42/96




It is possible that the real displacements to have the medium value ofdisplacements Dminand Dmaxas well.

PKEE

DDD 1

maxmin

maxmin )11

(2

1

2

+=+

= (19)

PKEE

EED 1

minmax

minmax

2

+= (20)

In order to the symmetrical normal distribution the moduli Emax and Emin can bewritten:

mediummedium EEE +=max (21)

mediummedium EEE +=min (22)

and

)1(1

2 2minmaxminmax

=+

mediumEEEEE (23)

If the maximum amplitude is admitted to be 20% of the average value (this means

an acceptable concentration), results that these ratios10

1= , and

100

12 = can

be negligible in ratio with unit. This result demonstrates that the medium elasticitymodulus is the most appropriate value that should be used in the design. Theconclusion is that for designing has to be used the medium value of elasticitymodulus, namely:

PKPKE

Dmedium

111 == (24)

the matrix K being determined withEmedium value. If in designing and execution, thereal medium values of elasticity modulus and Ecalculation< Emedium real, are not takeninto consideration, this fact is equivalent with the amplification of actions, and the

Ecalculation> Emedium realis equivalent with the diminishing the actions, the situationsmore difficult to control can be unfavourable. Actually, some finite elements willhaveE < Emedium, and otherE > Emedium, the concentration is around the mean, butdue to the symmetry of normal statistical distribution, compensations are produced.

4. CASE STUDY

SSN 1582-3024Article No.23, Intersections/Intersecii, Vol.2, 2005, No.1, Structural Mechanics 41


43/96


F. Luca, S.G. Luca


It was considered a metallic structure having three opens and five levels (fig. 1)with the load from figure 2. The weight of the structure used in the seismiccalculation is G=275323, 38Kg.Geometrical characteristics:- for the beams were used profiles IIPE300 with moment of inertia I=16712cm4;

- for the columns were used profiles IIPE400 with moment of inertia I=46260cm4.

1 2 3 4

5 6 7 8

9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25 26 27

28 29 30

31 32 33

34 35 36

37 38 39

40 41 42

6 m 6 m 6 m

3m

3m

3m

3m

3m

3m

1 2 3 4

5

9

13

17

21

25

6

10

14

18

22

26

7

11

15

19

23

27

8

12

16

20

24

28

Fig. 1. Geometry of structure



44/96


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F. Luca, S.G. Luca


Table 3 Erandom

E (N/mm2) Columns and beams

2.1x105 2, 4, 7, 8, 9, 12, 14, 18, 21, 22, 28, 31, 33, 35, 37, 39

1.995x105 1, 3, 6, 25, 27, 29

2.205x105 10, 11, 19, 26, 34, 36

1.932x105 32, 38, 40, 422.268x105 17, 20, 23

1.89x105 5, 15, 16

2.31x105 13, 24, 30

Fig. 3. Deformation of structure

A calculation for four cases it was performed and were written displacements (table3).

Table 3 Displacements

D5(cm) D9 (cm) D13 (cm) D17 (cm) D21 (cm) D25 (cm)

Emedium 0.790 2.266 3.784 5.096 077 734

Emin 0.8573 2.4583 4.105 5.5278 5911 7.3044

Emax 0.7329 2.1018 3.5105 4.7278 5.6375 2465

Erandom 0.802 2.292 3.8094 5.1234 098 7639

From the performed numerical analyses result a very good approach between thecalculated displacements Emedium and Erandom. The difference obtained being under2%.


The conclusion is that in practice the conformitys verification of medium elasticitymodulus achieved with elasticity modulus considered in designing should be acompulsory requirement.


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5. CONCLUSIONS

The variations of elasticity modulus lead to the redistributions of effortsproportional with rapports between elemental and calculation moduli, such as theelements more rigid are loaded more and the elements less rigid are loaded lessaccording to situation in which the elasticity modulus is considered the same.

