linear static seismic lateral force procedures

247

Chapter 5

Linear Static Seismic Lateral Force Procedures

Roger M. Di Julio Jr., Ph.D., P.E.Professor of Engineering, California State University, Northridge

Key words: Code Philosophy, Design Base Shear, Design Story Forces, Design Drift Limitations, Equivalent StaticForce Procedure, Near Fault Factors, Seismic Zone Factors, UBC-97, IBC-2000, Regular and IrregularStructures, Torsion and P-delta Effects, Site Soil Factors, Importance Factors

Abstract: The purpose of this chapter is to review and compare the sections of current seismic design provisions,which deal with the specification of seismic design forces. Emphasis will be on the equivalent static forceprocedures as contaned in the 2000 edition of the International Building Code and the 1997 Edition of theUniform Building Code. There are two commonly used procedures for specifying seismic design forces: The"Equivalent Static Force Procedure" and "Dynamic Analysis". In the equivalent static force procedure, theinertial forces are specified as static forces using empirical formulas. The empirical formulas do notexplicitly account for the "dynamic characteristics" of the particular structure being designed or analyzed.The formulas were, however, developed to adequately represent the dynamic behavior of what are called"regular" structures, which have a reasonably uniform distribution of mass and stiffness. For such structures,the equivalent static force procedure is most often adequate. Structures that do not fit into this category aretermed "irregular". Common irregularities include large floor-to-floor variation in mass or center of massand soft stories. Such structures violate the assumptions on which the empirical formulas, used in theequivalent static force procedure, are based. Therefore, its use may lead to erroneous results. In these cases,a dynamic analysis should be used to specify and distribute the seismic design forces. Principles andprocedures for dynamic analysis of structures were presented in Chapter 4.

248 Chapter 5

5. Linear Static Seismic Lateral Force Procedures 249

5.1 INTRODUCTION

In order to design a structure to withstand anearthquake the forces on the structure must bespecified. The exact forces that will occurduring the life of the structure cannot be known.A realistic estimate is important, however, sincethe cost of construction, and therefore theeconomic viability of the project depends on asafe and cost efficient final product.

The seismic forces in a structure depend ona number of factors including the size and othercharacteristics of the earthquake, distance fromthe fault, site geology, and the type of lateralload resisting system. The use and theconsequences of failure of the structure mayalso be of concern in the design. These factorsshould be included in the specification of theseismic design forces.

There are two commonly used proceduresfor specifying seismic design forces: The"Equivalent Static Force Procedure" and"Dynamic Analysis". In the equivalent staticforce procedure, the inertial forces are specifiedas static forces using empirical formulas. Theempirical formulas do not explicitly account forthe "dynamic characteristics" of the particularstructure being designed or analyzed. Theformulas were, however, developed toadequately represent the dynamic behavior ofwhat are called "regular" structures, which havea reasonably uniform distribution of mass andstiffness. For such structures, the equivalentstatic force procedure is most often adequate.

Structures that do not fit into this categoryare termed "irregular". Common irregularitiesinclude large floor-to-floor variation in mass orcenter of mass and soft stories. Such structuresviolate the assumptions on which the empiricalformulas, used in the equivalent static forceprocedure, are based. Therefore, its use maylead to erroneous results. In these cases, adynamic analysis should be used to specify anddistribute the seismic design forces.

A dynamic analysis can take a number offorms, but should account for the irregularitiesof the structure by modeling its "dynamic

characteristics" including natural frequencies,mode shapes and damping.

The purpose of this chapter is to review andcompare the sections of current seismic designprovisions, which deal with the specification ofseismic design forces. Emphasis will be on, asin the documents discussed, the equivalentstatic force procedure.

The following seismic design provisions areincluded in the discussion, which follows:

1. The Uniform Building Code, Volume 2,“Structural Engineering Design Provisions”issued by the International Conference ofBuilding Officials, 1997 edition, referred to asUBC-97.

2. “International Building Code”, IBC2000Edition, Published by the International CodeCouncil, INC., referred to as IBC2000

IBC2000 is based on "NEHRPRecommended Provisions for SeismicRegulations for New Buildings and OtherStructures, Part I Provisions, prepared by theBuilding Seismic Safety Council for the FederalEmergency Management Agency (FEMA),1997 edition, referred to as FEMA-302. Thecommentary on this document is contained inPart 2 Commentary, designated FEMA-303.

5.2 CODE PHILOSOPHY

The philosophy of a particular documentindicates the general level of protection that itcan be expected to provide. Most codedocuments clearly state that their standards areminimum requirements that are meant toprovide for life safety but not to insure againstdamage.

The code-specified forces are generallylower than the actual forces that would occur ina large or moderate size earthquake. This isbecause the structure is designed to carry thespecified loads within allowable stresses anddeflections, which are considerably less thanthe ultimate or yield capacity (when usingworking stress design) of the materials andsystem. It is assumed that the larger loads thatactually occur will be accounted for by thefactors of safety and by the redundancy and

250 Chapter 5

ductility of the system. Life safety is therebyinsured but structural damage may be sustained.

5.3 UBC-97 PROVISIONS

UBC-97, basically provides for the use ofthe equivalent static force procedure or adynamic analysis for regular structures under240 feet tall and irregular structures 65 feet orless in height. A dynamic analysis is requiredfor regular structures over 240 feet tall,irregular structures over 65 feet tall, andbuildings that are located on poor soils (type SF)and have a period greater than 0.7 seconds.

Although UBC-97 allows for both workingstress design and alternately strength or loadand resistance factor design, the earthquakeloads are specified for use with the latter. Thisis a departure from previous editions where theearthquake loads were specified at the workingstress level.

5.3.1 Design Base Shear V

The design base shear is specified by theformula:

WRT

ICV v= (5-1)

Where, T is the fundamental period of thestructure in the direction under consideration, Iis the seismic importance factor, Cv is anumerical coefficient dependent on the soilconditions at the site and the seismicity of theregion, W is the seismic dead load, and R is afactor which accounts for the ductility andoverstrength of the structural system.Additionally the base shear is dependent on theseismic zone factor, Z. The base shear asspecified by Equation 5-1 is subject to threelimits:

The design base shear need not exceed:

WR

ICV a5.2

= (5-2)

And cannot be less than:

IWCV a11.0= (5-3)

Where Ca is another seismic co-efficientdependent on the soil conditions at the site andregional seismicity.

Additionally in the zone of highestseismicity (zone 4) the design base shear mustbe greater than:

WR

IZNV v8.0

= (5-4)

Where Nv is a near-source factor thatdepends on the proximity to and activity ofknown faults near the structure. Faults areidentified by seismic source type, which reflectthe slip rate and potential magnitude ofearthquake generated by the fault.

The near source factor Nv is also used indetermining the seismic co-efficient Cv forbuildings located in seismic zone 4.

5.3.2 Seismic Zone Factor Z

Five seismic zones, numbered 1 2A, 2B, 3and 4 are defined. The zone for a particular siteis determined from a seismic zone map (SeeFigure 5-1). The numerical values of Z are:

ci

5

tfsieHo

Zone 1 2A 2B 3 4Z 0.075 0.15 0.2 0.3 0.4

The value of the coefficient thus normalizedan be viewed as the peak ground acceleration,n percent of gravity, in each zone.

.3.3 Seismic Importance Factor I

The importance factor I is used to increasehe margin of safety for essential and hazardousacilities. For such structures I=1.25. Essentialtructures are those that must remain operativemmediately following an earthquake such asmergency treatment areas and fire stations.azardous facilities include those housing toxicr explosive substances (See Table 5-1).


