-"ComputING", DT. Il, Mai 2000, pag. 21-28 ISSN 1221-43-37"A 5-a Conferinta de Elemente Finite si de Frontiera" - ELFIN5, Proceedings, Section 2.2- Application of the Structural Analysis in Civil Engineering, pag.f-B, Universitate a Oradea,25-27 Mai 2000 (General Chairman).

1!DYNAMICALRESPONSE OF "SCALA"BUILDING

BUCHAREST TO EARTHQUAKE MOTION WITH FINITEELEMENT METHOD.•=

Mircea IEREMIA, prof. Dr. ing., Technical University of Civil Engineering Bucharest,Silviu GINJU, ing., Acta Pro Construct srlAdina TUDORACHE, stud., Technical University of Civil Engineering Bucharest.

Abstract The paper presents the comparatively analysis between the frame structure response to the horizontalexcitation produced by an earthquake and the case ofvertical earthquake appearance. The analysis has been effectedfor 2variants ofmodeling. In the first variant the structure has been modeled al the ends ofthe co/umns with concentrated mass.In the second variant the mass has been considered as being uniform distributed on the jloor. The distribution of massesallows the appearance ofvertical vibrations of beams andjloors. The paper points the necessity of taking into account thevertical components of earthquake and influences of the Rayleigh viscous damping in the dynamic response of thestructure.

Key words: mesh, earthquake, modal analysis, spectral analysis, damping

1. Introduction

The paper presents the modaI and spectral dynamicanalysis of the strength structure of "Scala" building.

The building is placed in the center ohBucharest City, within a seismic zone of 8 degreesMSK; earthquake coefficient ks=O.20.

I\N.

Figure 1. "Scala" building

The structure has a system with basement,ground floor and 10 floors. The height of floors1-10 is 2.75m, and the height of the groundfloor is: 3.61m. At the last two levels, thestructure is presenting withdrawals. Thedestination of this building being flats, with theexception of the ground floor, where isfunctioning a market.

The strength structure of the building ismade up of frames of monolithic reinforcedconcrete. The concrete floors are made up of 5cm flagstone with connectors andmonolithissing 8 cm breadth top concrete. Thefoundation of the building consists of a generalfoundation plate that together with the wallsfrom basement and concrete ground floor makeup a rigid box.

2. Consideration regarding the modeling

The structure has been analyzed from dynamicpoint of view, through the method of finiteelement, by means of the computationalprogram ANSYS 5.4 produced by SwansonAnalysis Systems, lnc.

The structure has been discrete made intwo variants.

In the first variant have been generatedthe finite elements, type Beam and Shell, havingthe sizes of the elements whose demeanor aremodeling between the geometrical crossingpoints of the columns with the beams.

It resulted: 788 nodes, 2167 elements typeBeam4 and 549 elements type She1l63.

In the second variant of modeling, eachbeam has been divided in two finite elements,type Beam, and each loop from concrete floorhas been modeled by means of flour-finiteelements, type Shell. In this case, the masseswere considered as actioning in the junctionsresulting in the field of the concrete floors. Itresulted: 3147 elements type Beam-t, 2305elements type She1l65 and 2888 nodes.

J\N"~"'. 010- _oi,~·t t .

Figure 2 "Scala" structure. Dense mesb.

3. Considerations regarding the dynamic analysis3.1. Undamped modal analysis

The moving equation for an undarnped system,expressed in matrix notation, using the aboveassumptions, is:

[M]{ii}+[K){U}={O} (1)where: [MJ - the structure mass matrix;

[K) - the structure stiffness matrix.

For a linear system, free vibrations will be harmonicofthe form:

{u}={<!>}iCOS())it (2)where: {<!>L - eigenvector representing the modeshape ofthe jth natural frequency.

