Spectral Modal Modeling By FEM Of Reinforced Concrete Framed Buildings .

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Spectral modal modeling by FEM of reinforcedconcrete framed buildings irregular in elevationNehar Kheira CamelliapInternational Review ofApplied Sciences andEngineeringMechanical and Materials Development Laboratory (LDMM), University of Djelfa, PB 3117,Djelfa, Algeria12 (2021) 2, 183–193Received: December 24, 2020 Accepted: April 2, 2021Published online: May 15, 2021DOI:10.1556/1848.2021.00229 2021 The Author(s)ORIGINAL RESEARCHPAPERABSTRACTThe irregular buildings constitute a large part of urban infrastructure and they are currently adopted inmany structures for architectural or esthetic reasons. In contrast, the behavior of these buildings duringan earthquake generates a detrimental effect on their regularity in elevation which leads to the totalcollapse of these structures.The objective of this work is essentially to model reinforced concrete framed buildings irregular inelevation subjected to seismic loads by the Finite Element Method (FEM). This modeling aims toevaluate several parameters: displacements, inter-storey drifts and rigidities, using two dynamiccalculation methods; one modal and the other spectral modal. The latter is widely used by engineers.For this purpose, a detailed study of the frames which have several setbacks in elevation is carriedout to validate the correct functioning of our FEM calculation code in both cases of modal and modalspectral analyses. The performance, accuracy and robustness of the FEM calculation code produced inthis study is shown by the good correlation of the obtained results for the treated frames with thoseobtained using the ETABS software.KEYWORDSmodeling, finite element method, elevation irregularity, reinforced concrete frame, modal analysis,spectral modal analysis, dynamic calculation1. INTRODUCTIONpCorresponding author.E-mail: camellia90@hotmail.fr;c.nehar@univ-djelfa.dzWith the current evolution, an increase in the demand for sustainable development has beenrecorded, the people are interested to stay in urban areas rather than in rural areas. In theseurban areas, the structures are usually in the form of buildings which have a safe behaviorunder the effect of the seismic action which depends on several factors: rigidity, an adequatelateral resistance, ductility as well as simple and regular configurations [1]. There is much lessrisk to buildings with regular geometry and uniformly distributed mass and stiffness in planand elevation than with irregular configurations [2]. But nowadays, the need and demand ofthe latest generation have made it inevitable for engineers to plan buildings with irregularconfigurations [3].Furthermore, in this kind of buildings, there are different types of irregularities dependingon their location and extent, but mainly they are classified into two types, in plan andelevation [3]. Irregular structures in the plan are those in which the seismic response is notonly in translation but also in torsion and is the result of the rigidity and/or the eccentricityof the mass in the structure [4]. While irregular structures in elevation are those in whichmass or stiffness or geometric regularity are not uniform throughout the structure [4].These structures present weaknesses that can be produced between adjacent storeys bydiscontinuities of rigidity, strength or mass [5]. Such discontinuities between floors are oftenrelated to abrupt variations in the geometry of the frame along with the height. There aremany examples [6, 7] of building failures in past earthquakes due to discontinuities inelevation.Unauthenticated Downloaded 08/10/22 12:16 AM UTC

