Nonlinear Static Analysis To Assess Seismic Performance .

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICSFrom the safety point of view the seismic resistantdesign of moment resisting building frames areclassified as Ordinary Moment Resisting Frames,(OMRF), Intermediate Moment Resisting Frames,(IMRF) and Special Moment Resisting Frames,(SMRF) as referred [4,21]. The yield mechanismsadopted in earthquake resistant design are (i)strong column and weak beam, (ii) flexural yielding in beams, (iii) prevent shear failure or yieldingin beams and columns and flexural yielding atbase of beams. The performance based designwhich ensures safety under a specified earthquakeby estimating the capacity against the demand, isbetter approach than conventional code based design. This paper aims to study the behaviour ofmodern code-conforming OMRF and SMRF underdesigned earthquake condition for low and medium rise buildings.5seismic demand for low- and medium-rise momentresisting frames. In the present study, the pushoveranalysis is carried out for load patterns proportional to fundamental mode. A 100% dead loadplus 50% live load is applied prior to the lateralload on the structure.6 Development of user-defined hingeproperties and nonlinear staticanalysisThe analyses had performed using “SAP2000”,adopting a member-by-member modelling approach. Inelastic beam and column members aremodelled as elastic elements with plastic hinges attheir ends, the effective rigidity of beams is takenequal to 40% of the gross section rigidity (EIg)while for columns as 80% [3]. The moment rotation characteristics of the plastic hinges are estimated from section analysis using appropriatenon-linear constitute laws for concrete and steel.Generally the deformations are quantified and expressed in terms of chord rotations. The lumpedplasticity approach is commonly used in SAP2000for modelling deformation capacity estimates. Thevarious parameters which are directly related withthese deformations are i) steel ductility, ii) bar pullout from the anchorage zone, iii) axial load ratio,iv) shear-span ratio and v) concrete strength. Ananalytical procedure based on Eurocode-8 is usedto study the deformation capacity of beams andcolumns in terms of yield, plastic and ultimate rotations (θu, θpl, θy) and it defines the state of damage in the structure through three limit states of theNEHRP Guidelines (1997) and FEMA 356 (2000),namely i) Limit State “Near Collapse” (NC) level,corresponding to the “Collapse prevention”(CP)level ii) Limit State of “Significant Damage” (SD)level , corresponding to the “ Life Safety”(LS)level and to the single performance level for whichnew structures are designed according to currentIndian seismic design code iii) Limit State ofDamage Limitation (DL) level, corresponding tothe “ Immediate Occupancy” (IO) level. The driftor chord rotation of a member over the shear span(Ls) is a primary parameter which captures the macroscopic behaviour of the member. FEMA guidelines imply values of yield rotation approximatelyequal to 0.005 rad for RC beams and columns, orto 0.003 rad for walls, to be added to plastic hingerotations for conversion into total rotations, whichare approximately equal to the chord rotation θ ordrift of the shear span. According to these codeschord rotation θ is the summation of yield rotationNonlinear Static AnalysisThe understanding of structural behaviour is greatly facilitated by a study of the static loaddeformation responses that identify the elastic andinelastic behaviour characteristics of the structures.The nonlinear static analysis (pushover analysis) isgaining popularity for this purpose. In the pushover analysis, non-linear finite element model of agiven structure (eg. a building frame) subjected togravity loads, is laterally loaded until either a predefined target displacement is met, or the modelcollapses. The reliable post-yield material modeland inelastic member deformations are extremelyimportant in nonlinear analysis. The evaluation isbased on an assessment of important parameters,including global drift, inter-storey drift, inelasticelement deformations (either absolute or normalized with respect to yield value), deformations between elements, and element and connectionforces (for elements and connections that cannotsustain inelastic deformations). The inelastic staticpushover analysis can