5.3.4 Building Period T

The building period may be determined byanalysis or using empirical formulas. A singleempirical formula may be used for all framingsystems:

4

3

nthCT = (5-5)

where

=

buidlingsotherallfor020.0

frames braced eccentricfor 0.030

framesmoment concretefor 0.030

framesmoment steelfor 0.035

tC

hn = the height of the building in feet.

If the period is determined using Rayleigh'sformula or another method of analysis, thevalue of T is limited. In Seismic Zone 4, theperiod cannot be over 30% greater than thatdetermined by Equation 5-5 and in Zones 1, 2and 3 it cannot be more than 40% greater. Thisprovision is included to eliminate the possibilityof using an excessively long period to justify anunreasonably low base shear. This limitationdoes not apply when checking drifts.

5.3.5 Structural System Coefficient R

The structural system coefficient, R is ameasure of the ductility and overstrength of thestructural system, based primarily onperformance of similar systems in pastearthquakes.

The values of R for various structuralsystems are found in Table 5-2. A highernumber has the effect of reducing the designbase shear. For example, for a steel specialmoment resisting frame the factor has value of8.5, while and ordinary moment resisting framethe value is 4.5. This reflects the fact that aspecial moment resisting frame is expected toperform better during an earthquake.

5.3.6 Seismic Dead Load W

The dead load W, used to calculate the baseshear, includes not only the total dead load ofthe structures but also partitions, 25% of thefloor live load in storage and warehouseoccupancies and the weight of snow when thedesign snow load is greater than 30 pounds persquare foot. The snow load may be reduced byup to 75% if its duration is short.

The rationale for including a portion of thesnow load in heavy snow areas is the fact thatin these areas a significant amount of ice canbuild up and remain on roofs.

5.3.7 Seismic Coefficients Cv and Ca

The seismic coefficients Cv & Ca aremeasures of the expected ground acceleration atthe site. They may be found in Tables 5-3 and5-4.

The co-efficient, and hence the expectedground accelerations are dependent on theseismic zone and soil profile type. Theytherefore reflect regional seismicity and soilconditions at the site.

Additionally in seismic zone 4 they alsodepend on the seismic source type and nearsource factors Na and Nv. These factors reflectlocal seismicity in the region of highest seismicactivity.

5.3.8 Soil Profile Type S

The soil profile type reflects the effect ofsoil conditions at the site on ground motion.They are found in Table 5-5 and are labeled SA,through SF.

252 Chapter 5

Figure 5-1. Seismic Zone Map of the United States

Table 5-1 Seismic Importance Factor

OccupancyCategory Occupancy or Functions of Structure

SeismicImportantce

Factor, I1. Essential facilities Group I, Division 1 Occupancies having surgery and emergency treatment areas.

Fire and police stations.Garages and shelters for emergency vehicles and emergency aircraft.Structures and shelters in emergency-preparedness centers.Aviation control towers.Structures and equipment in government communication centers and other facilities requiredfor emergency response.Standby power-generating equipment for Category 1 facilities.Tanks or other structures containing housing or supporting water or other fire-suppressionmaterial or equipment required for the protection of Category 1, 2 or 3 structures.

1.25

2. Hazardousfacilities

Group H, Divisions 1, 2, 6 and 7 Occupancies and structures therein housing or supportingtoxic or explosive chemicals or substances.Nonbuilding structures housing, supporting or containing quantities of toxic or explosivesubstances that, if contained within a building, would cause that building to be classified asa Group H, Division 1, 2 or 7 Occupancy.

1.25

3. Special occupancystructures

Group A, Divisions 1, 2 and 2.1 Occupancies.Buildings housing Group E, Divisions 1 and 3 Occupancies with a capacity greater than 300students.Buildings housing Group B Occupancies used for college or adult education with a capacitygreater than 500 students.Group I, Divisions 1 and 2 Occupancies with 50 or more resident incapacitated patients, butnot included in Category 1.Group I, Division 3 Occupancies.All structures with an occupancy greater than 5,000 persons.Structures and equipment in power-generating stations, and other public utility facilities notincluded in Category 1 or Category 2 above, and required for continued operation.

1.00

4. Standardoccupancy

All structures housing occupancies or having functions not listed in Category 1, 2 or 3 andGroup U Occupancy towers.

1.00

5. Miscellaneous Group U Occupancies except for towers. 1.00

Table 5-2. Structural SystemsBasic Structural

System Lateral-Force-Resisting System Description R1. Bearing wall system 1. Light-framed walls with shear panels

a. Wood Structural panel walls for structures three stories or lessb. All other light-framed walls

2. Shear wallsa. Concreteb. Masonry

3. Light steel-framed bearing walls with tension only bracing4. Braced frames where bracing carries gravity load

a. Steelb. Concretec. Heavy timber

5.54.5

4.54.52.8

4.42.82.8

2. Building frame system 1. Steel eccentrically braced frame (EBF)2. Light-framed walls with shear panels

a. Wood structural panel walls for structures three stories or lessb. All other light-framed walls

3. Shear wallsa. Concreteb. Masonry

4. Ordinary braced framesa. Steelb. Concretec. Heavy timber

5. Special concentrically braced framesa. Steel

7.0

6.55.0

5.55.5

5.65.65.6

6.43. Moment-resisting frame system

1. Special moment-resisting frame (SMRF)a. Steelb. Concrete

2. Masonry moment-resisting wall frame (MMRWF)3. Concrete intermediate moment-resisting frame (IMRF)4. Ordinary moment-resisting frame (OMRF)

a. Steelb. Concrete

5. Special truss moment frames of steel (STMF)

8.58.56.55.5

4.53.56.5

4. Dual systems 1. Shear wallsa. Concrete with SMRFb. Concrete with steel OMRFc. Concrete with concrete IMRFd. Masonry with SMRFe. Masonry with steel OMRFf. Masonry with concrete IMRFg. Masonry with masonry MMRWF

2. Steel EBFa. With steel SMRFb. With steel OMRF

3. Ordinary braced framesa. Steel with steel SMRFb. Steel with steel OMRFc. Concrete with concrete SMRFd. Concrete with concrete IMRF

4. Special concentrically braced framesa. Steel with steel SMRFb. Steel with steel OMRF

8.54.26.55.54.24.26.0

8.54.2

6.54.26.54.2

7.54.2

5. Cantilevered column building systems

1. Cantilevered column elements 2.2

6. Shear wall-frame interaction systems

1. Concrete 5.5

254 Chapter 5

The soil profile types are broadly defined ingeneric terms, for example “Hard Rock” fortype SA. They are also defined by the physicalproperties of the soil determined by standardtests including; shear wave velocity, standardpenetration test, and undrained shear strength.

5.3.9 Seismic Source Type A, B and C

The seismic source type is used to specifythe capability and activity of faults in theimmediate vicinity of the structure. It is usedonly in seismic zone 4.

The seismic source types, labeled A, B or C,are found in Table 5-6. They are defined interms of the slip rate of the fault and themaximum magnitude earthquake it is capable ofgenerating. For example, the highest seismicrisk is posed by seismic source type A, which isdefined by a maximum moment magnitude of7.0 or greater and a slip rate of 5mm/year orgreater.

