11 CAPACITY DESIGN STRONG COLUMNS & WEAK BEAMS IN FRAMES REDUCED BEAM SECTIONSLINK BEAMS IN ECCENTRICALLY BRACED FRAMESBUCKLING RESISTANT BRACES AS FUSESRUBBER-LEAD BASE ISOLATORSHINGED BRIDGE COLUMNSHINGES AT THE BASE LEVEL OF SHEAR WALLSROCKING FOUNDATIONSOVERDESIGNED COUPLING BEAMSOTHER SACRIFICIAL ELEMENTS
17 ASCE 41 BEAM MODELThese capacities are for compact, laterally braced beams. The capacities are smaller for non-compact beams.The capacities in ASCE 41 are the same for all steel strengths and all depth-to-span ratios. Intuitively this looks like an over-simplification. The ASCE 41 recommendations are an excellent starting point, but the values will probably be revised as more experience is gained.Note that these capacities assume a symmetrical beam with no transverse load. They must be used with caution if the beam is unsymmetrical or if there are significant transverse loads.
18 ASCE 41 ASSESSMENT OPTIONS Linear Static AnalysisLinear Dynamic Analysis (Response Spectrum or Time History Analysis)Nonlinear Static Analysis (Pushover Analysis)Nonlinear Dynamic Time History Analysis(NDI or FNA)
19 STRENGTH vs DEFORMATION ELASTIC STRENGTH DESIGN - KEY STEPSCHOSE DESIGN CODE AND EARTHQUAKE LOADSDESIGN CHECK PARAMETERS Stress/BEAM MOMENTGET ALLOWABLE STRESSES/ULTIMATE– PHI FACTORSCALCULATE STRESSES – Load Factors (ST RS TH)CALCULATE STRESS RATIOSINELASTIC DEFORMATION BASED DESIGN -- KEY STEPSCHOSE PERFORMANCE LEVEL AND DESIGN LOADS – ASCE 41DEMAND CAPACITY MEASURES – DRIFT/HINGE ROTATION/SHEARGET DEFORMATION AND FORCE CAPACITIESCALCULATE DEFORMATION AND FORCE DEMANDS (RS OR TH)CALCULATE D/C RATIOS – LIMIT STATES
20 STRUCTURE and MEMBERS For a structure, F = load, D = deflection. For a component, F depends on the component type, D is the corresponding deformation.The component F-D relationships must be known.The structure F-D relationship is obtained by structural analysis.
21 FRAME COMPONENTS For each component type we need : Reasonably accurate nonlinear F-D relationships.Reasonable deformation and/or strength capacities.We must choose realistic demand-capacity measures, and it must be feasible to calculate the demand values.The best model is the simplest model that will do the job.
29 ASCE 41 DEFORMATION CAPACITIES This can be used for components of all types.It can be used if experimental results are available.ASCE 41 gives capacities for many different components.For beams and columns, ASCE 41 gives capacities only for the chord rotation model.
31 PLASTIC HINGE MODELIt is assumed that all inelastic deformation is concentrated in zero-length plastic hinges.The deformation measure for D/C is hinge rotation.
32 PLASTIC ZONE MODELThe inelastic behavior occurs in finite length plastic zones.Actual plastic zones usually change length, but models that have variable lengths are too complex.The deformation measure for D/C can be :- Average curvature in plastic zone.- Rotation over plastic zone ( = average curvature x plastic zone length).
38 PANEL ZONE ELEMENTDeformation, D = spring rotation = shear strain in panel zone.Force, F = moment in spring = moment transferred from beam to column elements.Also, F = (panel zone horizontal shear force) x (beam depth).
39 BUCKLING-RESTRAINED BRACE The BRB element includes “isotropic” hardening.The D-C measure is axial deformation.
40 BEAM/COLUMN FIBER MODEL The cross section is represented by a number of uni-axial fibers.The M-y relationship follows from the fiber properties, areas and locations.Stress-strain relationships reflect the effects of confinement
76 STRAIN CONCENTRATION STUDY Compare calculated strains and rotations for the 3 cases.
77 STRAIN CONCENTRATION STUDY No. of elemsRoof driftStrain in bottom elementStrain over story heightRotation over story height12.32%2.39%1.99%23.66%2.36%1.97%34.17%2.35%1.96%The strain over the story height is insensitive to the number of elements.Also, the rotation over the story height is insensitive to the number of elements.Therefore these are good choices for D/C measures.
79 PIER AND SPANDREL FIBER MODELS Vertical and horizontal fiber models
80 RAYLEIGH DAMPINGThe aM dampers connect the masses to the ground. They exert external damping forces. Units of a are 1/T.The bK dampers act in parallel with the elements. They exert internal damping forces. Units of b are T.The damping matrix is C = aM + bK.
81 RAYLEIGH DAMPINGFor linear analysis, the coefficients a and b can be chosen to give essentially constant damping over a range of mode periods, as shown.A possible method is as follows :Choose TB = 0.9 times the first mode period.Choose TA = 0.2 times the first mode period.Calculate a and b to give 5% damping at these two values.The damping ratio is essentially constant in the first few modes.The damping ratio is higher for the higher modes.
82 & & & √ √ √ √ & & + & & ( b ) & a b a b u M + C u Ku = u M + a M + K u RAYLEIGH DAMPINGtuM&+Cu&+Ku=tuM&+&(aM+bK)u+Ku=u&Cu&K++u=MMu&+u&22xw+wu=wx2M=C;aMbKxCCC====+2Mw√MK√2M2KM2√MK2√MKabwx=+2w2
83 Higher Modes (high w) = b RAYLEIGH DAMPINGHigher Modes (high w) = bLower Modes (low w) = aTo get z from a & b for anyw√MK=2pT=;wTo get a & b from two values of z1& z2z1=a2w1+b w12Solve for a & bz2=a2w2+b w22