It is demonstrated that the results of calculation are the most proximate to reality ifa medium modulus is used, the using of lower elasticity modulus leading to theequivalent effect with the amplification of actions and the using of a superiormodulus leads to an equivalent effect with diminishing of loads.

Accordingly, the conformitys verification of elasticity moduli through designingand execution is necessary.

References

1. 1. Cuteanu E., Marinov R., Metoda elementelor finite n proiectarea structurilor, Editura

Facla, Timioara, 1980 (in Romanian)2. 2. Jerca t., Ungureanu N., Diaconu D., Metode numerice n proiectarea construciilor,

Universitatea Tehnic Iai, 1997

3. 3. Ungureanu N., Chira Florentina, Studies about the Modulus of Elasticity in Design andElastic Safety of Structures, National Symposium, with international participation: "Innovative

Solutions for Complying to Essential Requirements in Civil Engineering", Editura SocietiiAcademice Matei-Teiu Botez Iasi, Romania, October 7-8, 2004, pp. 146-152.


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Formal similarities between unidimensional finite elements used

in deformability and diathermacy analysis

Dan Diaconu-otropa1and Mihaela Ibnescu11Faculty of Civil Engineering, Technical University Gh. Asachi, Iai, 700050, Romnia

Abstract

The paper demonstrates the perfect analogy between the finite element analysis of

deformability problems and heat transfer problems. For each case, the main steps

of the procedure are followed in order to obtain the finite element equations, which

have the same shape. The important conclusion of the study is the possibility of

extending such design methods to different engineering problems.

KEY WORD: heat transfer, diathermancy, finite element.



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Formal similarities between one-dimensional finite elements used in the deformability

1. INTRODUCTION

The analysis of a system, which takes into account all the operating parameters, is

difficult, being almost impossible from practical point of view. That is way, a

series of hypotheses concerning the geometry and the constitutive materials of the

system are admitted. There are also adopted assumptions for phenomenon causes.This procedure leads to two types of models: the cause model and the system

model.

The models can be analytical models or numerical models and they consist in

continuous or discrete virtual systems.

The most powerful numerical method for the analysis of both structural

deformability and conductive heat transfer is the finite element method.

The present paper approaches the perfect similarity between the problem of axial

elastic deformability for a finite element of bar type (1D) and the diathermancy

problem for the same finite element.

2. MATRIX EQUILIBRIUM EQUATION FOR OF AN AXIAL ELASTIC

DEFORMABLE BAR

The finite element of bar type is the classic one, having two nodes at its ends, at

each node the nodal displacement along x axis being defined, (vector) and the

corresponding nodal force, (vector). There are defined:

xid

xif

{ } { 21 xxT

x ddd = } - the vector of nodal displacements for axial elasticdeformable element,

{ } { }21 xxT

x fff = - the vector of nodal forces for the axial elastic deformableelement.

For any current section of the element, located at a distance from the

axis origin, Hookes law can be expressed. It states the relation between the axial

stress, )(xx /axial force and the corresponding displacement gradient

:

)(xfx

)(xgxxd

{ } [ ] { } [ ]

==

dx

xddExEx xxxxx

))(()()( (1)

or the equivalent relation:


{ } [ ]xxdxx

gAExf =)( (2)


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{ } { } [ ]{ } [ ] { } ==+= )(1)( 2121 xxxxdT

x (3)

SSN 1582-3024


D. Diaconu, M. Ibnescu



The displacement field is:

For the element nodes can be shortly written that:

{ } [ ]{ }

A

x

x

x

xd

dxd

dxdx

xx

xx =

=

++==

==

2

1

2

1

221

121

22

11

1

1

)(

)( (4)

where

{ } [ ] { xdA = 1

} (5)

and then:

{ } [ ] [ ] { } [ ] { }

[ ]{ }Txx

xxx

ddxNxN

dxNdAxxd

2121

1

)()(

)()()(

=

=== (6)

where represent the shape functions, that for type of element have the form)(xNi

l

xxN =1)(1 ,

l

xxN =)(2 (7)

and their derivatives with respect to have the expressions:

lxN

dx

d 1)(1 = ,

lxN

dx

d 1)(2 = (8)

The displacement gradient function can be related to nodal displacements.

