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Nonlinear static analysisPUSHOVERAdrian DOGARIUEuropean Erasmus Mundus Master CourseSustainable Constructions under Natural Hazardsand Catastrophic Events520121-1-2011-1-CZ-ERA MUNDUS-EMMC

Structural analysis for seismic assesment Elastic analysis– Lateral force method– Modal response spectrum analysis(spectral analysis)– Modal time history analysis– Linear dynamic analysis Inelastic analysis– Nonlinear static analysis (pushover)– Nonlinear dynamic analysisConventional designAdvanced design

Structural model Frames structures can be model using linear elements (beams,columns, braces) connected in nodes Modelling of inelastic behavior of structural components must beaccounted to perform a inelastic structural analysis Software:– SAP 2000,

Dissipative zones Different plastic mechanisms are possible, depending on thetype of structural action developed Plastic hinge:– Bending– Shear– Tension Concentrated plasticity model can be generalized for all typesof action (bending, shear, axial)

Beam-column with plastic hinges Axial force affects the moment capacity of the cross-section itis necessary to account for the axial force – bending momentinteraction for members subject to bending moments and axialforces

Beam-column with plastic hinges Bending moment – axial force interaction inplastic hinges (concentrated plasticity):bending moment capacity affected by the axialforce, but only plastic rotations are assumedto occur– Interaction neglected: elements with smallaxial force– Interaction curve (surface): steel memberssubjected to bending moment and axialforce (A 0.1Py for bending about thestrong axis of double T cross-sections)– Interaction curve (surface): reinforcedconcrete members subjected to bendingmoment and axial force

Nonlinear static analysis (pushover) Nonlinear static analysis under constant gravity loading andmonotonically increasing lateral forces Control elements:pushover– Base shear forcecurve– Control displacement (top displacement) Provides the capacity of the structure, and does not give directlythe demands associated with a particular level of seismic actionDGGGGF43FF2F11234

Nonlinear static analysis (pushover) Assumes that response is governed by a single mode of vibration,and that it is constant during the analysis Distribution of lateral forces (applied at storey masses):– modal (usually first mode – inverted triangle)– uniform: lateral forces proportional to storey massesFmFm

Shape of Lateral force Total base forceFb γ I ,e Sd (T1 ) mλ– Sd(T1) - ordinate of the design response spectrumcorresponding to fundamental period T1;– m - total mass of the structure;– γI,e – importance factor;– λ - correction factor (contribution of the fundamental mode ofvibration using the concept of effective modal mass):Lateral force at storey i– Triangular shapemi ziFi Fb N– Uniform shape mi zii 1

Nonlinear static analysis (pushover) Applicable to low-rise regular buildings, where the response isdominated by the fundamental mode of vibration. Application of loading:– Gravity loading: force control– Lateral forces: displacement control Modelling of structural components: inelastic monotonic forcedeformation obtained from envelopes of cyclic response

Nonlinear static analysis (pushover) Represents a direct evaluation of overall structural response, not onlyon an element by element basis Allows evaluation of inealstic deformations – the most relevantresponse quantity in the case of inelastic responseF Allows evaluation of the plastic mechanismα u Fand redundancy of the structure (αu/α1 ratio)α u/α1α1 F "Local" checks:– Interstorey drifts– Strength demands in non-dissipative components– Ductility of dissipative components "Global" checks – failure at the structure level– Failure to resist further gravity loading– Failure of the first vertical element essential for stability of thestructured

Deformation controlled (ductile) /force-controlled actions (brittle) actions Type 1 curve (deformation-controlled or ductile)Type 2 curve (deformation-controlled or ductile)Type 3 curve (brittle or nonductile behavior)Type of response (ductile / brittle) affects:– Modelling of structural component– Performance criteria