5.3.10 Near Source Factors Na and Nv

The near source factors Na and Nv are foundin Tables 5-7 and 5-8. In seismic zone 4, they

are used in conjunction with the soil profiletype to determine the seismic coefficients Cv

and Ca (See Tables 5-3 and 5-4). For example,for seismic source type A at a distance to thefault of less than 2km, Na = 1.5 (See Table 5-7).This is then used with Table 5-4 to determinethe seismic co-efficient, Ca.

5.3.11 Distribution of Lateral Force Fx

The base shear V, as determined fromEquations 5-1 through 5-4 are distributed overthe height of the structure as a force at eachlevel Fi, plus an extra force Ft at the top:

∑+==

n

iit FFV

1(5-6)

The extra force at the top is:

.sec7.025.007.0 >≤= TifVTVFt (5-7a)

.sec7.00.0 ≤= TifFt (5-7b)

Ft accounts for the greater participation ofhigher modes in the response of longer periodstructures.

Table 5-4. Seismic Coefficient Ca

Seismic Zone Factor, Z

Soil Profile Type Z = 0.075 Z = 0.15 Z = 0.2 Z = 0.3 Z = 0.4SA 0.06 0.12 0.16 0.24 0.32Na

SB 0.08 0.15 0.20 0.30 0.40Na

SC 0.09 0.18 0.24 0.33 0.40Na

SD 0.12 0.22 0.28 0.36 0.44Na

SE 0.19 0.30 0.34 0.36 0.36Na

SF Site-specific geotechnical investigation and dynamic site response analysis shall be performed.

Table 5-3. Seismic Coefficient CV

Seismic Zone Factor, Z

Soil Profile Type Z = 0.075 Z = 0.15 Z = 0.2 Z = 0.3 Z = 0.4SA 0.06 0.12 0.16 0.24 0.32NV

SB 0.08 0.15 0.20 0.30 0.40NV

SC 0.13 0.25 0.32 0.45 0.56NV

SD 0.18 0.32 0.40 0.54 0.64NV

SE 0.26 0.50 0.64 0.84 0.96NV

SF Site-specific geotechnical investigation and dynamic site response analysis shall be performed.


The remaining portion of the total baseshear (V - Ft) is distributed over the height,including the top, by the formula:

( )( )∑

−=

=

n

iii

xxtx

hw

hwFVF

1

(5-8)

Where, w is the weight at a particular leveland h is the height of a particular level abovethe shear base. At each floor, the force islocated at the center of mass.

For equal story heights and weights,Equation 5-8 distributes the force linearly,increasing towards the top. Any significantvariation from this triangular distributionindicates an irregular structure.

Table 5-5. Soil Profile TypesAverage Soil Properties for Top 100 Feet (30 480 mm) of Soil Profile

Soil ProfileType

Soil Profile Name/GenericDescription

Shear Wave Velocity,feet/second (m/s)

Standard PenetrationTest, (blows/foot)

Undrained ShearStrength, psf (kPa)

SA Hard Rock > 5,000(1,500)

SB Rock 2,500 to 5,000(760 to 1,500)

__ __

SC Very Dense Soil and Soft Rock 1,200 to 2,500(360 to 760)

> 50 >2,000(100)

SD Stiff Soil Profile 600 to 1,200(180 to 360)

15 to 50 1,000 to 2,000(50 to 100)

SE Soft Soil Profile < 600(180)

< 15 < 1,000(50)

SF Soil Requiring Site-specific Evaluation.

Table 5-7. Near-Source Factor Na

Closest Distance to Known Seismic SourceSeismic Source Type

≤ 2 km 5 km ≥ 10 kmA 1.5 1.2 1.0

B 1.3 1.0 1.0

C 1.0 1.0 1.0

Table 5-8. Near-Source Factor NV

Closest Distance to Known Seismic SourceSeismic Source Type

≤ 2 km 5 km 10 km ≥ 15 kmA 2.0 1.6 1.2 1.0

B 1.6 1.2 1.0 1.0

C 1.0 1.0 1.0 1.0

Table 5-6. Seismic Source TypeSeismic Source DefinitionSeismic

SourceType Seismic Source Description

Maximum MomentMagnitude, M

Slip Rate, SR(mm/year)

AFaults that are capable of producing large magnitude events and that have ahigh rate of seismic activity.

M ≥ 7.0 SR ≥ 5

B All faults other than Types A and C.M ≥ 7.0M < 7.0M ≥ 6.5

SR < 5SR > 2SR < 2

CFaults that are not capable of producing large magnitude earthquakes andthat have a relatively low rate of seismic activity.

M < 6.5 SR ≤ 2

256 Chapter 5

5.3.12 Story Shear and OverturningMoment Vx and Mx

The story shear at level x is the sum of allthe story forces at and above that level:

∑+==

n

xiitx FFV (5-9)

The overturning moment at a particular levelMx is the sum of the moments of the storyforces above, about that level. Hence:

( ) ( )∑ −+−==

n

xixiixntx hhFhhFM (5-10)

Design must be based on the overturningmoment as well as the shear at each level.

5.3.13 Torsion and P-Delta Effect

Accidental torsion, due to uncertainties inthe mass and stiffness distribution, must beadded to the calculated eccentricity. This isdone by adding a torsional moment at eachfloor equal to the story shear multiplied by 5%of the floor dimension, perpendicular to thedirection of the force. This procedure isequivalent to moving the center of mass by 5%of the plan dimension, in a directionperpendicular to the force.

If the deflection at either end of the buildingis more than 20% greater than the averagedeflection, it is classified as torsionally irregularand the accidental eccentricity must beamplified using the formula:

0.32.1

2

≤

=

AVG

MAXxA

δδ

(5-11)

where

δavg = the average displacement at level xδmax = the maximum displacement at level x

P-Delta effects must be included indetermining member forces and storydisplacements where significant.

5.3.14 Reliability / Redundancy Factor ρρρρ

The seismic design forces and hence thebase shear as determined from Equations 5-1through 5-4, must be multiplied be a reliability/redundancy factor for the lateral load resistingsystem:

5.120

21max

≤−=≤BAr

ρ (5-12)

Where, AB is the ground floor area of thestructure in square feet and rmax is the maximumelement-story shear ratio.

The element story shear ratio (ri) at aparticular level is the ratio of the shear in themost heavily loaded member to the total storyshear. The maximum ratio, rmax is defined as thelargest value of ri in the lower two-thirds of thebuilding.

Special provisions for calculating r, fordifferent lateral load resisting systems, aredemonstrated in the examples that follow.

For special moment-resisting frames, if ρexceeds 1.25, additional bays must be added.For the purposes of determining drift(displacement), and in seismic zones 0, 1 and 2,ρ =1.0.

5.3.15 Drift Limitations

The deflections due to the design seismicforces are called the design level responsedisplacements, ∆s. The seismic forces used todetermine ∆s may be calculated using areliability/redundancy factor equal to one,ignoring the limitation represented by Equation5-3, and using an analytically determinedperiod greater than the limits outlined in section5.3.4.

The maximum inelastic response is definedas:


sM R∆=∆ 7. (5-13)

Where, R is the structural system co-efficient defined in Table 5-2.

Deflection control is specified in terms ofthe story drift, which is defined as the lateraldisplacement of one level relative to the levelbelow. The story drift is determined from themaximum inelastic response as defined byEquation 5-13.

The displacement must include bothtranslation and torsion. Hence, the drift must bechecked in the plane of the lateral load resistingelements, generally at the ends of the building.P-Delta displacements must be included wheresignificant.