be viewed as a method forpredicting seismic force and deformation demands,whichaccounts in an approximate manner forthe redistribution of internal forces occurring dueto inertia forces that no longer can be resistedwithin the elastic range of structural behavior. Thetwo key steps in applying this method, i.e. lateralforce distribution and target displacement arebased on the assumption that the structural response is mainly from the fundamental mode, andthat the mode shapes remain unchanged afterstructure gets into the inelastic region. The nonlinear static analysis provides accurate estimate ofE-ISSN: 2224-3429A. Cinitha, P. K. Umesha, Nagesh R. Iyer42Issue 1, Volume 7, January 2012

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS(θy) plus plastic rotation (θP). Acceptable limitingvalues of these plastic rotations have been specified for primary or secondary components of thestructural system under collapse prevention earthquake as a function of the type of reinforcement,axial and shear force levels and detailing of RCmembers. For primary components acceptablechord rotations or drifts for collapse preventionearthquake are taken as 1.5 times lower than theultimate drifts or rotations. For life safety earthquake, the acceptable chord rotations or drifts forprimary and secondary components are taken asabout 1.5 or 2 times lower than the ultimate rotations or drifts. The yield, plastic and ultimate rotation capacities in terms of non-dimensionalnumbers is estimated. User defined P-M-M (P-MM hinges are assigned at the ends of columnmembers which are subjected to axial force andbending moments) and M3 (M- hinges are assigned at the ends of beam members which aresubjected to bending moments) curves are developed using the rotation capacities of members/elements. The default-hinge option inSAP2000 assumes average values of hinge properties instead of carrying out detailed calculation foreach member. The default-hinge model assumesthe same deformation capacity for all columns regardless of their axial load and their weak andstrong axis orientation. Hence nonlinear static analyses are carried out using user- defined plastichinge properties. Definition of user- defined hingeproperties requires moment rotation characteristicsof each element. Panagiotakos and Fardis, 2001defined the yield curvature φy as the point thatmarks onset of nonlinearity in the momentcurvature diagram (owing to either yielding of tension reinforcement or nonlinearity in concrete- forcompressive strains exceeding 90% of the strain atpeak stress of uni-axially loaded concrete):ϕ y min{fyE s ( 1 k y )dA ρ ρ' ρv 1.8f c}E ck ydA ρ ρ' ρ v bdf yNbdf y(2)NN ρ ρ' ρ v ε c Es bd1.8nbdfc 'B ρ ρ ' δ ' 0 .5 ρ v ( 1 δ ' )(3)Considering the lower yield curvature, the yieldmoment is computed asMybd 3 k y φy Ec 22k E ρ 0.5( 1 δ ' ) y s 1 k y ρ k y δ ' ρ' v 1 δ '63 2 ) (()( ) (1 δ ) ' (4) The deformation corresponding to chord rotationat yield, plastic and ultimate rotations areε y d bf y h L αvz θy φy v 0 .00135 1 1 .5 d d ' 6 f3L v c(5)plθum() max 0.01; ω' 1 0.0145 0.25 ν γel max (0.01; ω ) ()0.3 L f c0.2 v h 0.3525f yw αρ sx f c (1.275100 ρ d)(6)θ um () max 0.01; ω 10.016 0.3ν fc γ el max(0.01; ω) ()'0.225 Lv h 0.3525f αρ sx yw f c (1.25100 ρ d)(7)The confinement effectiveness factor is2 s s bi α 1 h 1 h 1 2bc 2 hc 6 bc hc (8)The moment-rotation analysis are carried outconsidering section properties and a constant axialload on the structural element. In the developmentof user-defined hinges for beams, axial forces areassumed to be zero and for the columns they areassumed to be equal to maximum load due to several possible combinations considered while designing. Following, the calculation of the ultimaterotation capacity of an element, acceptance criteriaare defined and labeled as IO, LS and CP as shownin Fig 2 .The typical user defined M3 and P-M-Mhinge used for the analysis are shown in Fig 3.This study defines these three points as 0.2 , 0.5 and 0.9 . Where, is the length of plastic hingeplateau.(1)The compression zone depth at yield ky (normalized to d) is k y ( n 2 A2 2nB )1 / 2 nA , in whichn Es/Ec and A, B are given by Eq. (2) or (3) , depending on whether yielding is controlled by theyielding of tension steel or by nonlinearity in thecompression zone;E-ISSN: 2224-3429NB ρ ρ ' δ ' 0 .5 ρ v ( 1 δ ' ) ';A. Cinitha, P. K. Umesha, Nagesh R. IyerThe acceptance criteria for performance with inthe damage control performance range are obtained by interpolating the acceptance criteria provided for the IO and the LS structural performancelevels. Acceptance Criteria for performance within the limited safety structural performance rangeare obtained by interpolating the acceptance criteria provided for the life safety and the collapse43Issue 1, Volume 7, January 2012