184International Review of Applied Sciences and Engineering 12 (2021) 2, 183–193In reality, structures are often irregular because theperfect regularity is an idealization that rarely occurs [8].Therefore, the configurations irregular in elevation haveoften been recognized as one of the main causes of failure inearthquakes [5].The seismic regulations, such as the Algerian EarthquakeRegulations 1999 Version 2003 (RPA 99V2003) [9] and theEurocode 8 [10], recommended the use of spectral modaldynamic analysis or temporal dynamic analysis as thepreferred calculation methods to assess the seismic responseof irregular buildings.Several investigations have been carried out to study theseismic behavior of irregular structures in elevation. The firststudies on the seismic behavior of buildings characterized bydiscontinuities along with the height in terms of mass, rigidity and resistance date back to the 1980s [11, 12], startingwith the work of Moehle and Asce (1984) [11]; Costa et al.(1988) [13]; Shahrouz and Moehle (1990) [14]; Wood (1992)[15]; Wong and Tso (1994) [16]; Cassis and Cornejo (1996)[17]; Valmundsson and Nau (1997) [18]; Magliulo et al.(2002) [19]; Rom ao et al. (2004) [1]; they found differencesin the response of irregular and regular frames. Among themost notable differences, they noted increases in ductilitydemands at the storey where the setback in elevation islocated and also at the storeys immediately above.The seismic efficiency of vertically irregular structureshas recently been studied in many previous studies (Karavasilis et al. (2008) [20]; Sarkar et al. (2010) [21]; Rajeevand Tesfamariam (2012) [22]; Varadharajan et al. (2013)[23]; Pirizadeh and Shakib (2013) [24]; Roy and Mahato(2013) [25]; Hamdani (2015) [26]).More recently, Darshale and Shelke (2016) [27], Bhosaleet al. (2017) [28] , Shelke et Ansari (2017) [29], Siva Naveenet al. (2019) [30], Etli et G uneyisi (2020) [31], examinedstudies on the analysis of irregular structures under seismicloads, they concluded that the irregular distribution ofmass, stiffness and vertical geometric irregularity showed adifferent response from the regular one.In addition, several techniques have been used in thenumerical simulation of reinforced concrete frames buildings. Among them, we have the most used methods in thefield of engineering, we can recall: the finite differencesmethod (FDM) [32] for diffusion problems (heat transfer,fluid flow, etc.), the boundary element method (BEM) [33]for the problems interfaces and interactions; the finite volume method (FVM) [34] and the Finite Element Method(FEM) [35]. In this work, we will choose this last method asa method of solving problems of irregular structures inelevation. The concept of this method is used for numericalanalysis wherein the model to be studied is divided intoelements known as finite elements and each element’sresponse is expressed in terms of a finite number of degreesof freedom [35]. Moreover, the finite element method isproving to be a very powerful tool for the analysis ofstructures which remain continuous, subjected to violentaccidental aggression such as an earthquake [36].Various researchers have conducted studies on numerical modeling using FEM to study regular steel gantries underseismic excitation such as Ozturk and Catal [37], Sekulonicet al. [38], Shousuke and Yasuhiro [39] et Sharbane andNiraj Kumar [40].In the present work, we have modeled the reinforcedconcrete framed buildings irregular in elevation subjected toa seismic excitation by the FEM. To find the displacements,two dynamic calculation methods were used: the methodof modal analysis and the method of spectral modalanalysis. The numerical modeling of these irregular framesrepresents the originality of this study compared to the workmentioned above.2. THE IRREGULARITY IN ELEVATION OFREINFORCED CONCRETE STRUCTURESThe seismic action is an accidental action which is definedin the Algerian Earthquake Regulations [9]. Indeed, to avoidthe damages resulting from this seismic action, a judiciousearthquake-resistant design must be carried out which ensures adequate seismic behavior. For example, discontinuities in stiffness and strength should be avoided, whichshould ideally be distributed evenly over the height of thestructure [18, 41].However, in practice, reinforced concrete frame buildings are generally found to be irregular in shape whose irregularity is characterized by a setback in elevation (Fig. 1).Fig. 1. Irregular buildings in elevationUnauthenticated Downloaded 08/10/22 12:16 AM UTC