Thus, equation (1) becomes:

(-o/[M]+ [K]){<I>l = {o} (3)

Figure 3 Mode sbape 1 - transversal vibrations

AJi'SY$ 5.4~.c30H:~2:o-l

sn~lSUB -1

rREO=.9~ZHUY (f.Vt;.)'R!'Jys:OPOWl!:cGreplu.c.l.'AC!'f!II:1AVJl! 5=HeI e:D)(X "".060804

SMX".OiOO3,

- ~OOG73Î.01'3414.0:20212.02 &949.03)69&.040423.G4716.053898.060035

This equality is satisfied if either {<I>l = {O} or if thedeterminant of [x]- ro 2 [MU is zero. The firstoption is the trivial one and, is not of interest. Thus,the second one gives the solution:

This is an eigenvaJue probJem, which may besolved for up to n values of, OJ 2 and n

eigenvectors ~ l,which satisfy equation (3) where"11" is the number ofDOF's.

The eigenvalue and eigenvector problemneeds to be solved for mode-frequency analyses. Ithas the form of:

[K]{<I>J= ÂJM K<I>J (4)

Figure 4 Mode shape 2 -longitudinal vibrations

FoIlowing the modal survey, we have beenobtained the eigenmodes of vibration through themethod of iterations on subspace, it reached in 20mode shape up to 86% participating factors of

where: Ai - the eigenvalue.AISr.5 5.4J).JJ 5 199921:13:2S.on.1L !V,)LU'IIOWSTUa:'1

SUB =zr~1.1l3UX (AVG)l<SIS"OpO"-XGraphl.CIII

l!p.lar-lAVRJ!5=1oI..a.t

OllJ[".0192.4~ =.05362S

o.005958.011917.017875.023833.02979Z.035"1,5

.041708

.04766"6

.053625

masses on longitudinal "x"direction; up to 87% ontransversal "y" direction and up to 60% on verticalgravitational "z" direction.

The fundamental period of vibration of thestructure resulted T1=1.2s. From the comparativesurvey of the modes shape of vibration of thestructure within the two variants of modeling it canbe observed that the horizontal vibrations of the

structure are not influenced by the mash density andthe situation of masses, while the vertical vibrationsamphfy, in case of modeling of structure, the secondvariant.

AJr5Y5 5.4

JU 5 1999.1013:54VODAI, SOLUTTOIi

S"",,",1lSU3 =3l'eC,.l=l.163UX (AVG)ons=OPo••erGraphl.~!p.lar=l

A\"Re5=ltattim! =- .080il23r>CI=-.OtP055_ -.04;055

-.04l005-.01fH54-.0327Q3

-.021Z~-.01180.-.016352-.01Q901-.005451o

Figure 5 Mode sbapc 3 - longitudinal vibrations

Thus in the first variant of modeling, themeaningful vibrations on vertical line appear in the14 mode shape of vibration, driving approximately5000 KN, to which it corresponds a participatingfactor of 22%. In the second variant of modeling, the

meaningful mode of vibration on vertical line isquicker obtained, on the 13 mode shape of vibration,the driven mass in vibration being 6760 KN, towhich it corresponds a participating factor of 26%.

ANSYS 5.4,fU 5 1999ll: 14: 18IODAI. ~.UTIOJl:il"I!P=1SUB =4r~o:2. 78gtJT (AVG)

M=OpOloll!rGraph.1cs

!"'-C!T=1A~t.oMX =.0612S1S·l3I =-.056454SMX =.061029ElI -.05645'

-.043401-.030347

-.011Z91-.004239.008814.021060.034922.047;76.06l029

Figure 6 Mode shape 4 - transversaJ vibrations

On the whole of the 20 vibration modeswhich have been analyzed, the total sum of thedriven mass on vertical line is approximately thesame. In the case of dense variant of mesh, a trend of

concentration of the vibrating mass in 3 modes ofvibration (nr.13,14 and 15), while in rare variant ofmesh, the meaningful vibrating mass can be foundon 5 modes of'vibrations (nr.13,14,16,17 aud 18).