For structures with a period less than 0.7seconds, the maximum story drift is limited to:

ha 025.≤∆ (5-14)

Where, h is the story height.For structures with a period greater than 0.7

seconds:

ha 020.≤∆ (5-15)

5.3.16 Irregular Structures

UBC-97 quantifies the notion of irregularity,which it breaks into two broad categories:

Table 5-9. Vertical Structural IrregularitiesIrregularity Type and Definition1. Stiffness irregularity-soft story

A soft story is one in which the lateral stiffness is less that 70 percent of that in the story above or less than 80 percent of theaverage stiffness of the three stories above.

2. Weight (mass) irregularityMass irregularity shall be considered to exist where the effective mass of any story is more than 150 percent of the effectivemass of an adjacent story. A roof that is lighter than the floor below need not be considered.

3. Vertical geometric irregularityVertical geometric irregularity shall be considered to exist where the horizontal dimension of the lateral-force-resisting systemin any story is more than 130 percent of that in an adjacent story. One-story penthouses need not be considered.

4. In-plane discontinuity in vertical lateral-force-resisting elementAn in-plane offset of the lateral-load-resisting elements greater than the length of those elements.

5. Discontinuity in capacity-weak storyA weak story is one in which the story strength is less than 80 percent of that in the story above. The story strength is the totalstrength of all seismic-resisting elements sharing the story shear for the direction under consideration.

Table 5-10. Plan Structural IrregularitiesIrregularity Type and Definition1. Torsional irregularity-to be considered when diaphragms are not flexible

Torsional irregularity shall be considered to exist when the maximum story drift, computed including accidental torsion, at oneend of the structure transverse to an axis is more than 1.2 times the average of the story drifts of the two ends of the structure.

2. Re-entrant cornersPlan configurations of a structure and its lateral-force-resisting system contain re-entrant corners, where both projections of thestructure beyond a re-entrant corner are greater than 15 percent of the plan dimension of the structure in the given direction.

3. Diaphragm discontinuityDiaphragms with abrupt discontinuities or variations in stiffness, including those having cutout or open areas greater than 50percent of the gross enclosed area of the diaphragm, or changes in effective diaphragm stiffness of more than 50 percent fromone story to the next.

4. Out-of-plane offsetsDiscontinuities in a lateral force path, such as out-of-plane offsets of the vertical elements.

5. Nonparallel systemsThe vertical lateral-load-resisting elements are not parallel to or symmetric about the major orthogonal axes of the lateral-force-resisting system.

258 Chapter 5

vertical structural and plan structuralirregularity. Vertical irregularities include softor weak stories, large changes in mass fromfloor to floor and large discontinuities in thedimensions or in-plane locations of lateral loadresisting elements. Plan irregular buildingsinclude those which undergo substantial torsionwhen subjected to seismic loads, have re-entrant corners, discontinuities in floordiaphragms, discontinuity in the lateral forcepath, or lateral load resisting elements whichare not parallel to each other or to the axes ofthe building.

The precise definitions of these irregularitiesare found in Tables 5-9 and 5-10. For a moredetailed discussion of irregularity, see ChapterSix.

5.3.17 Dynamic Lateral Force Procedure

UBC-97 requires that, if the base sheardetermined by a dynamic analysis using a site-specific spectra is less than that specified by thestatic lateral force procedure, it must be scaledto equal that determined by the equivalent staticforce procedure. Similarly, if the base shearobtained from a dynamic analysis is greaterthan that specified by the static lateral forceprocedure, it may be scaled down. In thismanner, the dynamic characteristics of thestructure are modeled, and thus the forces aredistributed properly, while the code level forcesare maintained. If a site-specific spectrum is notavailable, the spectra provided in UBC-97 (seeFigure 5-2) can be used.

Figure 5-2. Design Response Spectra


5.3.18 Examples

Example 5-1:

Determine the UBC-97 design seismicforces for a three-story concrete shear walloffice building. It is located in SoutheasternCalifornia on rock with a shear wave velocity of3000 ft/ sec. The story heights are 13 feet forthe first floor and 11 feet for the second andthird floors. The story dead loads are 2200,2000 and 1700 kips from the bottom up. Theplan dimensions are 180 feet by 120 feet. Thewalls in the direction under consideration are120 feet long and are without openings. Theshear walls do not carry vertical loads. Samplecalculations are presented and a completetabulation is found in Table 5-11.

• Base Shear:

RT

IWCV v= Equation 5-1

I=1.0 Table 5-1R=5.5 (Shear Walls) Table 5-2Seismic zone 3 Figure 5-1Z = .3 Section 5.3.2Soil Profile Type SB Table 5-5Cv = .3 Table 5-3

T = ( ) 43

3502. = .29 Seconds Equation 5-5W = 1700 + 2000 + 2200 = 5900 k

V = ( )( )29.5.5

0.13.(5900) = 1109.2 k

V ≤ 2.5 WR

ICa Equation 5-2

Ca = .3 Table 5-4

V ( )( )

5.5

13.5.2≤ (5900) = 804.5 k < 1109.2

V ≥ .11 Ca IW Equation 5 –3

V ≥ .11 (.3) (1) (5900) = 194.7 < 804.5

V = 804.5 k

• Vertical Distribution:

T< 0.7 sec Equation 5-7bFt = 0.0

Fx = ( ) ∑−=

n

iiixxt hwhwFV

1Equation 5-8

F3 = 804.5 (59.5) / 136.1 = 351.7 kF2 = 804.5 (48) / 136.1 = 283.7 kF1 = 804.5 (28.6) / 136.1 = 169.1 k

• Story Shear:

∑+==

n

xiitx FFV Equation 5-9

V3 = 351.7 kV2 =351.7 +283.7 = 635.4 kV3=351.7 +283.7 = 169.1 = 804.5 k

• Overturning Moment:

( ) ( )∑ −+−==

n

xixiixntx hhFhhFM Eq. 5-10

M3 = 351.7 (11) = 3869 ft-kM2 = 351.7 (22) +283.7 (11) = 10,858 ft-kM1 = 351.7 (35) +283.7(24)+ 169.1(13)=21,317

Table 5-11: Example 5-1

Levelhx

(ft)

wx

(k)wXhX x10-3

Fi+Ft

(k)

VX

(k)

Mx

(ft-k)

3 35 1700. 59.5 351.7 351.7 3869.

2 24 2000. 48.0 283.7 635.4 10858

1 13 2200. 28.6 169.1 804.5 21317.

Σ 5900. 136.1 804.5

• Allowable Inelastic Story Displacement:

T ≤ .7 seconds

260 Chapter 5

∆a ≤ .025 h Equation 5-14

2nd & 3rd Floors:∆a ≤ .025 (11x12) = 3.3 inches

1st Floor:∆a ≤ .025 (11 x13) = 3.56 inches

• Equivalent Elastic Story Displacement:

( ) hhR

h0065.

5.57.025.

7.025. ==≤∆ Eq. 5-13

2nd & 3rd Floor:∆ ≤ .0065 (11 x 12) = .858 inches

1st Floor:∆ ≤ .0065 (13 x 12) = 1.01 inches

• Reliability / Redundancy Factor:

For shear walls, ri is the maximum value ofthe product of the wall shear and 10/lw, dividedby the total shear, where lw is the length of wallin feet (120 ft).

An approximation of rmax can be obtained byassuming that half the story shear is carried byeach wall.

( )( )04.