185International Review of Applied Sciences and Engineering 12 (2021) 2, 183–193v2v1θ1u1θ2u2LFig. 3. Bar element in the plane with six degrees of freedom [35]dynamic system (Eq. (2)). Finally, the damping matrix ½C isexpressed as a linear combination of the two matrices ½K and ½M , using the Rayleigh damping, which is written as:½C ¼ a½M þ b½K Fig. 2. Damage in re-entrant angles in the vertical plane due todifferential oscillations: (a) Boumerdes earthquake, Algeria 2003[42], (b) Kobe earthquake, Japan 1995 [43]This vertical irregularity is one of the main factors ofstructural failures [41]. The changes in height of stiffness andmass make the dynamic characteristics of these buildingsdifferent from those of the normal or regular building (Fig. 2).3. DYNAMIC MODELING BY FEM OFIRREGULAR FRAMESTo model the frame buildings in reinforced concrete, thefinite element approximation is given as follows [35]:Xuðx; tÞ ¼Ni ðxÞqi ðtÞ(1)i nwhere uðx; tÞ represents the field of approximate displacements, Ni ðxÞ represents the standard interpolation functionsand qi ðtÞ the vector of nodal displacements, n the set ofnodes of the discretized domain [35]. The use of the Rayleigh-Ritz method allows us to write the linear matrixequation describing the behavior of a dynamic system:½M f qðtÞg þ ½C fqðtÞgþ ½K fqðtÞg ¼ fFðtÞgwith:(2)Zr½N T ½N dΩ½M ¼(3)Z½B T ½D ½B dΩ½K ¼where a and b are the Rayleigh coefficients.To numerically model the frames and establish thematrices already mentioned we used a linear element (bar)that has two nodes with three degrees of freedom per node[35] (Fig. 3).The stiffness and mass matrices are calculated byanalytical integration for the bending bar element composedas follows [44]:23EAEA00 00 76L6 L767612EI 6EI12EI 6EI 7670 3676L3L2LL2 76764EI6EI2EI 7670 26767LLL7½K e ¼ 667EA67600 767L6767612EI6EI 767 (6) 6L3L2 7676744EI 5Sym:L321 0 0 0 0 060 1 0 0 0 07767676001000rAL 67;½M e ¼62 60 0 0 1 0 0777640 0 0 0 1 050Ω(4)Ωwhere r is the density, ½B is the strain matrix derived fromthe interpolation functions and ½D the elasticity matrix. Thematrices ½M , ½C and ½K are respectively the matrices ofmass, of damping and elastic rigidity. The vector fFg is thevector of dynamic excitations.The integration of Eqs. (3) and (4) on the domain Ωbrings us back to an assembly of elementary matrices.The boundary conditions are imposed before solving the(5)00001For how to calculate the displacement, there are severaldynamic methods. In this study, we have chosen the methodof modal and modal spectral analysis because they are themost recommended by the RPA 99V2003 [9] in the case ofirregular structures.3.1. Modal analysisThe solution of displacement uðtÞ of a linear dynamicproblem can be approximated by its decomposition on afrustoconical basis of eigenmodes Fj [45, 46]. Then it can beexpressed as follows:Unauthenticated Downloaded 08/10/22 12:16 AM UTC