A.•S".lS 5.4.n..lI 5 1999

U:H:S9:aOOAl. SOLlrtlOli"smFI:~nm '36rreO--l.1SI-= IA"")",,,,,,zOPO'«!rerapb..l.c;"EPAc..E'1=l

"'\~!S=M.tl),.'G =.091Q09

"'!li =-.OOl9?V"". -.002185

-.003971\-.OOJ227-.OOl47'-.001724-.9'73!!·03-.221Z-03.530.-00.001282.OO2Q33.002185

Figure 7 Mode shape 6 - torsional vibrations

3.2. Damped moda) analysis

In the damped method the eigen problem becomes aquadratic eigenvalue problem given by:

[K ]{<I>i}+ Ai [cK<I>i}= -Ai 2 [MK<I>i} (5)

where: Â; = ~- Â; - ilie complex eigenvalue;[C] - ilie damping matrix.

The dynamic response ofthe system is given by:{uJ = {<I>Je(o;+jC1)J (6)

where: (5; - real part of the eigenvalue;

imaginary part ofthe eigenvalue; j= ~ .

OJ -I

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'MX =.06l13!ffl5--.050963SNI z.OG1i46

-.050963-.038~4-.025917-.013)94--.8105--D3

.011653

.02U7';

.03.699

.049222

.n61'746

Figure 8 Mode shape 10 - transversal vibrations

The general damping matrix ([C]) may be used fi

damped modal analyses:NMA't NEL

[c]= a[M] +$+ f3cXK]+ Lf3AKj]+ L[Ck]+ [cd (7)j=I k=l

where: a -ilie constant mass matrix multiplier;P - ilie constant stiffuess matrix rnultiplier;

Pc - the variable stiffness matrix multiplier;NMAT - the number of material;

Pj - the constant mass matrix multiplier formaterial i:

[Kj] - the portion of structure stiffuess matrixbased on material j;

NEL - the number of elements with specifieddamping;

[Ck] - the element darnping matrix;

[Cţ] - the frequency-dependent dampingmatrix.

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STl!Fl!HlB :.14

r~.ltJ5113 (AVGl

""~Po•••:e.rGra.ph.1e.5!rACl~lA.'~!.S=Nattl.-a: =.114392""" =-.OO299l5n.'t -.09-49157

-.002991.007802.018155.019628.040502.(151375.06:2-\8.07312J.08399'1.094367

Figure 9 Mode shape J.t - vertical vibrations

In the Rayleigh model the damping matrix[C] is calculated by using these constants to multiplythe mass matrix [M] and stiffness matrix [K]:

[C]=a [M]+ j3 [K] (8)The values of a and j3 are not generally knowndirectly, but are calculated from modal dampingratios, Si. Si isthe ratio of actual damping for aparticular mode of vibration,i. If O)i is the naturalcircular frequency of mode i, a and j3 satisfy

Damping Ratio

relation:

~j = ~+ proj2roj 2

To specify both a and j3 forgiven dampingratio S, it is comrnonly assumed that the sum of thea. and j3 tems is nearly constant over a range offrequencies (see figure 10). There forgiven S and acircular frequency range O)i to O)j. two simultaneousequations can be solved for a and j3.

(9)

o-damping

•ro! ro2Figure 10 Damping - circular frequencie relation

In our computation we used thefollowing values for the fraction of criticaldamping, corresponding to eigenmodes 1 and 2:

O)

~!=2% and ~2=5%. Solving the equation system(9) we computed the values for the constants ofproportionality, which are presented in table 2.

Table 2

I ~\9 I ~.~4 I ~201 I :.\5 I ~.87 1....J...~-.0-3-9-----1

Figure 11 Mode shape 17 - vertical vibrations

The influence of damping decreases thedynamic response of the structure with 14%.