120102max ==

S

S

V

Vr

AB = 120 x 180 = 21,600 ft2

Bmax Ar

20-2ρ = Equation 5-12

ρ= 2 – 20 /. 04 600,21 = –1.4 < ρmin = 1.0

ρ = 1.0

Example 5-2:

Determine the UBC-97 design seismic forcesfor a nine story ductile moment resisting steel

frame office building located in Los Angeles,California on very dense soil and soft rock. Thebuilding is located 5km from a fault capable oflarge magnitude earthquakes and that has amoderate slip rate (M>7, SR>2mm/yr). Thestory heights are all thirteen feet. The plan areais 100 feet by 170 feet. The total dead load is100 pounds per square foot at all levels. Themoment frames consist of two four bay framesin the transverse direction and two seven bayframes in the longitudinal direction. Samplecalculations are presented and a completetabulation is found in Table 5-12.

• Base Shear:

RT

IWCV v= Equation 5-1

I = 1.0 Table 5-1R = 8.5 (SMRF) Table 5-2Seismic Zone 4 Figure 5-1Z = .4 Section 5.3.2Soil Profile Type Sc Table 5-5Seismic Source Type B Table 5-6Nv = 1.2 Table 5-8Cv = .56 Nv = .56 (1.2) = .67 Table 5-3

T =.035 43

)117( = 1.25 seconds Equation 5-5

W = .1 (170) 100 = 1700 k / floorW = 9 (1700) = 15,300 k

( )( )( ) ( ) KWWV 8.964300,15063.063.

25.15.8

0.167. ====

WR

ICV a5.2≤ Equation 5-2

Na = 1.0 Table 5-7Ca = .4 Na = .4 (1.0) = .4 Table 5-4

( )( )( ) k 8.964k 1800300,155.8

0.14.5.2 >=≤V

V ≥ .11 Ca IW Equation 5-3


V ≥ .11(.4)(1.0)(15,300)= 673.2k < 964.8k

Since the building is in zone 4:

V WR

IZNv8.≥ Equation 5-4

V ≥ ( )( )( )( ) k 8.964k 2.691300,155.8

0.12.14.8. <=

V = 964.8 k


T > .07 secFt=.07TV=.07(1.25)(964.8)= 84.4k Eq.5-7a.25V = .25 (964.8) = 241.2 > 84.4Ft = 84.4 k(V-Ft) = 964.8 – 84.4 = 880.4

( )( )∑

−=

=

n

iii

xxtx

hw

hwFVF

1

Equation 5-8

F9+Ft= 880.4(1700)(117)/996,500+84.4=260.1kF8 = 880.4 (1700) (104) / 996,500 = 156.2 k(See Table 5-12)

• Story Shear:

∑+==

n

xiitx FFV Equation 5-9

V9 = 260.4 kV8 = 260.4 + 156.2 = 416.6 k

Overturning Moment

( ) ( )∑ −+−==

n

xixiixntx hhFhhFM Eq. 5-10

M9 = 260.1 (13) = 3381 ft.-k,M8 = 260.1 (26) + 156.2 (13) = 8,793 ft-k

• Allowable Inelastic Story Displacement:

T > 0.7 seconds

∆a<.02h= .02(13x12)= 3.12 inches Eq. 5-15


( ) hh

R

ha 00336.

5.87.

02.

7.

02. ==≤∆ Eq. 5-13

∆ ≤ .00336 (13 x 12) = .52 inches

• Reliability / Redundancy Factor

For moment frames, ri is normally 70% ofthe shear in two adjacent interior columns. Anapproximation for ri can be obtained byassuming all interior columns carry equal shearand external columns carry half as much.

BArmax

202ρ −= Equation 5-12

AB = 100 x 170 = 17,000 ft2

Transverse Direction:Two 4 Bay Frames

( ) 175./887.max =+= VVVr

25.112.1000,17175.

202ρ ≤=−=

ρmax = 1.25 for special moment frame ok.

ρ = 1.12

Longitudinal Direction:Two 7 Bay Frames


Level hx

(ft)

wx

(k)

wxhx

x10-3

Fi+Ft

(k)

Vx

(k)

Mx

(ft-k)

9 117 1700. 198.9 260.1 260.1 3381.

8 104 1700. 176.8 156.2 416.6 8793.

7 91 1700. 155.7 137.6 553.9 15994.

6 78 1700. 132.6 117.2 671.1 24718.

5 65 1700. 110.5 97.6 768.7 34711.

4 52 1700. 88.4 78.1 846.8 45720.

3 39 1700. 66.3 58.6 905.4 57490.

2 26 1700. 45.2 39.9 945.3 69779.

1 13 1700. 22.1 19.5 964.8 82321.

Σ 15300. 996.5 964.8

262 Chapter 5

( ) 1./14147.max =+= VVVr

0.1ρ47.000,171.

202ρ min =<=−=

ρ = 1.0

5.4 IBC2000 PROVISIONS

IBC2000 is broadly similar to UBC-97, butdoes contain significant differences. Theseinclude ground accelerations specified on alocal basis by a set seismic risk maps.

The concept of a seismic use group, whichis related to the importance factor in UBC-97, isintroduced. In addition to defining theimportance factor it is used to designate theseismic design category and to establish theallowable story drift.

The seismic design category determines theanalysis procedures to be used and height andsystem limitations.

5.4.1 Seismic Use Group I, II, III

Each structure is assigned to a seismic usegroup based on the occupancy of the buildingand the consequences of severe earthquakedamage. Three seismic hazard groups aredefined:

GROUP III...."having essential facilities thatare required for post-earthquake recovery andthose containing substantial quantities ofhazardous substances ". These facilities includefire and police stations, hospitals, medicalfacilities having emergency treatment facilities,emergency preparedness centers, operationcenters, communication centers, utilitiesrequired for emergency backup, and structurescontaining significant toxic or explosivesubstances.

GROUP II...."have a substantial publichazard due to occupancy or use...". Theseinclude high occupancy buildings and utilitiesnot required for emergency backup.

GROUP I -- All other buildings.

5.4.2 Occupancy Importance Factor I

An occupancy importance factor is assignedbased on the seismic use group. This factor isused to increase the design base shear forstructures in seismic use groups II and III.

The values of the importance occupancyfactor are:

Seismic Use Group II 1.0II 1.25III 1.50

5.4.3 Maximum Considered EarthquakeGround Motion

Regional seismicity is specified by a seriesof maps. The maps provide the spectralresponse accelerations at short periods, Ss andat a period of one second, S1 (see Figures 5-3and 5-4).

In areas of low seismic activity (Ss ≤ .15g,S1 ≤ .04g) the acceleration need not bedetermined.

5.4.4 Site Class

The soil conditions at the site determine thestructures “site class”. These are virtuallyidentical to the soil profile types in UBC-97(see Table 5-5).

