INCREMENTAL DYNAMIC ANALYSIS AND PUSHOVER ANALYSIS. A PROBABILISTIC COMPARISON

Transcription

1 COMPDYN 2 III ECCOMAS Thematc Conference on Computatonal Metho n Structural Dynamcs and Earthquake Engneerng M. Papadrakaks, M. Fragadaks, V. Plevrs (e.) Corfu, Greece, May 2 INCREMENTAL DYNAMIC ANALYSIS AND PUSHOVER ANALYSIS. A PROBABILISTIC COMPARISON Yeudy F. Vargas *, Llus G. Pujades, Alex H. Barbat, Jorge E. Hurtado 2 Unversdad Poltécnca de Cataluña Department of Geotechncal Engneerng and Geoscences, Jord Grona -3, Buldng D2, Campus Norte UPC, 834 Barcelona Span yeudy.felpe.vargas@upc.edu 2 Unversdad Naconal de Colomba Unversdad Naconal de Colomba, Apartado 27, Manzales, Colomba jehurtadog@unal.edu.co Keywor: Pushover analyss, Incremental dynamc analyss, Monte Carlo smulaton, Vulnerablty. Abstract. Capacty-spectrum-based-metho are also used for assessng the vulnerablty and rsk of exstng buldngs. Capacty curves are usually obtaned by means of nonlnear statc analyss. Incremental Dynamc Analyss s another powerful tool based on nonlnear dynamc analyss. Ths method s smlar to the pushover analyss as the nput s ncrementally ncreased but t s dfferent as t s based on dynamc analyss. Moreover, t s well known that the uncertantes assocated to the structural response can be sgnfcant, because the uncertantes nvolved n the mechancal propertes of the materals and the expected sesmc actons are also hghly uncertan. In ths work selected mechancal propertes are consdered as random varables and the sesmc hazard s consdered n a probablstc way. A number of accelerograms of actual European sesmc events have been selected n such a way that ther response spectra ftted well the response spectra provded by the sesmc codes for the zone where the target buldng s constructed. In ths work a fully probablstc approach s tackled by means of Monte Carlo smulaton and t s appled to a detaled study of the sesmc response of a renforced concrete buldng. The buldng s representatve for offce buldngs n Span but the metho used and the results obtaned can be extended to other types of buldngs. The man purposes of ths work are ) to analyze the dfferences when statc and dynamc technques are used and 2) to obtan a measure of the uncertantes nvolved n the assessment of the vulnerablty of structures. The results show that statc based procedures are somehow conservatve and that uncertantes ncrease wth the severty of the sesmc actons and wth the damage. Low damage state fraglty curves have lttle uncertanty whle hgh damage grades fraglty curves show great scatterng.

2 INTRODUCTION Amng to prevent the sesmc rsk, t s necessary to assess the vulnerablty of exstng structures. To do that, several metho have been proposed, startng from dfferent approaches. One s the vulnerablty ndex method n whch the acton s defned from the EMS-98 by macrosesmc ntenstes and structural behavour through a vulnerablty ndex [, 2]. Another hghly used method s based on the capacty spectrum. In ths, the sesmc acton s defned by means of the elastc response spectra and the vulnerablty or fraglty of the buldng by means of the capacty curve; the latter s calculated from an ncremental nonlnear statc analyss, commonly known as "Pushover Analyss" [3, 4 5]. Another tool used to evaluate the performance of structures aganst sesmc actons s the Incremental Dynamc Analyss (IDA) proposed by Vamvatskos & Cornell [6]. The purpose of IDA s to obtan a measure of damage n the structure by ncreasng the ntensty of the acton record, n ths case the peak ground acceleraton. Vamvatskos & Cornell make an nterestng analogy between the PA and the IDA, because both procedures ncreases the load on the structure and measure the response of the system n terms of a control varable whch may be the maxmum placement at the roof, the maxmum nter storey drft, etc. Ths procedure allows obtanng the dynamc response of a structure when the sesmc acton s ncreased. On the other hand, the mechancal propertes of the materals whch consttute the structure and the sesmc acton are random varables and, therefore, the vulnerablty of the buldng s also a random varable. To take nto account the nherent randomness of the problem, t s approprate to use the Monte Carlo method. Therefore, n ths paper, a probablstc comparson between the PA and the IDA s performed when calculatng the vulnerablty of an exstng renforced concrete buldng. The man concluson from ths comparson hghlghts the mportance of measurng the vulnerablty of structures takng nto account that the varables nvolved are random. Ths approach, mxed wth powerful tools to analyze the structure such as the PA and the IDA, provde valuable nformaton that can hardly be obtaned wth other methodologes. 2 BUILDING DESCRIPTION Ths paper analyzes a renforced concrete structure, consstng of columns and waffle slabs, whch s part of the North Campus of the Unversdad Poltécnca de Cataluña. It has 7 levels and 4 spans, the heght s m and the wdth s 22.5 m (see Fgure ). The fundamental perod of the buldng s.97 secon. Ths value s hgher when compared to that of conventonal renforced concrete buldngs, because n the numercal model, the waffle slabs are approxmated wth beams of equvalent nerta and, therefore, are structural elements wde and flat leadng to a reducton of the lateral stffness of the structure. In the calculaton model, the structural elements (equvalent beams and columns) follow an elastc-plastc consttutve law, whch does not take nto account ether hardenng or softenng. Yeldng surfaces are defned by the moment-axal load nteracton dagram n columns and by the moment-angular deformaton nteracton dagram n beams. 2