186International Review of Applied Sciences and Engineering 12 (2021) 2, 183–193STARTData introductionGeometric, mechanical,TREATMENT- Initialization of the elementary matrix of mass M andstiffness K.- Assembly: Construction of the global matrices M and K.- Introduction of boundary conditions.STATICDYNAMICMake the force vector f.Solution of the linear equation KU fEigenvalues and vectors calculationModalSpectralModalSpectral displacementModal displacementcalculationcalculationENDFig. 4. Flowchart of the digital modeluðtÞ ¼NXyj ðtÞFj(7)j¼1where yj ðtÞ are the generalized displacements which areevaluated by the Duhamel integral [47, 48].The pulsations uj and the eigenmodes Fj are determinedfrom the following relation [47, 48]:ihih(8)det K u2j M ¼ 0; et K u2j M Fj ¼ 0 :3.2. Spectral modal analysisSpectral modal analysis is a dynamic method based on themodal analysis method. Once the periods of oscillation ofthe structures have been established (modal analysis), onereads the spectral acceleration assumed to be a maximumresponse (spectral analysis). This last method uses theresponse spectrum which is a tool to estimate the responseof a seismic building [49].Due to its simplicity, this method is frequently used bymost computer structural analysis engineers dealing withseismic design [50].Thus, for a given earthquake, the maximum displacement of the structure is defined as the quadratic combination of modal displacements [50, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiu N 2uXumax ¼ t(9)yjmax Fj :j¼1All the formulas explained above have been programmedin MATLAB [52] and organized in the form of the flowchartpresented in Fig. 4.4. NUMERICAL APPLICATIONTo model and evaluate the irregular systems in elevationadopted in this work, a framed building in reinforced concrete was chosen. It is a structure of 06 levels, selected fromthe literature [8, 30]. This structure is made up of 3 spans of5.00 m each and a height of 4.00 m for the first level and3.00 m for the other levels (Fig. 5 (a)). We modified thisstructure to study the effect of irregularity in elevation byremoving some structural elements from each level(See Fig. 5 (b), (c), (d), (e) et (f)).The frames were designed in accordance with the provisions of the calculation code for Reinforced Concrete inLimit States BAEL91 [53] and the RPA 99V2003 [9]. Thedata of the frame, in particular, the sections of the beamsand the columns are of 40 cm 3 40 cm.In the design procedure, the permanent and operatingloads are distributed evenly over the floor and their values are: 2 2for standard flooring : G ¼ 5:10 KNand Q ¼ 1:50 KNm: m for terrace flooring : G ¼ 6:20 KN m2 and Q ¼ 1:00 KN m2 :The modal and modal spectral analysis methods are usedto study the seismic response of these frames which areexcited by the same acceleration and spectral response loadUnauthenticated Downloaded 08/10/22 12:16 AM UTC

1874.00 m5 @ 3.00 mInternational Review of Applied Sciences and Engineering 12 (2021) 2, 183–1933 @ 5.00 m(a)(b)(d)(e)(c)(f)Fig. 5. The geometric configuration of the studied structure: (a) Regular, (b) Irregular N8 01, (c) Irregular N8 02, (d) Irregular N8 03,(e) Irregular N8 04, (f) Irregular N8 05recorded during the Boumerdes earthquake (Algeria) in2003 and which are presented in Fig. 6 (a and b).The mechanical characteristics of the material areYoung’s modulus E 5 32164195 KPa, the Poisson’s ratio isν 5 0.18 and the density r 5 2,400 Kg/m3, the dampingcoefficient is taken equal to 0.05.to show the performance of the developed program, acomparison is made with the ETABS software [54].The obtained results in regular and irregular structures interms of natural periods, storey displacement profiles, interstorey drift and finally the rigidity of each storey are presented.5. RESULTS AND DISCUSSIONS5.1. Preliminary resultsThe theory presented previously has been programmedand implemented with our cases of irregular frames. AndThe results in terms of natural periods obtained by ourcalculation code FEM and ETABS [54] for the differentstudied models are shown in Table 1. It is clear that the10382Acceleration x g (m/s )2Spectral Acceleration x g (m/s )210-1-2642-3005101520Times (s)(a)25303501234Périod T (s)(b)Fig. 6. Boumerdes earthquake: (a) Accelerogram, (b) Response spectrumUnauthenticated Downloaded 08/10/22 12:16 AM UTC

