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**CALCULATED vs MEASURED**

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ENERGY DISSIPATION

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**IMPLICIT NONLINEAR BEHAVIOR**

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**STEEL STRESS STRAIN RELATIONSHIPS**

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INELASTIC WORK DONE!

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**CAPACITY DESIGN STRONG COLUMNS & WEAK BEAMS IN FRAMES**

REDUCED BEAM SECTIONS LINK BEAMS IN ECCENTRICALLY BRACED FRAMES BUCKLING RESISTANT BRACES AS FUSES RUBBER-LEAD BASE ISOLATORS HINGED BRIDGE COLUMNS HINGES AT THE BASE LEVEL OF SHEAR WALLS ROCKING FOUNDATIONS OVERDESIGNED COUPLING BEAMS OTHER SACRIFICIAL ELEMENTS

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**STRUCTURAL COMPONENTS**

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**MOMENT ROTATION RELATIONSHIP**

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**IDEALIZED MOMENT ROTATION**

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PERFORMANCE LEVELS

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**IDEALIZED FORCE DEFORMATION CURVE**

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ASCE 41 BEAM MODEL These 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.

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**ASCE 41 ASSESSMENT OPTIONS**

Linear Static Analysis Linear Dynamic Analysis (Response Spectrum or Time History Analysis) Nonlinear Static Analysis (Pushover Analysis) Nonlinear Dynamic Time History Analysis (NDI or FNA)

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**STRENGTH vs DEFORMATION**

ELASTIC STRENGTH DESIGN - KEY STEPS CHOSE DESIGN CODE AND EARTHQUAKE LOADS DESIGN CHECK PARAMETERS Stress/BEAM MOMENT GET ALLOWABLE STRESSES/ULTIMATE– PHI FACTORS CALCULATE STRESSES – Load Factors (ST RS TH) CALCULATE STRESS RATIOS INELASTIC DEFORMATION BASED DESIGN -- KEY STEPS CHOSE PERFORMANCE LEVEL AND DESIGN LOADS – ASCE 41 DEMAND CAPACITY MEASURES – DRIFT/HINGE ROTATION/SHEAR GET DEFORMATION AND FORCE CAPACITIES CALCULATE DEFORMATION AND FORCE DEMANDS (RS OR TH) CALCULATE D/C RATIOS – LIMIT STATES

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**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.

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**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.

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F-D RELATIONSHIP

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DUCTILITY LATERAL LOAD Partially Ductile Ductile Brittle DRIFT

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**ASCE 41 - DUCTILE AND BRITTLE**

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**FORCE AND DEFORMATION CONTROL**

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BACKBONE CURVE

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**HYSTERESIS LOOP MODELS**

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STRENGTH DEGRADATION

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**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.

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**BEAM END ROTATION MODEL**

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PLASTIC HINGE MODEL It is assumed that all inelastic deformation is concentrated in zero-length plastic hinges. The deformation measure for D/C is hinge rotation.

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PLASTIC ZONE MODEL The 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).

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**ASCE 41 CHORD ROTATION CAPACITIES**

LS CP Steel Beam qp/qy = 1 qp/qy = 6 qp/qy = 8 RC Beam Low shear High shear qp = 0.01 qp = 0.005 qp = 0.02 qp = 0.025

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**COLUMN AXIAL-BENDING MODEL**

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**STEEL COLUMN AXIAL-BENDING**

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**CONCRETE COLUMN AXIAL-BENDING**

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SHEAR HINGE MODEL

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PANEL ZONE ELEMENT Deformation, 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).

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**BUCKLING-RESTRAINED BRACE**

The BRB element includes “isotropic” hardening. The D-C measure is axial deformation.

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**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

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WALL FIBER MODEL

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**ALTERNATIVE MEASURE - STRAIN**

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**NONLINEAR SOLUTION SCHEMES**

ƒ iteration ∆ ƒ u 1 2 iteration CONSTANT STIFFNESS ITERATION 3 4 5 6 1 2 ∆ ƒ ∆ u u NEWTON – RAPHSON ITERATION

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CIRCULAR FREQUENCY +Y Y q Radius, R -Y

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THE D, V & A RELATIONSHIP u . t ut1 .. Slope = ut1 t1

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**UNDAMPED FREE VIBRATION**

) cos( t u w = + k m & where

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RESPONSE MAXIMA u t ) cos( w = 2 - & sin( max

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**BASIC DYNAMICS WITH DAMPING**

u & - = w + x 2 M Ku C t C K g u &

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**RESPONSE FROM GROUND MOTION**

ug .. 2 ug2 ug1 1 t1 t2 Time t & A B t u g = + - x w

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**DAMPED RESPONSE { [ & ] cos ( )] sin } e u B t A = - + w Bt x [u ) xw**

1 2 Bt x [u ) 3

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SDOF DAMPED RESPONSE

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**RESPONSE SPECTRUM GENERATION**