3 Fgure. Pcture of the buldng omega located n the Unversdad Poltécnca de Cataluña, Barcelona, Span. 3 DAMAGE INDEX BASED ON PUSHOVER ANALYSIS A tool often used to evaluate the behavour of the structures aganst sesmc loa, s the nonlnear statc analyss, commonly called Pushover Analyss (PA). Ths numercal tool conssts n apply a horzontal load to the structure, accordng to a certan pattern of forces, and n ncreasng ts value untl the structural collapse s reached. From ths procedure one obtan a relatonshp between the placement at the roof of the buldng and the base shear, called capacty curve. In ths artcle, due to the probablstc approach, the PA s performed repeatedly, therefore, t s approprate to apply a procedure for obtanng automatcally the horzontal load lmt. For ths, Satyarno [7] proposes the adaptve ncremental nonlnear analyss that establshes the horzontal load lmt as a functon of the tangent fundamental frequency,.e. the frequency assocated wth the frst vbraton mode, whch s beng calculated for each load ncrement. Therefore, n each step s calculated the frst mode of vbraton to determne the shape of the load n heght. A detaled descrpton of ths procedure s found n the manuals of the program Ruaumoko [8] used for calculatng the statc and dynamc nonlnear structural response. As mentoned n the ntroducton, the mechancal propertes of materals are consdered as random varables. To do ths, the values used n the structural desgn for concrete compressve strength fc, and the tensle strength assocated wth steel yeld strength fy, are treated as random varables and usng the nverson method of the cumulatve probablty trbuton curve are generate random samples of these varables. In ths paper, t s assumed that the generated random varables follow a Gaussan probablty functon whose mean and standard devatons are shown n Table. Mean Value (kpa) Standard devaton (kpa) Coeffcent of varaton fc fy 5 5. Table. Features of the Gaussan random varables consdered. For the generated samples, the PA s performed tmes and the capacty curve s obtaned as the random varable shown n Fgure 2. 3

4 9 8 7 Base shear (kn) Dsplacement at the roof (m) Fgure 2. Capacty curves obtaned from the PA, takng nto account the uncertanty n the mechancal propertes of materals. The capacty curves shown n Fgure 2 are transformed nto capacty spectra, whch relate the spectral placement to spectral acceleraton by means of the followng equatons [9]: sd sa PF V W () The subscrpt n equatons () s referred to the appled load ncrements on the structure durng the PA; sd s the spectral placement; s the placement at the roof of the buldng; PF s the modal partcpaton factor of the frst mode of vbraton; sa s the spectral acceleraton; s the base shear; W s the weght of the buldng and s the modal V mass coeffcent of the frst mode of vbraton. On the other hand, the capacty spectrum can be represented n a blnear form, whch s useful for defnng damage states. Assumptons to buld the blnear capacty spectrum are: ) The area under the blnear curve must be equal to the area of the orgnal curve. 2) The coordnates of the pont of maxmum placement must be the same n both curves. 3) The slope of the ntal branch should be equal n both curves. Fgure 3 shows an example of the blnear representaton of the capacty spectrum. Ths can 4

