Chapter 3 Direct Displacement Based Design

8/18/2019 Chapter 3 Direct Displacement Based Design

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CHAPTER 3DIRECT DISPLACEMENT-BASED

DESIGN

FUNDAMENTAL CONSIDERATIONSMuhammad Noman

15-UET/PhD-civ-78

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3.1 INTRODUCTION

DDBD (Direct Displacement Based Design)developed recently

Aim: mitigating deficiencies in force-baseddesign (explained by Dr. Syed Saqib already)

Fundamental Philosophy: To design astructure which would achieve a givenperformance limit state under given seismicintensity

The Design Procedure determines the strengthrequired at designated plastic hinges locations toachieve the design aims in terms of defineddisplacement objectives

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3.2 BASIC FORMULATION OF THEMETHOD

The design method isexplained with figure,which considersSDOF representation of frame

buildingKi = Initial ElasticStiffness

rKi= Post Yield

StiffnessKe= Secant stiffness atmaximum displacement∆d

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Force based Design characterizes a structure interms of Elastic, Pre-Yield properties (Initialstiffness Ki, Elastic Damping)

DDBD characterizes the structure by secantstiffness Ke at maximum displacement ∆d, andviscous damping ξ (zeta).

Thus representative of the combined elasticdamping and hysteretic energy absorbed during

inelastic response.

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The effective stiffness K e of equivalent SDOFsystem at maximum displacement can be foundby inverting the normal equation:

K e = 4 ̂ 2 m e / T ê 2

Where me is effective mass T e is effective period

Design base shear force

Vbase =K e ∆d

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3.3 DESIGN LIMIT STATES ANDPERFORMANCE LEVELS

In recent years there has been increased interestin defining seismic performance objectives forstructures.

In vision 2000 document, four performance levelsand four levels of seismic excitation areconsidered.

The performance levels are designated as:

1. Fully Operational 2. Operational

3. Life Safe 3. Near Collapse

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In California, the following levels of Earthquakeare defined:

EQ-I:87% probability in 50 yearsEQ-II: 50% probability in 50 years

EQ-III:10% probability in 50 years

EQ-IV: 2% probability in 50 years

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3.3.1 SECTION LIMIT STATES

A) Cracking Limit State:For Concrete & Masonrymembers the onset of cracking generally marks the pointfor a significant change in stiffness

B) First-Yield Limit State: A second significant change

in stiffness of concrete and masonry members occurs atonset of yield in the extreme tension reinforcement

C) Spalling Limit State: Associated with onset ofnegative incremental stiffness and possibly suddenstrength loss

D) Buckling Limit State:Beyond this state, removaland replacement of member is required

E) Ultimate Limit State: inability to carry imposedloads 10


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3.3.2 STRUCTURE LIMIT STATES

A) Serviceability limit state:This correspondsto the “fully functional” seismic performance level

B) Damage Control limit state: At this limitstate, a certain amount of repairable damage isacceptable, but the cost should be significantlyless than the cost of replacement

C) Survival limit state:Extensive damage mayhave to be accepted, to the extent that it may not

be economically or technically feasible to repairthe structure after the earthquake

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3.4 SINGLE-DEGREE-OF-FREEDOM

STRUCTURES3.4.1 DESIGN DISPLACEMENT FOR A SDOF STRUCTURE

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It is comparatively straight forward to compute designdisplacement from strain limits

Consequently there are two possible limit state curvatures,based on concrete compression and reinforcement tensionrespectively

The lesser will govern the structural design

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3.4.2 YIELD DISPLACEMENT

The yield displacement is required for two reasons:

1: If structural considerations define limitdisplacement

2: In order to calculate viscous damping

D, hc, lw, hs, hb are section depths

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Lb is beam span and hb is beam depth

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3.4.3 EQUIVALENT VISCOUSDAMPING

The design procedure requires relationshipsbetween displacement ductility and equivalentviscous damping. Figure 3.1 (c)

The damping is sum of Elastic damping andhysteretic damping

Hysteretic Damping depends upon hysteresisrule

Normally elastic damping used for:

Concrete structures: 0.05

Steel Structures: 0.02 17


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(A) HYSTERETIC DAMPING

Initial work on hysteretic damping was done byJacobsen who formulated the following equation:

Ah = Area of one complete cycle of forcedisplacement response

Fm and∆m are maximum force and displacement

achieved in stabilized loopThe above equation is related toKe (Secantstiffness) to maximum response. figure 3.8

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A) characteristicsof some isolationsystem,incorporating

frictional sliders, B) variousisolation systems

C) DuctileReinforcedconcrete wall of

column D) ductilereinforcedconcrete framestructures

E) Ductile steelstructure

F) unboundedPost tensionedwith littledamping

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The Dwairi & Kowalsky study represented thehysteretic component of response in the form:

Grant et al. considered a wider range ofhysteretic rules and used complex formulationbetween ductility and viscous damping:

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B) ELASTIC DAMPING

For SDOF systems, elastic damping is used in thedynamic equation of equilibrium

Grant et al., compared results of elasticsubstitute structure analysis with inelastic timehistory results to determine the correction factorto be applied to the elastic damping coefficient forassumptions of either initialstiffness elasticdamping. The equation 3.9 is thus slightlychanged to:

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C) DESIGN CONSIDERATIONS

Example: If a reinforced concrete wall was to be designed for adisplacement ductility of µ = 5 and effective period Te = 2.0 sec.,and an elastic damping ratio of 0.05 (5%) related to tangentstiffness elastic damping. Calculate the appropriate equivalentviscous damping

Solution: from table 3.1: for TT, a= 0.215, b= 0.642, c= 0.824, d=6.444

Hence from equation 3.12

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We list down equations for tangent stiffness elasticdamping only, since this is felt to be the correct structural

simulation, further these equations are for coefficient ofdamping 0.05. for different coefficients, complex equation(like equation 3.12) are used

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E) GENERATION OF INELASTICDISPLACEMENT SPECTRA

Te = Secant Period, Ti = Elastic period, Ru = responsemodification factor

Substituting the reduction factor for elastic damping valuesgreater than 0.05 from eq. (2.8) into the damping ductility

equations eq.(3.17), spectral displacement reduction factors in theform:

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3.4.4 DESIGN BASE SHEAREQUATION

It will be clear that the above approach described can besimplified to a single equation, once the design displacement anddamping have been determined

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3.4.5 DESIGN EXAMPLE 3.3: DESIGNOF A SIMPLE BRIDGE PIER

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3.5 MULTI-DEGREE-OF-FREEDOMSTRUCTURES

For Multi Degree of Freedom (MDOF) structures, theinitial part of design process requires thedetermination of the characteristics of theEquivalent SDOFSubstitute structures

The required characteristics areEquivalent Mass,theDesign Displacement, and theEffective Damping

When these have been determined, the design baseshear for the substitute structure can be determined

The base shear is then distributed between masselements of real structure as inertia forces, andstructure analyzed under these forces to determinethe design moments at locations of potential plastichinges

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3.5.1 DESIGN DISPLACEMENT

The characteristic design displacement of thesubstitute structure depends on the:

• Limit Stateof Displacement orDrift of the mostcritical members of real structure

• and an assumeddisplacement shapeThis displacement shape is that which corresponds tothe in-elastic first mode at the design level of seismicexcitation

Wheremi and∆i are masses and displacements of nsignificant mass locations respectively 31


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The displacements of individual masses are givenby:

Where∂i is the inelastic mode shape and∆c is thedesign displacement at the critical massc,∂c isthe value of the mode shape at massc

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3.5.2 DISPLACEMENT SHAPES A) FRAME BUILDINGS

For regular frame buildings, the followingequations, through approximation have beenshown to be adequate for design purposes:

Hi andHn are the heights of leveli and roof(leveln) respectively

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B) CANTILEVER WALL

For cantilever wall buildings the maximum driftwill occur in the top storey

The value of this drift may be limited by codemaximum drift limit, or by plastic rotationcapacity of the base plastic hinge

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The yield driftθyn at top of wall will be

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B) MULTI SPAN BRIDGES

With bridges it is less easy to determine the designdisplacement profile, particularly for transverseseismic response

The transverse displacement profile will depend

strongly onrelative column stiffness, and moresignificantly onrelative displacement restraintprovided at the abutment, andsuper structure lateralstiffness

For each bridge type three possible displacement

profiles are shown in fig. 3.19 corresponding toabutmentfully restrainedagainst displacement, acompletely unrestrainedabutment and one whereabutment isrestrained, but has significant transverseflexibility

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3.5.3 EFFECTIVE MASS

The effective mass of structure is

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354EQUIVALENTVISCOUS


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3.5.4 EQUIVALENT VISCOUSDAMPING A) VISCOUS DAMPING

This requires determination of displacementductility demand of the structure.

Since the design displacement d has already∆been determined, from (eq. 3.26). The effective

yield displacement y needs to be interpolated∆from the profile displacements at yield

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B)INFLUENCEOFFOUNDATION


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B) INFLUENCE OF FOUNDATIONFLEXIBILITY ON EFFECTIVEDAMPING

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Area based equivalent viscous damping forfoundation for foundation and for the structurecan be expressed as

Hence system equivalent viscous damping is

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EXAMPLE3.5EFFECTIVEDAMPING


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EXAMPLE 3.5 EFFECTIVE DAMPINGFOR A CANTILEVER WALLBUILDING

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3.5.8 DESIGN EXAMPLE 3.6: DESIGN MOMENTSFOR A CANTILEVER WALL BUILDING

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3.6.1: CURRENT DESIGN APPROACHES

When a model is loaded, it deflects. The deflections in themembers of the model may induce secondary moments dueto the fact that the ends of the member may no longer bevertical in the deflected position. These secondary effectsfor members can be accurately approximated through theuse of P-Delta analysis

This type of analysis is called “P-Delta” because themagnitude of the secondary moment is equal to “P”, theaxial force in the member, times “Delta”, the distance oneend of the member is offset from the other end

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Llk\

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It is apparent from the figure that P- effect not∆only reduces the lateral force, but also modifiesthe entire lateral force-displacementcharacteristics. The effective initial stiffness is

reduced, and the post yield stiffness may benegative

P- effect is typically quantified by some form of∆“stability index”, θ∆ which compares the

magnitude of P- effect at either nominal yield or∆at expected maximum displacement, to thedesign base moment capacity of the structure.