5.4.5 Site Coefficients Fa and Fv

The regional seismicity, as expressed by themaximum considered earthquake groundmotion, Ss and S1, must be modified for the soilconditions at the site. These are defined by thesite class. The maximum considered earthquakespectral response accelerations adjusted for siteclass effects, are:

SMS = Fa Ss (5-16 a)

SM1 = Fv S1 (5-16 b)

Figure 5-3. IBC 2000 Spectral Map for Short Period Range (T=0.3 Sec)

264 Chapter 5

Figure 5-4 IBC 2000 Spectral Map for Intermediate Period Range (T=1.0 Sec).

d

osp

5

e

d

5

dp

Table 5-13. Values of Fa as a Function of Site Class and Mapped Short- Period Maximum Considered Earthquake

Spectral AccelerationMapped Maximum Considered Earthquake Spectral Response Acceleration at Short PeriodsSite

Class SS ≤ 0.25 SS=0.5 SS=0.75 SS=1.00 SS ≥ 1.25A 0.8 0.8 0.8 0.8 0.8

B 1.0 1.0 1.0 1.0 1.0

C 1.2 1.2 1.1 1.0 1.0

D 1.6 1.4 1.2 1.1 1.0

E 2.5 1.7 1.2 0.9 a

F a a a a aaSite-specific geotechnical investigation and dynamic site response analyses shall be performed.

Table 5-14. Values of Fv as a Function of Site Class and Mapped 1 Second Period Maximum Considered Earthquake

Spectral AccelerationMapped Maximum Considered Earthquake Spectral Response Acceleration at 1 SecondPeriodSite

Class S1 ≤ 0.1 S1=0.2 S1=0.3 S1=0.4 S1 ≥ 0.5A 0.8 0.8 0.8 0.8 0.8

B 1.0 1.0 1.0 1.0 1.0

C 1.7 1.6 1.5 1.4 1.3

D 2.4 2.0 1.8 1.6 1.5

E 3.5 3.2 2.8 2.4 a

F a a a a a

Where, Fa and Fv are the site coefficientsefined in Tables 5-13 and 5-14.

For site class F, and site class E in regionsf high seismicity (Ss>1.25g or S1>5g), a site-pecific geotechnical investigation must beerformed.

.4.6 Design Spectral ResponseAccelerations SDS and SD1

The spectral accelerations for the designarthquake are:

SDS =2/3SMS (5-17a)

SD1 =2/3SM1 (5-17b)

These are the accelerations used toetermine the design base shear.

.4.7 Seismic Design Category

The structure must be assigned a seismicesign category, which determines theermissible structural systems, limitations on

height and irregularity, those components of thestructure that must be designed for seismicloads, and the types of analysis required.

The seismic design categories, designated Athrough F, are presented in Tables 5-15 and 5-16. They depend on the seismic use group andthe design spectral acceleration coefficients, SDS

and SD1. The structure is assigned the moresevere of the two values taken for these tables.

5.4.8 Design Base Shear V

IBC2000 specifies the design base shear bythe formula:

V = C s W (5-18)

The base shear is a percentage, Cs of thetotal dead load W.

5.4.9 Total Dead Load W

The seismic dead load consists of the totalweight of the structure, plus partitions andpermanent equipment. It also includes 25% offloor live load in areas used for storage, and the

266 Chapter 5

snow load if it is greater than 30 lb/ft2. Thesnow load may be reduced by up to 80% if itsduration is short.

5.4.10 Seismic Response Coefficient CS

The seismic response coefficient isdetermined from the formula:

IR

SC DS

s = (5-19)

where

SDS = the design spectral acceleration in theshort period range

R = the response modification factor fromTable 5-17 and defined below

I = the occupancy importance factor

The coefficient Cs, as specified by Equation5-19, is subject to three limits.

It need not exceed:

ITR

SC D

s1= (5-20)

It must be greater than:

ISC DSs 044.= (5-21)

Additionally for structures in seismic designcategories E and F, and for structures with a 1second spectral response greater than or equalto .6g, it cannot be less than:

IR

SCs

15.= (5-22)

where

SD1 = the design spectral response at a 1.0second period

T = the fundamental period of the structureS1 = the maximum considered earthquake

spectral response acceleration at a 1second period

5.4.11 Building Period T

The building period can be estimated usingthe empirical formula:

Ta = Ct hn3/4 (5-23)

where

Table 5-15. Seismic Design Category Based on Short Period Response AccelerationsSeismic Use Group

Value of SDSI II III

SDS < 0.167g A A A

0.167g ≤ SDS < 0.33g B B C

0.33g ≤ SDS < 0.50g C C D

0.50g ≤ SDS Da Da Da

aSeismic Use Group I and II structures located on sites with mapped maximum considered earthquake spectral response acceleration at 1 secondperiod, S1, equal to or greater than 0.75g shall be assigned to Seismic Design Category E and Seismic Use Group III structure located on such sites shall beassigned to Seismic Design Category F.

Table 4-16. Seismic Design Category Based on 1Second Period Response AccelerationsSeismic Use Group

Value of SDSI II III

SD1 < 0.067g A A A

0.067g ≤ SD1 < 0.133g B B C

0.133g ≤ SD1 < 0.20g C C D

0.20g ≤ SD1 Da Da Da


=

buidlingsotherallfor020.0

frames braced eccentricfor 0.030

framesmoment concretefor 0.030

framesmoment steelfor 0.035

tC

hn = the height of the building in feet.

An alternate formula is provided for steeland concrete moment frame buildings twelvestories or less in height and with story heightsten feet or greater:

Ta =0.1 N (5-24)

where, N is the number of stories.

The period may also be determined by ananalysis. The period used to determine the baseshear is subject to an upper limit, which isbased on the design spectral responseacceleration at a period of one second, SD1. Therelationship between SD1 and the maximum

allowable period used to specify the base shearis:

SD1 Tmax/Ta

≥ 0.4 1.20.3 1.30.2 1.40.15 1.5≤ 0.1 1.7

This provision insures that an excessivelylong analytically determined period is not usedto justify an unrealistically low design baseshear. When determining drifts these limits donot apply.

5.4.12 Response Modification Factor R

The response modification factor, R servesthe same function as the structural systemcoefficient in UBC–97. It reduces the designloads to account for the damping and ductilityof the structural system. An abbreviated set forvalues for R is found in Table 5-17.

Table 5-17 Design Coefficients and Factors for Basic Seismic-Force-Resisting Systems

Basic Seismic-Force-Resisting System Response Modifications Coefficient, R Deflection Amplication Factor, Cd

Bearing Wall Systems

Special reinforced concrete shear walls 5.5 5

Ordinary reinforced concrete shear walls 4.5 4

Building Frame Systems

Special steel concretrically braced frames 6 5

Special reinforced concrete shear walls 6 5

Moment Resisting Frame Systems

Special steel moment frames 8 5.5

Ordinary steel moment frames 4 3.5

Dual Systems with Intermediate Moment Frames Capable of Resisting at Least 25% of Prescribed Seismic Forces

Special reinforced concrete shear walls 6 5

Ordinary reinforced concrete shear walls 5.5 4.5

Inverted Pendulum Systems and Cantilevered Column Systems

Special steel moment frames 2.5 2.5

Ordinary steel moment frames 1.25 2.5

268 Chapter 5

5.4.13 Vertical Distribution of Force FX

The seismic force at any level is a portion ofthe total base shear:

Fx = Cvx V (5-25)

where

∑=

=n

i

kii

kxx

vx

hw

hwC

1

(5-26)

where

wi,wx= the portion of the dead load at orassigned to level i or x

hi,hx= height above the base to level i or xk = an exponent related to the building

period as follows:

For buildings with a period of 0.5 secondsor less, k=1.0. If the period is 2.5 seconds ormore, k=2.0. For buildings with a periodbetween 0.5 and 2.5 seconds, it may be taken as2.0 or determined by linear interpolationbetween 1.0 and 2.0.

For k=1.0 the distribution is a straight line.This is reasonable for short buildings with aregular distribution of mass and stiffness.Hence, k=1.0 for buildings with a period of 0.5seconds or less.