188International Review of Applied Sciences and Engineering 12 (2021) 2, 183–193Table 1. The natural periods of the regular and irregular structureCase of the structureIrregularModesMode 1Mode 2Mode 3Periods (s)Our code FEMETABSOur code FEMETABSOur code FEMETABSRegularN8 01N8 02N8 03N8 04N8 991.0351.0420.3450.3480.2010.203results acquired by our calculation code FEM are in goodcorrespondence with those obtained by ETABS.We can also notice that the natural periods for the regular structure are higher than those where the structure isirregular. Which means that the irregularity in elevationmade the structure more flexible. Also, it is noted from thevalues of the natural periods of the irregular structures that areduction of these values occurred by changing the type ofstructure. For the irregular structure N8 05, the first modereaches 1.035.those determined by the modal analysis for the same type ofstructure and number of the storeys, except for special cases.This is due to the principle of the spectral modal analysismethod which uses the max accelerations.To better understand the shape of the spectral displacements we have presented the displacement of thesetback point of each type of structure (Fig. 9). It is clearthat this setback in elevation will influence the results ofdisplacements, this is perhaps due to the reduction in therigidity of the structures.5.2. Modal displacement profiles5.4. Inter-storey driftThe frames are excited by the Boumerdes earthquake 2003.The results of modal displacements in the X directionobtained by our code FEM and ETABS [54] are presentedin Fig. 7.These displacement curves are represented for themost stressed node of each frame case. Remarkably, theobtained results by our code FEM are very close to thoseobtained by ETABS. So, we can conclude that the obtainedresults in terms of modal displacements are satisfactoryand promising.We can also notice from the obtained curves that thedisplacement has increased considering the irregularity inelevation. This confirms that the irregular structure isdangerous because the setbacks in elevation can lead toserious damage, especially in cases of major earthquakes.The calculation of inter-storey drifts is carried out takinginto account the obtained results by our calculation codeFEM using spectral modal analysis and which are presentedin Fig. 10. The variations in elevation of these displacementsare compared with an inter-storey drift of 1% of the storeyheight, which is required by the Algerian Earthquake Regulations RPA 99V2003 [9].We can notice that the computed relative displacementsdo not exceed the displacement threshold admitted by theRPA 99V2003 [9]. It should be noted that the inter-storeydrift in the case where the structure is irregular decreasescompared to the regular structure.5.3. Profiles of spectral modal displacementsThe frames are solicited by the response spectrum of thesame Boumerdes earthquake (Algeria) 2003. The displacements obtained by our code FEM are presented with themax modal displacements and those obtained by ETABS[54] and this for each level of the regular and irregularframes N8 01, 02, 03, 04 and 05 (See Fig. 8).We noted initially, a good correspondence between theresults of spectral displacements and the max modal displacements obtained by our code FEM as well as those obtained by ETABS, and that the displacements increaseaccording to the level of the storey.Secondly, we can conclude that the displacementscalculated by the spectral modal analysis are greater than5.5. Storeys StiffnessesFigure 11 shows the rigidity shapes of each storey bychanging the frame type (regular – irregular). It should benoted that these shapes are reached using the obtained results by our calculation code FEM. We can see from theobtained curves that the stiffness increases with the storeylevel and decreases by changing the irregularity of the frame.6. CONCLUSIONThis study presents a computational procedure to modelnumerically the structures that present a setback in elevationusing the FEM. The dynamic results are obtained by themodal and modal spectral analysis method. After a comparison between the obtained results by our computer codeand those obtained by ETABS we concluded that there is aUnauthenticated Downloaded 08/10/22 12:16 AM UTC

189International Review of Applied Sciences and Engineering 12 (2021) 2, 183–193Regular Frame0.04Irregular Frame N 010.03Our code FEM ModalETABS ModalOur code FEM ModalETABS Modal0. (m)Ux 0246810121416182002468Times (s)Irregular Frame N 020.02101214161820Times (s)Irregular Frame N 030.015Our code FEM ModalETABS ModalOur code FEM ModalETABS Modal0.0100.01Ux (m)Ux 10121416180202468101214161820Times (s)Times (s)Irregular Frame N 040.008Our code FEM ModalETABS Modal0.010Irregular Frame N 05Our code FEM ModalETABS Modal0.0060.0040.005Ux (m)Ux .008024681012141618Times (s)2002468101214161820Times (s)Fig. 7. The modal dynamic displacements Ux for the regular and irregular frame N8 01, 02, 03, 04 and 05good correlation between these results for several configurations processed. So, we may say that our code gives us fullsatisfaction in terms of the accuracy and richness of theresults obtained.Also, we have concluded some remarks which are summarized as follows:1. Given the irregularity in elevation, the modal displacements have increased.2. With the exception of special cases, the displacementsmeasured by the spectral modal analysis are greater thanthose determined by the modal analysis for the samestructure form and storey level.Unauthenticated Downloaded 08/10/22 12:16 AM UTC