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICSprevention structural performance levels. A targetperformance is defined by a typical value of roofdrift, as well as limiting values of deformation ofthe structural elements. To determine whether abuilding meets performance objectives, responsequantities from the pushover analysis are considered with each of the performance levels.8A. Cinitha, P. K. Umesha, Nagesh R. IyerPlastic Hinge LengthPlastic hinges form at the maximum moment regions of RC members. The accurate assessment ofplastic hinge length is important in relating thestructural level response to member level response.The length of plastic hinge depends on many factors. The following is a list of important factorsthat influence the length of a plastic hinge 1) levelof axial load 2) moment gradient 3) level of shearstress in the plastic hinge region 4) mechanicalproperties of longitudinal and transverse reinforcement 5) concrete strength and 6) level of confinement and its effectiveness in the potentialhinge region. From the literature the followingexpressions are adopted for the present studyL p 0.18L s 0.025a sl d b f y(9)Fig. 3 Typical User- Defined moment-rotationhinge propertiesNote:SF is scale w.r.t yield pointL p 0.8h 0.025a sl d b f y(10)L p 0 .5 h(11)7.Lp 0.08L 0.022 fy dbl 0.044 fy dbl(12)Evaluation of Seismic Performance of BuildingsThe seismic performance of a building is measuredby the state of damage under a certain level ofseismic hazard. The state of damage is quantifiedby the drift of the roof and the displacement of thestructural elements. Pushover analysis is a nonlinear static analysis in which the magnitude of thelateral load is gradually increased, maintaining apredefined distribution pattern along the height ofthe building. At each step, the base shear and theroof displacement relationship are plotted to generate the pushover/capacity curve. It gives an insight into the maximum base shear that thestructure is capable of resisting. Building performance level is a combination of the performancelevels of the structure and the non-structural components. The performance level describes a limiting damage condition which may be consideredsatisfactory for a given building with specificground motion. The three global performance levels (FEMA356) considered are as follows. i) Immediate Occupancy: Transient drift is about 1% ornegligible permanent drift, ii) Life Safety: Transient drift is about 2% or 1% permanent drift, iii)Collapse Prevention: 4% transient drift or permanent drift.E-ISSN: 2224-3429Lpl 0.1LV 0.17h 0.24d bl f yfc(13)The nonlinear static analyses are carried out fortwo designed cases of low and medium rise buildings, in each case separate analyses were carriedout by varying the plastic hinge length estimatedthrough the above mentioned expressions and thustotally five cases are studied. They are namelycase1, case2, case3, case4 and case5 corresponding to Eq.9-13. The capacity curves observed ineach case are shown in Fig.4-7.The roof displacement obtained in this studyobviously show that the demands of 4-storeybuildings are higher than those of 6-storey ones.Therefore, it is difficult to precisely estimatewhich building group is more vulnerable during aseismic event. However SMRF building showshigher capacity compared to OMRF. The studyalso reveals that the amount of transverse reinforcement plays an important role in seismic performance of buildings, as the amount of transversereinforcement increases the sustained damage decreases. A profound variation in capacity and displacement are brought out by varying the plastichinge length and designing the building as OMRF44Issue 1, Volume 7, January 2012