AJIS"tS5.4JU ~ l.99921,20,)5JtOWU. SCl.lr'tIOli'

~l~um=17rp-7(F!l.589uz (A"';)03=0Powe.rGrapb.icaEt'AC!!:'l"""l""~MatDMX =.100285

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-.072907.053779-.03465;-.015$24• OOJ 604.00ZTH.041959.Oro98?.ooOll'l.099242

The first ten periods of vibration are presentedin table 3.

Figure 12 Mode shape 18 - vertical vibrations

3.3 Spectral analysisThe strength structure of "Scala" block of

flats within the both variants of model ing wassubmitted to spectral analysis. Thus, it wasestablished the maximum reply of the structure athorizontal and vertical action of an earthquake.

In case of on horizontal earthquakeexcitation the values of displacement of structureand of sectional efforts in the component elements

.us'lŞ 5.4JJJI 5 191192:1:20:55JJOnlL SCIL ti"I IOI"snsP-lSUB =:18rRl!()IIIO.212ua (AVG)03=0PowecGraph.iC8IPAel]t="lA\I'JU~S:M&e

lWl: '" .0929SSS!'4."iI ;0;- .ORiS9S::s.1'If:iI: =.09214.7_ -.081595

-.06i!"09- .01159'-.021581-.001516• 0J2 <\28.0".12133.054439.07244.2.09244'

(columns aud beams) do not significantly differ inthe two variants of'modeling.

In case of vertical seismic excitationcorresponding to a 0.6 ratio from the horizontalcomponent part, the vertical oscillation of the beamscan alter the answer ofthe structure. In thisway, for the second variant of modeling with a Largenumber of finite elements, it has been obtained

tensile axial efforts in the central columns two timesmore rare than in the case of the first variant ofdiscretisation. Although these axial efforts have nothigh values in comparison with the value of thecompresion axial effort originated from the verticalloadings on the central columns, can decline thevalue ofthe efficient moment of columns.

For the opposite direction of action of thevertical component part of the earthquake, the axialefforts compress the columns. For the colurnns withreduced gravitational vertical loadings, the effect ofthe increasing ofthe compression is favorable, while

for the central columns with important gravitationalaxial efforts, the increasing of the axial efforts ofcompression can lead to outrunning of thestrengthening limit in compresion of concrete.

For what concems the beams with importantvertical stresses-leaning columns on beam at higherlevels - from the vertical action of the earthquake,resulted maximum sectional bending effort M=57.4KNm and shear effort T=46.7 KN, in case of themodel ing with dense mesh, with 33% higher than inthe case of the rare variant of model ing.

AJlSYS 5.4JAII 5 199921:21:39NODAL SOLUTIONSTEP=1SUB =20FRI!CF10.?82UY (AVG)RSY3=OPowerGraphic,"Hl ACE'1'= 1AVRE3=MatDMX =.115234SIDl =-.086701SMX =.093014

-.086701-.066733-.046764-.026796-.006828.013141.033109.053077.073046.093014

Figure 13 Mode shape 20 - vertical & transversal vibrations

4. ConclusionsThe dynamic analysis of the strength

structure of "Scala" building emphasizes thefollowing aspects:

- The analyzed structure has a meaningfulcapacity ofviscous damping ofthe vibrations (14%),taking into consideration that it takes part in thecategory of semi-flexible structure (T= l.2s).

- The type of mesh can distort the structuralanswer in case of an gravitational earthquake

component. Talking into account that in the lastyears on registered earthquake with importantvertical component, in comparison with thehorizontal component (see 17.01.19994 Northridge:ah=1.82g and av= 1.18g), it becomes a necessity theintroduction in the seismic dynamic analysis alsotowards the action ofthe vertical component part.

5. References[1] BATHE K. j, Finite Element Procedures, Prentiee HalI, New Jersey (1996), V.S.A.

[2] IEREMIA M. - Elasticity. Plasticity.Nonlinearity. ,Ed.PRlNTECH Bueharest (1998), Romania.

[3] ANSYS 5.4, Swanson Analisis Systems Ine., Houston (1997), U.SA