190International Review of Applied Sciences and Engineering 12 (2021) 2, 183–193Irregular Frame N 01Regular Frame665Our Code FEM Modal SpectralOur Code FEM Max ModalETABS Modal Spectral54Level (i)Level (i)433221100.000Our Code FEM Modal SpectralOur Code FEM Max ModalETABS Modal .005Our Code FEM Modal SpectralOur Code FEM Max ModalETABS Modal Spectral5Level (i)Level (i)Our Code FEM Modal SpectralETABS Modal SpectralOur Code FEM Max 00.0150.020Displacement (m)Displacement (m)Irregular Frame N 04Irregular Frame N 0566Our Code FEM Modal SpectralOur Code FEM Max ModalETABS Modal Spectral54Our Code FEM Modal SpectralOur Code FEM Max ModalETABS Modal Spectral4Level (i)Level (i)0.0304333221100.0000.0256450.020Irregular Frame N 03Irregular Frame N 02600.0000.015Displacement (m)Displacement (m)50.0100.0050.0100.015Displacement (m)0.02000.025 0.0000.0050.0100.0150.0200.025Displacement (m)Fig. 8. The spectral modal dynamic displacements for the regular and irregular frame N8 01, 02, 03, 04 and 05Unauthenticated Downloaded 08/10/22 12:16 AM UTC

International Review of Applied Sciences and Engineering 12 (2021) 2, 183–1933. The setbacks in elevation influence the results of displacements – this is perhaps due to the reduction in therigidity of structures.4. Compared with the regular structure, the inter-storeydrift decreases in the case of the irregular structure.5. The rigidity increases with the storey level and decreasesby changing the frame irregularity.0.035Displacement (m)0.0300.0250.0200.0150.0100.0050.000rullaReg 01 04 05 03 02ar Nar Nar Nar Nar NgulgulgulgulgulIrreIrreIrreIrreIrreFrameFig. 9. The spectral displacement of the setback point of each typeof regular and irregular frame N8 01, 02, 03, 04 and 056Our Code FEM Regular FrameOur Code FEM Irregular Frame N 01Our Code FEM Irregular Frame N 02Our Code FEM Irregular Frame N 03Our Code FEM Irregular Frame N 04Our Code FEM Irregular Frame N 055Level (i)432100. drift (%)Fig. 10. Inter-storey drifts of the regular and irregular structureN8 01, 02, 03, 04 and 056Our Code FEM Regular FrameOur Code FEM Irregular Frame N 01Our Code FEM Irregular Frame N 02Our Code FEM Irregular Frame N 03Our Code FEM Irregular Frame N 04Our Code FEM Irregular Frame N 055Level 0KLevel(i) (KN/m)Fig. 11. The Storeys Stiffnesses of the regular and irregular structureN8 01, 02, 03, 04 and 05Therefore, from these results, we can conclude that wewill have to be careful when using this type of irregularstructures, especially in high seismicity areas because thistype has caused enormous damage in past catastrophicearthquakes.Finally, we can develop and extend this work in thefuture so that we could study the finite element modeling ofreinforcement of irregular structures in elevation by anotherbracing system which would avoid their ruin and minimizerigidities.REFERENCES[1] X. Rom ao, A. Costa, and R. Delgado, “Seismic behavior ofreinforced concrete frames with setbacks,” in 13th World Conference on Earthquake Engineering Vancouver, Canada, Aug. 1–6,2004.[2] F. Alba, A. G. Ayala, and R. Bento, “Seismic performance evaluation of plane frames regular and irregular in elevation,” in Proc.,4th European Workshop on the Seismic Behaviour of Irregular andComplex Structures, 2005.[3] G. Ashvin, D. G. Agrawal, and A. M. Pande, “Effect of irregularities in buildings and their consequences,” Int. J. Mod. TrendsEng. Res. (IJMTER), vol. 2, no. 4, 2015.[4] S. K. Abid Sharief, M. Shiva Rama Krishna, and S. V. Surendhar,“A case study on seismic analysis of an irregular structure,” Int. J.Innovative Technol. Exploring Eng. (IJITEE), vol. 8, no. 4, 2019.[5] P. Sarkar, A. M. Prasad, and D. Menon, “Vertical geometric irregularity in stepped building frames,” Eng. Struct., vol. 32,pp. 2175–82, 2010.[6] M. Inel, H. B. Ozmen, and H. Bilgin, “Re-evaluation of buildingdamage during recent earthquakes in Turkey,” Eng. Struct., vol.30, pp. 412–27, 2008.[7] S. J. Kim and A. S. Elnashai, “Characterization of shaking intensitydistribution and seismic assessment of RC buildings for theKashmir (Pakistan) earthquake of October 2005,” Eng. Struct., vol.31, pp. 2998–3015, 2009.[8] A. Habibi and K. Asadi, “Seismic performance of RC framesirregular in elevation designed based on Iranian seismic code,” J.Rehabil. Civil Eng., vol. 1, no. 2, pp. 40–55, 2013.[9] R egles parasismiques Alg eriennes 1999 - Version 2003, DTR-BC248 - CGS. Alger, 2003.[10] Eurocode 8, EN 1998-1-1, Design of Structures for EarthquakeResistance-Part 1: General Rules, Seismic Actions and Rules forBuildings. Brussels (Belgium): CEN, European Committee forStandardization, 2015.[11] J. P. Moehle and A. M. Asce, “Seismic response of verticallyirregular structures,” J. Struct. Eng., vol. 110, no. 9, pp. 2002–14,1984.Unauthenticated Downloaded 08/10/22 12:16 AM UTC