Earthquake Record 0.40 0.20 16 GROUND ACC, g 0.00 14 -0.20 12 -0.40 10 0.00 1.00 2.00 3.00 4.00 5.00 6.00 DISPLACEMENT, inches 8 TIME, SECONDS 6 T= 0.6 sec 4 4.00 2 2.00 DISPL, in. 0.00 2 4 6 8 10 -2.00 PERIOD, Seconds -4.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 T= 2.0 sec 8.00 Displacement Response Spectrum 5% damping 4.00 DISPL, in. 0.00 -4.00 -8.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00

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**SPECTRAL PARAMETERS S PS w = V d a v DISPLACEMENT, in. PERIOD, sec**

4 8 12 16 2 6 10 PERIOD, sec DISPLACEMENT, in. 20 30 40 VELOCITY, in/sec 0.00 0.20 0.40 0.60 0.80 1.00 ACCELERATION, g d V S PS w = v a

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**Spectral Acceleration, Sa**

THE ADRS SPECTRUM ADRS Curve Spectral Displacement, Sd Spectral Acceleration, Sa 2.0 Seconds 1.0 Seconds 0.5 Seconds Period, T RS Curve

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THE ADRS SPECTRUM

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**ASCE 7 RESPONSE SPECTRUM**

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PUSHOVER

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THE LINEAR PUSHOVER

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**EQUIVALENT LINEARIZATION**

How far to push? The Target Point!

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**DISPLACEMENT MODIFICATION**

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**DISPLACEMENT MODIFICATION**

Calculating the Target Displacement d = C0 C1 C2 Sa Te2 / (4p2) C0 Relates spectral to roof displacement C1 Modifier for inelastic displacement C2 Modifier for hysteresis loop shape

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**LOAD CONTROL AND DISPLACEMENT CONTROL**

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P-DELTA ANALYSIS

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**P-DELTA DEGRADES STRENGTH**

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**P-DELTA EFFECTS ON F-D CURVES**

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**THE FAST NONLINEAR ANALYSIS METHOD (FNA)**

NON LINEAR FRAME AND SHEAR WALL HINGES BASE ISOLATORS (RUBBER & FRICTION) STRUCTURAL DAMPERS STRUCTURAL UPLIFT STRUCTURAL POUNDING BUCKLING RESTRAINED BRACES

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RITZ VECTORS

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FNA KEY POINT The Ritz modes generated by the nonlinear deformation loads are used to modify the basic structural modes whenever the nonlinear elements go nonlinear.

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**ARTIFICIAL EARTHQUAKES**

CREATING HISTORIES TO MATCH A SPECTRUM FREQUENCY CONTENTS OF EARTHQUAKES FOURIER TRANSFORMS

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**ARTIFICIAL EARTHQUAKES**

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MESH REFINEMENT

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**APPROXIMATING BENDING BEHAVIOR**

This is for a coupling beam. A slender pier is similar.

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**ACCURACY OF MESH REFINEMENT**

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PIER / SPANDREL MODELS

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**STRAIN & ROTATION MEASURES**

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**STRAIN CONCENTRATION STUDY**

Compare calculated strains and rotations for the 3 cases.

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**STRAIN CONCENTRATION STUDY**

No. of elems Roof drift Strain in bottom element Strain over story height Rotation over story height 1 2.32% 2.39% 1.99% 2 3.66% 2.36% 1.97% 3 4.17% 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.

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**DISCONTINUOUS SHEAR WALLS**

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**PIER AND SPANDREL FIBER MODELS**

Vertical and horizontal fiber models

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RAYLEIGH DAMPING The 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.

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RAYLEIGH DAMPING For 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.

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**& & & √ √ √ √ & & + & & ( b ) & a b a b u M + C u Ku = u M + a M + K u**

RAYLEIGH DAMPING t u M & + C u & + Ku = t u M & + & ( a M + b K ) u + Ku = u & C u & K + + u = M M u & + u & 2 2 x w + w u = w x 2 M = C ; a M b K x C C C = = = = + 2 M w √ M K √ 2 M 2 K M 2 √ M K 2 √ M K a b w x = + 2 w 2

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**Higher Modes (high w) = b**

RAYLEIGH DAMPING Higher Modes (high w) = b Lower Modes (low w) = a To get z from a & b for any w √ M K = 2p T = ; w To get a & b from two values of z1& z2 z1 = a 2w1 + b w1 2 Solve for a & b z2 = a 2w2 + b w2 2

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**DAMPING COEFFICIENT FROM HYSTERESIS**

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**DAMPING COEFFICIENT FROM HYSTERESIS**

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A BIG THANK YOU!!!

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UNIT - IV PLASTIC ANALYSIS OF STRUCTURES

UNIT - IV PLASTIC ANALYSIS OF STRUCTURES

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