5 be defned completely by the ponts (Dy, Ay) and (Du, Au). These ponts are useful to defne the states of damage, accordng to the procedure descrbed n Lantada et al (29)..25 Spectral acceleraton (g).2.5. (Dy,Ay) (Du,Au).5 Capacty spectrum Blnear representaton Spectral placement (m) Fgure 3. Capacty spectrum and the blnear representaton. Dfferent studes have been proposed to calculate the damage of the structure from the defnton of damage states (), whch are a descrpton of the damage n the structure for a gven spectral placement. For example, HAZUS 99 [] and Rsk EU [], defne 4, namely slght, moderate, extensve and complete. Descrpton of the damage states depen on the type of structure. For example, Accordng to HAZUS, n the case of renforced concrete structures, the slght s descrbed as the begnnng of crackng due to bendng moment or shear n beams and columns. Collapse state consders that the structure reaches an mmnent rsk of collapse. Rsk EU seeks to defne the damage states n smplfed form, startng from the capacty spectrum n a blnear representaton. Based on the values (Dy, Ay) and (Du, Au), the spectral placements for the four damage states threshold are obtaned accordng to the followng equatons: * Dy Dy Dy.25*( Du Dy) Du (2) 5

6 Therefore, after calculatng the capacty spectrum n blnear representaton and applyng equatons 2, t s possble to obtan the damage states threshol as random varables, as shown n Fgure Spectral acceleraton (g) Spectral placement (m) Fgure 4. Damage states as random varables. The mean, standard devaton and coeffcent of varaton of the damage states are shown n Table 2, n whch t can be seen that the coeffcent of varaton of the damage state 4 s greater than that of the nput varables. Ths s due to the fact that the problem s nonlnear and therefore t shows the mportance of the probablstc approach n ths type of analyss. (cm) 2 (cm) 3 (cm) 4 (cm) c.v Table 2. Mean value, standard devaton and coeffcent of varaton of the damage states. After obtanng the damage states as random varables t s possble to calculate the fraglty curves, whch represent the probablty of reachng or exceedng a damage state, n functon of a parameter representng the sesmc acton. In ths work, ths parameter s the spectral placement. To obtan the fraglty curves the followng assumptons must be consdered: ) The probablty that the spectral placements n each damage state threshold,, equals or 6

7 excee the damage state s 5%. 2) The fraglty curves follow a lognormal cumulatve probablty functon descrbed by the followng equaton: sd P / sd Ln (3) where sd s the spectral placement and s the standard devaton of natural logarthm of the varable. 3) The expected sesmc damage n buldngs follows a bnomal probablty trbuton. Fgure 5 shows all fraglty curves calculated after applyng the descrbed procedure P(/sd) Fraglty curves Fraglty curves 2 Fraglty curves 3 Fraglty curves Spectral placement (m) Fgure 5. Fraglty curves as random varables. Snce the probabltes of occurrence of each state of damage are easly obtaned from the fraglty curves, one can calculate the expected damage ndex, DI, whch s the normalzed mean damage state, whch can be nterpreted as a measure of the overall expected damage n the structure. DI n P( ED ) (4) n 7

8 where n s the number of damage states consdered, n ths case 4 and P( ) s the probablty of occurrence of. Fgure 6 shows the ID calculated from the fraglty curves of Fgure 5. The curves of Fgure 6 can be nterpreted as random vulnerablty curves Damage Index Spectral placement (m) Fgure 6. Damage ndex obtaned startng from the PA as random varable. 4 DAMAGE INDEX BASED ON THE INCREMENTAL DYNAMIC ANALYSIS Incremental dynamc analyss allows obtanng the dynamc response of a structure to an earthquake acton. Ths earthquake s scaled to varous PGA. As mentoned above, the purpose of ths artcle s to compare the results obtaned wth the methodology based on the capacty spectrum wth the ncremental dynamc analyss. Accordng to the probablstc approach t s necessary to obtan the sesmc acton as a random varable. To do that, 2 earthquakes have been selected from two databases, one from Span and the other from Europe, whose elastc response spectra are compatble wth elastc response spectrum taken from Eurocode 8. In ths case, the elastc spectrum type and sol D s selected. Fgure 9 shows the spectra of the selected earthquakes, ther average value, and the spectrum type sol D, taken from Eurocode 8. 8

9 6 5 Elastc spectrum selected from Eurocode 8 Mean of the actual spectra Actual elastc spectra 4 sa (g) Perod (s) Fgure 7. Selected spectra of the accelerograms that are compatble wth spectrum type sol D of Eurocode 8. After selectng the accelerograms, the dynamc response of the structure s calculated, for dfferent PGA untl a maxmum of.32 g, at ntervals of.4 g. In each run of the nonlnear dynamc analyss, the damage ndex proposed by Park & Ang [2] s calculated and, also, the maxmum placement at the roof of the buldng, allowng to compare these results wth those obtaned prevously from statc procedure. Fgure 8 shows the results obtaned wth both metho, and shows that the damage ndex obtaned wth the procedure based on the PA s conservatve compared to the results obtaned wth the procedure based on IDA. However, when the damage ndex s close to, smlar values are obtaned wth both procedures. On the other hand, t can be seen n the curves obtaned wth the PA procedure that the structural damage begns for a smaller spectral placement than n the case of the IDA procedure and that, n both cases, slopes are smlar. Ths means that the PA curves are shfted respectng the IDA curves what could be easly adjusted by changng the damage states coeffcents. It s mportant to note the large scatter n both cases, showng the mportance of assessng the vulnerablty of structures from a probablstc perspectve, whchever procedure s used. 9