{ } { }

[ ]{ }{ } [ ]{ } [ ]{ }xxx

xx

xd

dBdxNdx

ddxN

dx

d

xddxd

dxxddxg

x

===

==

=

)()(

)())(()(

(9)

By using Hookes law it can be stated:

{ } [ ]{ } [ ] [ ]{ } [ ] [ ] { }xxxxTxx dBDdBAExgAExf === )()( (10)

or

{ } [ ] { }xx dxNdx

dxN

dx

dDxf

= )()()( 21 =


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[ ] [ ]

=

2

1

21 )()(x

x

d

dxN

dx

dDxN

dx

dD (11)

SSN 1582-3024





where, by expansion, rearrangement, and substitution of known terms, it is

obtained:

=

2

1

2

1

x

x

x

x

xx

xx

f

f

d

d

l

AE

l

AEl

AE

l

AE

(12)

that represents the matrix equilibrium equation for an axial elastic deformable bar.

It can be shortly expressed as:

[ ] { } { }xx fdk = (13)

where is the stiffness matrix of the element.[ ]k

Equation (13) has been obtained by considering that the axial force at node 1 is a

negative one, while the axial force at node 2 is a positive one.

3. MATRIX EQUILIBRIUM EQUATION FOR A DIATHERMIC

BAR

The adopted finite element is a unidimensional one, having the cross sectional

areaA. At the nodes, which are provided at the element ends, the temperatures iT

}

are defined (a scalar quantity) and the corresponding heat flow, (a vector).

There are expressed:

xiQ

{ } { }21 TTTT= -the vector of nodal temperatures for the diathermic element and

{ } { 21 xxT

x QQQ = -the vector of heat flow for the same element.

For any current section of the element, located at a distance from the origin,

Fouriers relation can be written and it states the relation between the heat

flux, / heat flow, and the corresponding temperature gradient,)(xqx

)(x

)(xQx

gxT

{ } [ ]

=

dx

xTdxq

xx

))(()( (14)


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D. Diaconu, M. Ibnescu

where x represents the thermal permeability characteristic.

The equivalent relation is:

{ } [ ] { })()( xgAxQ xTxx = (15)

The function of temperature variation along the element is a first degree function:

{ } { } [ ]{ } [ ]{ } ==+= )(1)( 2121 xxxxTT

(16)

The temperature function can be related to nodal temperatures:

{ } [ ]{ }

=

=

+

+==

=

=A

x

x

x

xT

TxT

TxT

2

1

2

1

221

121

22

11

1

1

)(

)( (17)

It results that:

{ } [ ] { }TA = 1 (18)

So:

{ } [ ] [ ] { } [ ]{ } [ ]{ TTTxNxNTxNTAxxT 21211

)()()()()( === } (19)

where are the shape functions that for an element with two nodes have the

following form:

)(xNi

l

xxN =1)(1 ,

l

xxN =)(2 (20)

and their derivatives with respect to have the expressions:

lxN

dxd 1)(1 = , l

xNdxd 1)(2 = (21)

The function of temperature gradient is expressed in terms of nodal

temperatures:

{ } { }

[ ] { }{ } [ ] { } [ ] { }TBTxNdx

dTxN

dx

d

xTdx

d

dx

xTdxgxT

===

==

=

)()(

)())((

)(

(22)


By using Fouriers relation, it is obtained:


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{ } [ ]{ } [ ] [ ]{ } [ ] [ ] { }TBDTBAxgAxQ xxTxx ==== )()( (23)

or:

{ } [ ] { }TxNdx

dxN

dx

dDxQx

= )()()( 21 =

= [ ] [ ]

2

1

21 )()(T

TxN

dx

dDxN

dx

dD (24)

which, by expansion, rearrangement and substitution of known terms becomes:

=

2

1

2

1

x

x

xx

xx

Q

Q

T

T

l

A

l

Al

A

l

A

(25)

that represents the matrix equilibrium equation of the diathermic bar, expressed interms of temperatures, and which can be shortly expressed:

[ ] { } { }xQT = (26)

where [ ] is the permeability matrix of the diathermic bar.