Deformation controlled (ductile) /force-controlled actions (brittle) actions Examples of ductile / brittle actions:Type of structureSteel momentresisting framesSteelconcentricallybraced framesSteeleccentricallybraced framesComponentDuctile actionsBrittle actionsbeamsBending (M)Shear (V)columnsMAxial (N), VjointsV (in general)-bracesN-beams-Ncolumns-Nlinks (short)VM, Nbraces-M, N, (V)beams-M, N, Vcolumns-M, N, V

Modelling of components Modelling of ductile components:– A-B segment: elastic response– B-C segment: strain hardening– C-D segment: strength degradation– D-E segment: residual strength Modelling and performance criteria can be specified in terms of:– Absolute deformations (θ or ) or– Normalised deformations (θ/θy or / y)

Performance criteria: components The degree to which a structural components fulfil a performancecriteria is established based on the demand to capacity ratios. Generally, modelling of components and their performancecriteria are obtained from experimental tests, depending on:– Type of structural component(primary / secondary)– Performance level considered For usual materials andstructural types, data fromliterature or codescan be used (e.g. FEMA 356,EN 1998-3, P100-3)

Performance criteria: components Principle of checking the performance:Ed RdEffect of action (demand) Capacity of the component In case of plastic analysis methods, performance criteria arechecked in terms of deformations for ductile components and interms of forces for brittle components Codes for existing buildings (FEMA 356, EN1998-3):– Ductile components: design deformation capacity– Brittle components: design force strength determined usingcharacteristic material properties

Examples of modelling parameters and performancecriteria (FEMA 356)

Performance criteria: structure The structure shall be provided with at least one continuous loadpath to transfer seismic forces, induced by ground motion in anydirection, from the point of application to the final point ofresistance. All primary and secondary components shall be capable ofresisting force and deformation actions within the applicableacceptance criteria of the selected performance level.

Target displacement in a nonlinear static analysis:the N2 method

N2: performance evaluation procedure1. Initial data– Properties of the structure– Elastic pseudo-accelerationresponse spectrum Sae2. Determination of spectra in ADformat for constant values ofductility, e.g. µ 1, 2, 4, 6, etc. (only ifgraphical representation of themethod is needed)TRµ ( µ 1) 1TCT TCRµ µT TCRµµ10,0TCT

N2: performance evaluation procedure3. Nonlinear static analysis– Assume displacement shape {φ}Note: normalized in such a way thatthe component at the top is equalto 1– Determine vertical distribution oflateral forcesFi [m]{φ} mi φi– Determine base shear (V)–topdisplacement (Dt) relationship byperforming the nonlinear staticanalysisVD

N2: performance evaluation procedure4. Equivalent SDOF system– Determine mass m* mi φi– Transform MDOF quantities (Q) to SDOFquantities (Q*)m*Q* Q/ΓΓ 2 mφi i– Determine an approximate elasto-plasticforce – displacement relationship F*-D*– Determine strength Fy*, yield displacementDy*, and period T****T 2πm DyFy*– Determine capacity diagram Sa-Sd (only ifgraphical representation of the method isneeded)F*Sa *mS d D*

N2: performance evaluation procedure5. Seismic demand for SDOF system– Determine reduction factor Rµ– Determine displacementdemand Sd D*

N2: performance evaluation procedure6. Global seismic demand for MDOF system– Transform SDOF displacement demand tothe top displacement of the MDOF modelDt Γ Sd7. Local seismic demands for MDOF system– Perform pushover analysis of MDOF modelup to the top displacement Dt– Determine local response quantities (e.g.story drifts, rotations θ, etc.) correspondingto Dt8. Performance evaluation– Compare local and global seismic demandswith the capacities for the relevantperformance level

– Linear dynamic analysis Inelastic analysis – Nonlinear static analysis (pushover) – Nonlinear dynamic analysis Conventional design Advanced design. Structural model Frames structures can be model using linear elements (beams, columns, braces) connected in nodes Modelling of inelastic behavior of structural components must be accounted to perform a inelastic structural analysis Software ...