10 Damage ndex Damage ndex obtened wth the PA Damage ndex obtaned wth the IDA Spectral placement (m) Fgure 8. Damage ndex obtaned wth statc and dynamc procedures. 5 CONCLUSIONS In ths work, the vulnerablty of a real renforced concrete structure, wth columns and waffle slab has been assessed, takng nto account that the nput varables are random. Two approaches to evaluate the vulnerablty of the buldng have been used. The frst one s based on the pushover analyss and the second one s based on the ncremental dynamc analyss. An mportant concluson s that, despte workng wth advanced structural analyss, these procedures show sgnfcant uncertantes when takng nto account the randomness of the varables assocated wth the problem. It should be emphaszed that n ths work relatvely small coeffcents of varaton for nput varables have been consdered takng nto account the uncertantes that may exst n older structures that dd not have qualty control and have not been desgned accordng to the earthquake-resstant crtera. An mportant concluson s that the results obtaned wth the procedure based on the capacty curve are conservatve when compared wth the results obtaned wth the ncremental dynamc analyss. However, wthn the procedure based on the capacty curve, there are factors such as those gven n equaton 2, whch can be modfed to mprove the correlaton wth the results based on dynamc calculaton. The man concluson of ths paper s that whchever the procedure s used to evaluate the vulnerablty of a structure, t s mportant to note that the nput varables, such as the mechancal propertes of materals and the sesmc acton, are random varables and these generate large uncertantes n the sesmc response, whch can lead to overestmate or to underestmate the real damage that can occur n a structure.

11 6 ACKNOWLEDGEMENTS Ths work was partally funded by the Geographc Insttute of Catalona (IGC), through the mnstry of scence and nnovaton n Span, by the European Commsson and a scholarshp through research projects CGL C3-2/BTE, CGL28-869/BTE, INTERREG: POCTEFA 27-23/ 73/8 y MOVE FT7-ENV REFERENCES [] Barbat A. H., Yépez Moya F. & J.A Canas, Damage scenaros smulaton for rsk assessment n urban zones. Earthquake Spectra. 2(3), , 996. [2] Barbat A. H., Mena U. & F. Yépez, Evaluacón probablsta del resgo sísmco en zonas urbanas. Revsta nternaconal de métodos numércos para cálculo y eño en ngenería. 4(2), , 998. [3] Borz B., Phno R. & H Crowley, Smplfed Pushover analyss for large-scale assessment of RC buldngs. Engneerng Structures. 3:84-82, 28. [4] Barbat A.H., Pujades L.G., Lantada N. & R. Moreno, Sesmc damage evaluaton n urban areas usng the capacty spectrum method: applcaton to Barcelona. Sol Dynamcs and Earthquake Engneerng. 28, , 28. [5] Lantada N, Pujades LG & A.H. Barbat, Vulnerablty ndex and capacty spectrum based metho for urban sesmc rsk evaluaton. A comparson. Natural Hazar. 5:5-524, 29. [6] Vamvatskos D. & C.A. Cornell, The Incremental Dynamc Analyss. Earthquake Engneerng and Structural Dynamcs. 3(3): 49-54, 22. [7] Satyarno I., Pushover analyss for the sesmc assessment of renforced concrete buldngs. Doctoral Thess, Department of cvl engneerng, Unversty of Canterbury, 999. [8] Carr, A. J., Ruaumoko-Inelastc Dynamc Analsys Program. Dept. of Cvl Engneerng, Unv. of Canterbury, Chrstchurch, New Zealand, 2. [9] ATC-4, Sesmc evaluaton and retroft of concrete buldngs. Appled Technology Councl, Redwood Cty, Calforna, 996. [] HAZUS-99, Earthquake Loss Estmaton Methodology Earthquake. Techncal Manual, Vol, Federal Emergency Management Agency (FEMA). Washngton D.C, 999. [] RISK-UE, Project of the European Commsson, an advanced approach to earthquake rsk scenaros wth applcatons to dfferent European towns. Contract number: EVK4- CT-2-4, 24. [2] Park, Y-J & Ang, A.H-S, Mechanstc sesmc damage Model for Renforced Concrete. J. Struct. Dv. ASCE. Vol. No. 4. pp , 985.