192International Review of Applied Sciences and Engineering 12 (2021) 2, 183–193[12] J. P. Moehle, A. M. Asce, and L. F. Alarcon, “Seismic analysismethods for irregular buildings,” J. Struct. Eng., vol. 112, pp. 35–52,1986.[13] A. G. Costa, C. S. Oliveira, and R. T. Duarte, “Influence of verticalirregularities on seismic response of buildings,” in Proceedings ofthe Ninth World Conference on Earthquake Engineering WCEE,Tokyo, Japan, May, 1988.[14] B. M. Shahrouz and J. P. Moehle, “Seismic response and design ofsetback buildings,” J. Struct. Eng., vol. 116, no. 5, pp. 1423–39,1990.[15] S. L. Wood, “Seismic response of R/C frames with irregular profiles,” J. Struct. Eng., vol. 118, no. 2, pp. 545–66, 1992.[16] C. M. Wong and W. K. Tso, “Seismic loading for buildings withsetbacks,” Can. J. Civil Eng., vol. 21, no. 5, pp. 863–71, 1994.[17] J. H. Cassis and E. Cornejo, “Influence of vertical irregularities inthe response of earthquake resistant structures,” in Proceedings ofthe 11th World Conference on Earthquake Engineering WCEE,Acapulco, Mexico, 1996.[18] E. G. Valmundsson and J. M. Nau, “Seismic response of buildingframes with vertical structural irregularities,” J. Struct. Eng., vols123, no. 30, pp. 30–41, 1997.[19] G. Magliulo, R. Ramasco, and R. Realfonzo, “Seismic behaviorof irregular in elevation plane frames,” in The 12th EuropeanConference on Earthquake Engineering ECEE, London, England,2002.[20] T. L. Karavasilis, N. Bazeos, and D. E. Beskos, “Seismicresponse of plane steel MRF with setbacks: Estimation of inelasticdeformation demands,” J. Construct. Steel Res., vol. 64, no. 6,pp. 644–54, 2008.[21] P. Sarkar, A. M. Prasad, and D. Menon, “Vertical geometricirregularity in stepped building frames,” Eng. Struct., vol. 32, no. 8,pp. 2175–82, 2010.[22] P. Rajeev and S. Tesfamariam, “Seismic fragilities for reinforcedconcrete buildings with consideration of irregularities,” Struct.Saf., vol. 39, pp. 1–13, 2012.[23] S. Varadharajan, V. K. Sehgal, and S. Babita, “Determination ofinelastic seismic demands of RC moment resisting setbackframes,” Arch. Civil Mech. Eng., vol. 13, no. 3, pp. 370–93, 2013.[24] M. Pirizadeh and H. Shakib, “Probabilistic seismic performanceevaluation of non-geometric vertically irregular steel buildings,”J. Construct. Steel Res., vol. 82, pp. 88–98, 2013.[25] R. Roy and S. Mahato, “Equivalent lateral force method forbuildings with setback: Adequacy in elastic range,” EarthquakesStruct., vols 4, no. 6, pp. 685–710, 2013.[26] N. Hamdani, Comportement sismique de structures en portique enb eton arm e irr eguli eres en el evation. Bayonne, France: RencontresUniversitaires de G enie Civil, 2015.[27] S. D. Darshale and N. L. Shelke, “Seismic response control ofvertically irregular R.C.C. Structure using base isolation,” Int. J.Eng. Res., vol. 5, no. 2, pp. 683–9, 2016.[28] A. S. Bhosale, R. Davis, and P. Sarkar, “Vertical irregularity ofbuildings: regularity index versus seismic risk,” ASCE-ASMEJ. Risk Uncertainty Eng. Syst. A: Civil Eng., vol. 3, no. 3, pp. 189–95,2017.[29] R. N. Shelke and U. Ansari, “Seismic analysis of vertically irregularRC building frames,” Int. J. Civil Eng. Technol., vol. 8, no. 1,pp. 155–69, 2017.[30] E. Siva Naveen, N. M. Abraham, and S. D. Anitha Kumari,“Analysis of irregular structures under earthquake loads,” Proced.Struct. Integrity, vol. 14, pp. 806–19, 2019.uneyisi, “Seismic performance evaluation of[31] S. Etlia and E. M. G regular and irregular composite moment resisting frames,” LatinAm. J. Sol. Structures, vol. 17, no. 7, p. e301, 2020.[32] N. Ozisik, Finite Difference Methods in Heat Transfer. CRC Press,1994.[33] G. Beer, I. Smith, and C. Duenser, The Boundary ElementMethod with Programm