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3.6.2 THEORETICALCONSIDERATION

Inelastic time history analysis indicate that thesignificance of P- effects depends on the shape of∆the hysteretic response.

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3.6.3 DESIGN RECOMMENDATIONS


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OF DIRECT DISPLACEMENT BASEDDESIGN

There are significant difficulties in rationallyconsidering P- effects in forced based design∆

In chapter 1, estimation of maximum expecteddisplacement from different codified force-based

designs is subject to wide variabilityFurthermore, most force-based design codes seriouslyunderestimate the elastic and inelastic displacements,and hence underestimate the severity of P- effects∆

The treatment of P- effects in DDBD is comparatively∆

straight forwardUnlike FBD, the design displacement is known at thestart of the design process and hence the P- moment is∆also known before the required strength is determined 56


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DDBD is based on the effective stiffness atmaximum design displacement

The initial strength corresponding to zerodisplacement is thus given by:

For consistency with DDBD philosophy, we shouldtake C=1

Steel structures are likely to be more criticallyaffected than will concrete structures

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Steel Structures: when the structural stability indexdefined by Eq. 4.45 exceeds 0.05, the design base momentcapacity should be amplified for P- considerations as∆indicated in Eq. 3.49, taking C=1

Concrete Structures: when the structural stability indexdefined by Eq. 4.45 exceeds 0.10, the design base momentcapacity should be amplified for P- considerations as∆indicated in Eq. 3.49, taking C=0.5

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37COMBINATIONOFSEISMICANDGRAVITY


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3.7 COMBINATION OF SEISMIC AND GRAVITY

ACTIONS

3.7.1: A COMBINATION OF CURRENT FORCE-BASEDDESIGN APPROACHES

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The great differences in effective stiffness used todetermine gravity moments (gross sectionstiffness) and seismic moments (cracked-sectionstiffness reduced by ductility factors), resulting

the moment combination would be meaninglessTherefore, the gravity moments should bedetermined using the same effective stiffness asappropriate for seismic design

In DDBD the effects of gravity moments are verysmall in comparison with the seismic moments

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3.8 CONSIDERATION OF TORSIONAL RESPONSEIN DIRECT DISPLACEMENT-BASED DESIGN

Structures with asymmetry in plan are subjectedto Torsional rotations as well as directtranslation under seismic response

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CM = Centre of Mass

CR = Centre of Rigidity or Stiffness

C V = Centre of Shear Strength

In traditional elastic analysis of torsional effectsin buildings only the first two are considered, anda structure is considered to have plan eccentricitywhen CM and CR do not Coincide

But it has recently become apparent that forstructures responding in-elastically to seismicexcitation, the centre of strength is at least asimportant as the centre of rigidity

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3.8.3 DESIGN TO INCLUDETORSIONAL EFFECTS

A) Design to avoid Strength Eccentricity

B) Design to Minimize Strength Eccentricity

C) Modification of Design Displacement to Account for Torsion

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3.9 CAPACITY DESIGN FOR DIRECTDISPLACEMENT BASED DESIGN

Special measures are required to ensure thatunintended plastic hinges do not occur at otherlocations up the wall height, where adequatedetailing for ductility has not been provided

Unlike FBD in which only fundamental mode ofvibration is considered, actual structure will includeeffects of higher modes

A further factor to be considered is that conservativeestimates of material strengths will normally beadopted when determining the size of members andthe amount of reinforcing steel.

Amplification factors to be used in designing(dynamic amplification factors, design strength etc)

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310SOMEIMPLICATIONSOFDDBD


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3.10 SOME IMPLICATIONS OF DDBD3.10.1 INFLUENCE OF SEISMIC INTENSITY ONDESIGN BASE SHEAR STRENGTH

DDBD implies significantly different structuralsensitivity than found from current codified forcebased design procedures

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3.10.2 INFLUENCE OF BUILDING HEIGHT ONREQUIRED FRAME BASE SHEAR STRENGTH

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3.10.3 BRIDGE WITH DIFFERENTPIER HEIGHT

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BUILDINGWITHUNEQUAL


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3.10.4 BUILDING WITH UNEQUALWALL LENGTHS

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Chapter 3 Direct Displacement Based Design