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICSA. Cinitha, P. K. Umesha, Nagesh R. Iyerand SMRF. Table 1 shows the inelastic responsedisplacements of the frame. It is observed that inelastic displacement of all the frames are withincollapse prevention.Base Shear (kN)Table.1 Inelastic response displacements (storey drifts in 20.4Roof Displacement (m)Fig.6 Capacity curves of six storey –OMRFCase33000250020001500100050000.6Fig.7 Capacity curves of six storey-SMRFFig.4 Capacity curves of four storey –OMRFCase1Case2Case4Case59The vulnerability index is a measure of the damage in a building [11] obtained from the pushoveranalysis. It is defined as a scaled linear combination (weighted average) of performance measuresof the hinges in the components, and is calculatedfrom the performance levels of the componentsat the performance point or at the point of termination of the pushover analysis. The vulnerabilityindex of a building is assessed with the expressionas followsCase33000Base Shear (kN)Vulnerability Analysis2500200015001000500000.10.20.3VI bldg 0.4Displacement (m)Fig.5 Capacity curves of four storey- SMRFE-ISSN: 2224-34291 . 5 ic N ic x i bj N bj x j icN bj N(14)Where N ic and N bj are the numbers of hingesin colunmns and beams, respectively, for the ithand jth performance range. A weightage factor (xi )is assigned for columns and (xj) is assigned for45Issue 1, Volume 7, January 2012

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICSbeams to each performance range, the weightagefactor is shown in Table.2 .Table 3 Vulnerability Indexear Static AnalysisDetails4- sto- 4reystoreyOMRF ase40.2020.127Case50.0160.188VIbldg is a measure of the overall vulnerabilityof the building. A high value of VIbldgreflectspoor performance of the building. However, thisindex may not reflect a soft storey mechanism.Table.2 Weightage Factors for Performance RangeSerialPerformanceWeightageNumberRange (i)Factor(xi)1 -D,D-E, and E9 N icxic N ici(15)Where N ic is the number of column hinges in thestorey under investigation for a particular performance range. In a given building, the presenceof soft/weak storey is reflected by a relatively highvalue of VI storey for that storey, in relation to theother storeys. Thevulnerability Index of thebuildings studied is shown in the Table. 3. Thevulnerability index of storey (VI storey) is observedto be almost very neglibible in the case of fourstorey building. Where as it is considerable in thecase of 6-storey OMRF building, where columndamages are observed in the ground floor itself.From the study it is apparent that, the OMRFframed buildings are more vulnerable than SMRFand storey vulnerability index of zero indicate thatmost of the hinges are formed in beams rather thanin columns.E-ISSN: 2224-3429based on Nonlin6- F0.00110.0170.0170.05130.054ConclusionsThis study has illustrated the nonlinear static analysis responses of OMRF and SMRF buildingframes under designed ground motions. The capacity against demand is observed significantlyhigher for SMRF building frames compared toOMRF. The user defined hinge definition and development methodology is also described. Theuser- defined hinges takes into account the orientation and axial load level of the columns comparedto the default hinge. The influence of plastic hingeon capacity curve is brought out by deploying fivecases of plastic hinge length. The study revealsthat plastic hinge length has considerable effectson the displacement capacity of frames. Based onthe analysis results it is observed that inelastic displacement of the modern code-conforming building frames are within collapse prevention level.The vulnerability index which is a measure ofdamage is estimated for both SMRF and OMRFare presented for 4- and 6-storey buildings. Fromthe study it is apparent that, the OMRF framedbuildings are more vulnerable than SMRF. Thevulnerability index of the building quantitativelyexpress the vulnerability of the building as such,where as storey vulnerability index assist to locatethe columns in the particular storey in which significant, slight or moderate level of damages havetaken place.A soft storey mechanism is difficult to tracewith this method. A storey vulnerability index(VI storey ) defined to quantify the possibility of asoft/weak store

based on Eurocode 8 and incorporated the same in pushover analysis using SAP2000. The nonlinear static analysis is carried out for load patterns proportional to fundamental mode. The analysis gives an estimate of seismic capacity of the structural system and its components based on its material characteristics and detailing of member dimensions.