concrete framed buildings irregular in elevation subjected to a seismic excitation by the FEM. To find the displacements, two dynamic calculation methods were used: the method of modal analysis and the method of spectral modal analysis. The numerical modeling of these irregular frames represents the originality of this study compared to the work

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An Introduction to Modal Logic 2009 Formosan Summer School on Logic, Language, and Computation 29 June-10 July, 2009 ; 9 9 B . : The Agenda Introduction Basic Modal Logic Normal Systems of Modal Logic Meta-theorems of Normal Systems Variants of Modal Logic Conclusion ; 9 9 B . ; Introduction Let me tell you the story ; 9 9 B . Introduction Historical overview .

This work is devoted to the modal analysis of a pre-stressed steel strip. Two different complementary ap-proaches exist in modal analysis, respectively the theoretical and experimental modal analyses. On the one hand, the theoretical modal analysis is related to a direct problem. It requires a model of the structure.

"fairly standard axiom in modal logic" [3: 471]. However, this is not a "fairly standard" axiom for any modal system. More precisely, it is standard only for modal system S5 by Lewis. Intuitively, this is not the most clear modal system. Nevertheless, this system is typically has taken for the modal ontological proof.

At the Animal Nutrition Group (ANU), a student can conduct research for a thesis with a workload of 18, 21, 24, 27, 30, 33 (Minor thesis), 36 or 39 ECTS (Major thesis). The aim of this thesis research is to train the students’ academic skills by means of an in-depth, scientific study on a subject of interest. With completion of the thesis, you have demonstrated that you can conduct a .