Equation (26) has been obtained by considering that the heat flow which enters

node 1 is positive, while the heat flow which exits node 2 is negative.

4. CONCLUSIONS

By comparing the matrix equilibrium equation (12), stated for the axial elasticdeformability case, with equation (25), valid for diathermancy case, it can be

noticed that they have the same shape. In the second mentioned equation, the

longitudinal modulus of elasticity that occurs in the first equation, was

substituted by the thermal permeability characteristic,

xE

x .

Taking into account that the two equilibrium equations are identical from a formal

point of view, it can be concluded that these procedures concerning the elastic

deformability analysis can be also applied for heat transfer problems, at least for

the stationary case.


Such formal similarities between different phenomena offer to the analysts the

possibility to perform analogies, but also to use and / or extend design methods ofapparently different engineering problems.


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53


Synthesis of the problematics approach of the worship spacesdesign

Dragos Ciolacu1, Lucian Strat2

1Architectural Faculty, TU Gh. Asachi, Iai, 700050, Romania2Structural Mechanics Department, TU Gh. Asachi, Iai, 700050, Romania

SUMMARY

The main aspects of the Theory of Architecture dealing with ecclesiastical

buildings versus the structure forms in the design of different architecture functions

are briefly described in this paper, together with some technological aspects .

KEYWORDS: Theory of Architecture, worship symbolistic, structural functions,Christian church buildings

1. INTRODUCTION

The holly church is Gods image and icon

as one having the same doing as Him by imitation and imagination


Having in mind this beautiful thought so piously expressed in

mystagogia by Saint Maxim the Confessor, I tried to discover what it is left

today from the topics on the edified sacred which accompanied us for a

millennium. As we know, this topic, as integral and integrating part of the Christian

sacred, marked our passing through the European culture with the permanence of a

spiritual need, after it settled on the pagan ruins of the roman civilization, the

Christianity became, in time, the identifying matrix of a whole culture, united at the

beginning, shattered then into doctrinarian fragments, the Christian culture

preserved its entire seduction and modeling power in its various forms of

manifestation. One of them is the orthodox architecture dedicated to the

transposing of the idea of sacred into the earthly reality with the means offered by

the latter. The sacred architecture refers both to the emotion and intellect by

symbols and meanings, spatially and volumetrically configured. The sacrality

materializes through structure and acquires esthetical valences by the related artsaccompanying this practice. The purpose of this work under the sign of the divine

is to transcend the structural rational and the architectural functional in mystic


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D. Ciolacu, L. Strat


spatiality. From a historic point of view, the ecclesiastic architecture was always

there, being influenced in the structural component by the evolution of the engineer

thinking, and in the formal-aesthetic one, by the succession of the cultural centers,

but maintaining the articulation of the sacred archetypes unaltered. The timeless

laws establishing the in genius loci tradition give rise to volumes with a strong

identity-related content, which customize them both in the panorthodox field and in

the Romanian cultural regionalism. The Byzantine tradition, engrafted on the

popular culture on which the materials expression and the structural aesthetics are

added, is our legacy in the field of ecclesiastical architecture.

Along with the changes from the years 90 an extremely broad topic on

architecture became a current issue and namely, the ecclesiastical program. For 50

years, the sacred in architecture was prohibited enough to cancel a crystallized

tradition in centuries of explanatory practice. Today we are subjected to an

enormous pressure towards filling an empty space and the finding of an

architectural expression, synchronous to the age, proved rather difficult. A series of

competitions, symposiums and printed works having as topic the religious

architecture succeeded in outlining a directio

Revista Intersectiii No1_eng

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