For k=2.0 the distribution is a parabola withthe vertex at the base. This is reasonable for tallregular buildings where the participation ofhigher modes is significant. Hence, k=2.0 forbuildings with a period of 2.5 seconds or more.This effect is accounted for by the force Ft,placed at the roof in UBC-97.

5.4.14 Overturning Moment Mx

IBC2000 allows for a reduction in thedesign overturning moment:

( )∑ −==

n

xixiix hhFM τ (5-27)

where

τ = 1.0 for the top 10 storiesτ = 0.8 for the 20th story from the top and

below and is interpolated between 0.8and 1.0 for stories in between.

Part of the reasoning behind this reductionis that the design story forces are an envelope ofthe maximums at each floor, and it is unlikelythat they will all reach a maximumsimultaneously.

5.4.15 Drift Limitations

For buildings, other than masonry, over fourstories the allowable drifts are:

∆ ≤ ∆a=0.010 hsx Use Group III (5-28)∆ ≤ ∆a=0.015 hsx Use Group II (5-29)∆ ≤ ∆a=0.020 hsx Use Group I (5-30)

For buildings four stories or less and height,other than masonry, the allowable drifts are:

∆ ≤ ∆a=0.015 hsx Use Group III (5-28a)∆ ≤ ∆a=0.020 hsx Use Group II (5-29a)∆ ≤ ∆a=0.025 hsx Use Group I (5-30a)where

∆ = the design interstory displacement∆a = the allowable story displacementhsx= the height of the story below level x

The design interstory displacement ∆, is thedifference in the deflections δx, at the top andbottom of the story under consideration. It isbased on the calculated deflections and isevaluated by the formula:

I

δ Cδ d

xxe= (5-31)

where

Cd = the deflection amplification factorδxe = the deflections determined by an

elastic analysis.I =the occupancy importance factor


The deflection amplification factor Cd isassigned values from 1.25 to 5.5 and accountsfor the ductility of the system and the propertiesof the materials from which it is constructed(see Table 5-17).

In determining these deflections the perioddetermined by an analysis may be used tocalculate the base shear without considering thelimitation on the period discussed in Section5.4.11. This has the implication that lower storyforces may be used to determine deflectionsthan are used to determine member forces. Asimilar provision is contained in UBC-97.

Where significant, P-Delta and torsionaldeflections must be considered in satisfying thedrift limitation. This is discussed further in thenext section.

5.4.16 Torsion and P-Delta Effect

Torsion is accounted for in same manner asin UBC-97. The torsional moment resultingfrom the location of the center of mass plus thatresulting from an assumed movement of fivepercent of the plan dimension must beaccounted for.

For buildings with torsional irregularity, inseismic design categories C through F, the fivepercent accidental torsion must be amplifiedusing Equation 5-11. For this purpose abuilding is irregular if the diaphragm is rigidand the maximum interstory displacement ismore than 1.2 times the average.

The P-Delta effect must be included in thecomputation of story shears, story drifts andmember forces when the value of the "stabilitycoefficient" has a value, for any story, suchthat:

θ = Px∆/VxhsxCd > 0.10 (5-32)

where

∆= the design story driftVx= the seismic force acting between level x

and x-1hsx = the story height below level xPx = total gravity load at and above level xCd = the deflection amplification factor

The stability coefficient can be visualized asthe ratio of the P-Delta moment (Px∆) to thelateral force story moment (Vxhsx). Hence if the

Table5-18. Plan Structural IrregularitiesIrregularity Type and Description

1a Torsional Irregularity—to be considered when diaphragms are not flexibleTorsional irregularity shall be considered to exist when the maximum story drift, computed including accidental torsion, atone end of the structure transverse to an axis is more than 1.2 times the average of the story drifts at the two ends of thestructure.

1b Extreme Torsional Irregularity – to be considered when diaphragms are not flexibleExtreme torsional irregularity shall be considered to exist when the maximum story drift, computed including accidentaltorsion, at one end of the structure transverse to an axis is more than 1.4 times the average of the story drifts at the two endsof the structure.

2 Re-entrant CornersPlan configurations of a structure and its lateral force-resisting system contain re-entrant corners, where both projections ofthe structure beyond a re-entrant corner are greater than 15 percent of the plan dimension of the structure in the givendirection.

3 Diaphragm DiscontinuityDiaphragms with abrupt discontinuities or variations in stiffness, including those having cutout or open areas greater than 50percent of the gross enclosed diaphragm area, or changes in effective diaphragm stiffness of more than 50 percent from onestory to the next.

4 Out-of-Plane OffsetsDiscontinuities in a lateral force resistance path, such as out-of-plane offsets of the vertical elements.

5 Nonparallel SystemsThe vertical lateral force-resisting elements are not parallel to or symmetric about the major orthogonal axes of the lateralforce-resisting system.

270 Chapter 5

P-Delta moment is equal to 10 percent of thestory moment at any floor the P-Delta effectshould be considered. The code also specifiesan upper limit on the stability coefficient.

5.4.17 Irregularity

IBC2000 defines irregularity in a mannersimilar to UBC-97, but goes further byassigning a building to a seismic designcategory based on its irregularity. Itdistinguishes between the two broad categoriesof plan and vertical irregularity.

Plan irregularities include: a non-symmetrical geometric configuration, re-entrantcorners, significant torsion due to eccentricitybetween mass and stiffness, nonparallel lateralforce resisting elements, out of plane offsetsand discontinuous diaphragms.

Vertical irregularities include: soft and weakstories, large changes in mass-stiffness ratiosbetween adjacent floors, large changes in plandimension from floor to floor and significanthorizontal offsets in the lateral load system.

The definitions of plan and verticalstructural irregularities and their assignedseismic design categories are found in Tables 5-18 and 5-19.

5.4.18 Reliability Factor ρρρρ

The reliability factor ρ is identical to andserves the same function as in UBC-97 (Seesection 5.3.14, Equation 5-12). It is assigned avalue of 1.0 for seismic design categories A, Band C. For special moment resisting frames inSeismic Design Category D, ρ cannot exceed1.25. For special moment resisting frames inSeismic Design Categories E and F, ρ cannotexceed 1.1.

5.4.19 Analysis Procedures

The minimum level of structural analysis isdependent on the seismic design category. Forbuildings in category A, the design lateral forceat all floors is 1 % of gravity. Buildings incategories B and C, whether regular orirregular, may be analyzed using the equivalentlateral force procedure.

The analysis procedure for buildings incategories D, E & F is specified as follows.Regular buildings up to 240 feet in height maybe analyzed using the equivalent lateral forceprocedure. Buildings that are either over 240feet tall, irregular, located on poor soils, or

Table 5-19. Vertical Structural IrregularitiesIrregularity Type and Description

1a Stiffness Irregularity—Soft StoryA soft story is one in which the lateral stiffness is less than 70 percent of that in the story above or less than 80 percent of theaverage stiffness of the three stories above.

1b Stiffness Irregularity—Extreme Soft StoryAn extreme soft story in one in which the lateral stiffness is less than 60 percent of that in the story above or less than 70percent of the average stiffness of the three stories above.

2 Weight (Mass) IrregularityMass Irregularity shall be considered to exist where the effective mass of any story is more than 150 percent of the effectivemass of an adjacent story. A roof that is lighter than the floor below need not be considered.

3 Vertical Geometric IrregularityVertical geometric irregularity shall be considered to exist where the horizontal dimension of the lateral force-resisting systemin any story is more than 130 percent of that in an adjacent story.

4 In-Plane Discontinuity in Vertical Lateral Force Resisting ElementsAn in-plane offset of the lateral force-resisting elements greater than the length of those elements or a reduction in stiffness ofthe resisting element in the story below.

5 Discontinuity in Capacity—Weak StoryA weak story is one in which the story lateral strength is less than 80 percent of that in the story above. The story strength isthe total strength of all seismic-resisting elements sharing the story shear for the direction under consideration.


close to known faults in areas of high seismicityrequire various types of dynamic analysis.

5.4.20 Dynamic Analysis

Provisions are included for a simplified twodimensional version of modal analysis which isapplicable to regular structures withindependent orthogonal seismic force resistingsystems. For such structures the motion ispredominantly planar and a two dimensionalmodel may be appropriate.

For irregular structures or with interactingseismic force resisting systems a threedimensional model is required.

The required base shear is equal to thatdetermined by Equation 5-18, where the periodused may be 20 percent longer than themaximum period allowed in the equivalentlateral force procedure (see Section 5.4.11). Thejustification for this is that a modal analysis ismore accurate than a static analysis. Althoughthe total force on the building does not changeappreciably its distribution over the height ismore accurately modeled.

5.4.21 Examples

Example 5-3:

Rework Example 5-1 using IBC2000 andspecial reinforced concrete shear walls.

• Base Shear:

WCV S= Equation 5-18

seismic use group I Section 5.4.1I = 1.0 Section 5.4.2SS = .5g Figure 5-3S1 = .2g Figure 5-4site class B Table 5-5Fa = 1.0 Table 5-13Fv = 1.0 Table 5-14SMS = Fa SS = .5g Eq. 5-16aSM1 = FvS1 = .2g Eq. 5-16bSDS= 32 SMS = 32 (.5) = .333g Eq. 5-17a

SD1= 32 SM1 = 32 (.2) = .133g Eq. 5-17bseismic design cat. C Tables 5-15,16R = 6 (special shear walls) Table 5-17

Cs=IR

DSS=

16333.

=.0555 Equation 5-19

Ta =.02hn3/4=.02(35)3/4=.29 Sec Eq.5-23

( ) ( ) 076.1629.

133.T

SC D1

S ==≤IR

Eq.5-20

CS ≥ .044SD1 I = .044(.333)(1) = .0147 Eq.5-21

CS = .0555V = .0555 (5900) = 327.5k


Fx = CvxV Equation 5-25

∑=

=n

i

kii

kxx

vx

hw

hwC

1

Equation 5-26

T < 0.5 seck = 1.0Since k=1, the procedure is identical toExample 5-1. See Table 5-20.


( )∑ −==

n

xixiix hhFM τ Equation 5-27

τ = 1.0 for top ten storiesSince τ=1, the procedure is the same as forExample 5-1. See Table 5-20.

Table 5-20: Example 5-3Level hx

(ft)wx

(k)wxhx

x10-3Cvx

(k)Fx

(k)Vx

(k)Mx

(ft-k)

3 35 1700 59.5 0.44 144.1 144.1 15852 24 2000 48.0 0.35 114.6 258.7 44311 13 2200 28.6 0.21 68.8 327.5 8688Σ 5900 136.1 1.00 327.5

• Allowable Inelastic Story Displacements:

272 Chapter 5

seismic use group Iless than four stories

sxa h025.0=∆ Equation 5-30a

1st Floor:∆ = 0.025(13)(12) = 3.9 inches2nd and 3rd Floors:∆ = 0.025(11)(12)= 3.3 inches


IC xedδδ = Equation 5-31

Cd = 5 Table 5-17

xeδ5δ =

1st Floor:∆ = 5a∆ = 3.9/5 = 0.78 inches

2nd and 3rd Floors:∆ = 5a∆ = 3.3/5= 0.66 inches

• Reliability Factor:

seismic design category Cρ = 1.0 Section 5.4.18

Example 5-4:

Rework Example 5-2 using IBC2000.• Base Shear:

WCV S= Equation 5-18

seismic use group I Section 5.4.1I = 1.0 Section 5.4.2Ss = 2.05g Figure 5-3S1 = 0.81g Figure 5-4site class C Table 5-5Fa = 1.0 Table 5-13Fv = 1.3 Table 5-14SMS = Fa Ss = 1.0 (2.05) = 2.05 Eq. 5-16aSM1 = FV S1 = 1.3 (.81) = 1.05 Eq. 5-16bSDS= 2/3SMS= 2/3(2.05)= 1.37g Eq. 5-17aSD1= 2/3 SM1= 2/3 (1.05) = 0.7g Eq. 5-17b

gS 75.1 ≥ Tables 5-15,16; footnote a.

seismic design category ER = 8 (special moment frame) Table 5-17

Cs = 171.18

37.1 ==IR

SDS Equation 5-19

( ) sec25.1117035. 43

==T Equation 5-23

( ) 07.1825.1

7.1 ==≤ITR

SC D

s Equation 5-20

( )( ) 0603.137.1044.044. ==≥ ISC DSs Eq. 5-21

( )051.

1881.5.5. 1 ==≥

IR

SCs Equation 5-22

Cs = .07gV = Cs W = .07 (15,300) = 1071 k


∑=

=n

i

kii

kxx

vx

hw

hwC

1

Equation 5-26

Interpolate to find k:

k=1.0+(1.25-.5)/(2.5-.5)=1.375h9

1.375=1171.375=697.8Cv9=1700(697.8)/1700(3000.9)=.233F9=.233(1071)=250 kSee Table 5-21.

The story shear is determined by the sameprocedure as UBC-97.


( )∑ −==

n

xixiix hhFM τ Equation 5-27

τ = 1.0 for top ten storiesSince τ=1.0 the procedure is the same as forUBC-97. See Table 5-21.



Level hx(ft)

wx(k)

hx1.375 Cvx Fx

(k)

Vx(k)

Mx(ft-k)

9 117 1700 697.8 .233 250 250 3250

8 104 1700 593.5 .198 212 462 9256

7 91 1700 493.9 .165 177 639 17563

6 78 1700 399.6 .133 142 781 27716

5 65 1700 311.0 .104 111 892 39312

4 52 1700 228.8 .076 81 973 51967

3 39 1700 155.1 .051 55 1028 65325

2 26 1700 88.2 .029 31 1059 79092

1 13 1700 35.0 .011 12 1071 90870

Σ 15300. 3000.9 1.0 1071

• Allowable Inelastic Story Displacements:

seismic use group I∆a=.02hsx=.02(13)(12)=3.12 inches Eq. 5-30

• Equivalent Elastic Story Displacements:

Cd = 5.5 Table 5-17δ = Cdδxe/ I= 5.5 δxe Equation 5-31∆ ≤ 2.34/5.5 = 0.567 inches

• Reliability Factor:

The calculations are the same as for UBC-97(See example 5-2):

ρ = 1.0 Longitudinalρ = 1.12 Transverse

But in seismic design category E:

ρmax = 1.1 Section 5.4.18

Therefore, we need more transverse bays. Notethat ρ will be even higher using actual shears.

5.5 CONCLUSION

Basic linear static lateral force procedures ofthe 1997 UBC, the 1997 NEHRP, and the 2000IBC codes were discussed. Numerical exampleswere provided to highlight practicalapplications of these procedures.

274 Chapter 5

linear static seismic lateral force procedures

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