Sổ tay kết cấu thép - Section 9

LATERAL-FORCE DESIGN

9.1SECTION 9

LATERAL-FORCE DESIGNCharles W Roeder, P.E.

Professor of Civil Engineering, University of Washington,Seattle, Washington

Design of buildings for lateral forces requires a greater understanding of the load mechanismthan many other aspects of structural design To fulfill this need, this section provides abasic overview of current practice in seismic and wind design It also discusses recentchanges in design provisions and recent developments that will have an impact on futuredesign.

There are fundamental differences between design methods for wind and earth-quakeloading Wind-loading design is concerned with safety, but occupant comfort and service-ability is a dominant concern Wind loading does not require any greater understanding ofstructural behavior beyond that required for gravity and other loading As a result, the pri-mary emphasis of the treatment of wind loading in this section is on the loading and thedistribution of loading Design for seismic loading is primarily concerned with structuralsafety during major earthquakes, but serviceability and the potential for economic loss arealso of concern Earthquake loading requires an understanding of the behavior of structuralsystems under large, inelastic, cyclic deformations Much more detailed analysis of structuralbehavior is needed for application of earthquake design provisions, because structural be-havior is fundamentally different for seismic loading, and there are a number of detailedrequirements and provisions needed to assure acceptable seismic performance Because ofthese different concerns, the two types of loading are discussed separately in the following.

9.1DESCRIPTION OF WIND FORCES

The magnitude and distribution of wind velocity are the key elements in determining winddesign forces Mountainous or highly developed urban areas provide a rough surface, whichslows wind velocity near the surface of the earth and causes wind velocity to increase rapidlywith height above the earth’s surface Large, level open areas and bodies of water providelittle resistance to the surface wind speed, and wind velocity increases more slowly withheight Wind velocity increases with height in all cases but does not increase appreciablyabove the critical heights of about 950 ft for open terrain to 1500 ft for rough terrain Thisvariation of wind speed over height has been modeled as a power law:

nz

where V is the basic wind velocity, or velocity measured at a height zgabove ground and Vzis the velocity at height z above ground The coefficient n varies with the surface roughness.It generally ranges from 0.33 for open terrain to 0.14 for rough terrain The wind speeds Vzand V are the fastest-mile wind speeds, which are approximately the fastest average wind

speeds maintained over a distance of 1 mile Basic wind speeds are measured at an elevation

zgabove the surface of the earth at an open site Design wind loads are based on a statisticalanalysis of the maximum fastest-mile wind speed expected within a given recurrence interval,such as 50 years Statistical maps of wind speeds have been developed and are the basis ofpresent design methods However, the maps consider only regional variations in wind speedand do not consider tornadoes, tropical storms, or local wind currents The wind speed dataare maintained for open sites and must be corrected for other site conditions (Wind speedsfor elevations higher than the critical elevations mentioned previously are not affected bysurface conditions.)

Wind speeds Vware translated into pressure q by the equation

where the wind speed is in miles per hour and pressure, in psf.

The shape and geometry of the building have other effects on the wind pressure andpressure distribution Large inward pressures develop on the windward walls of enclosed

buildings and outward pressures develop on leeward walls, as illustrated in Fig 9.1a

Build-ings with openBuild-ings on the windward side will allow air to flow into the building, and internal

pressures may develop as depicted in Fig 9.1b These internal pressures cause loads on the

over-all structure and structural frame More important, these pressures place great demandson the attachment of roofing and external cladding Openings in a side wall or leeward wall

may cause an internal pressure in the building as illustrated in Fig 9.1c and d This buildup

of internal pressure depends on the size of the openings for all walls and the geometry of

the structure Slopes of roofs may affect the pressure distribution, as illustrated in Fig 9.1e.

Projections and overhangs (Fig 9.2) may also restrict the airflow and accumulate pressure.These effects must be considered in design.

The velocity used in the pressure calculation is the velocity of the wind relative to thestructure Thus, vibrations or movements of the structure occasionally may affect the mag-nitude of the relative velocity and pressure Structures with vibration characteristics whichcause significant changes in the relative velocity and pressure distribution are regarded assensitive to aerodynamic effects They may be susceptible to dynamic instability due tovortex shedding and flutter These may occur where local airflow around the structure causesdynamic amplification of the structural response because of the interaction of the structuralresponse with the airflow These undesirable conditions require special analysis that takesinto account the shape of the body, airflow around the body, dynamic characteristics of thestructure, wind speed, and other related factors As a result, dynamic instability is not in-cluded in the simplified methods included in this section.

The fastest-mile wind speed is smaller than the short-duration wind speed due to gusting.Corrections are made in design calculations for the effect of gusting through use of gustfactors, which increase design wind pressure to account for short-duration increases in windspeed The gust factors are largely affected by the roughness of the surface of the earth.They decrease with increasing height, reduced surface roughness, and duration of gusting.

of wind (a) High pressure on a solid wall on the windward side but outward or reduced inward pressure onthe leeward side (b) Wind entering through an opening in the windward wall induces outward pressure on theinterior of the walls (c) and (d ) Wind entering through openings in a side wall or a leeward wall produceinternal pressures in the building (e) On a slopng roof, high inward pressure develops on the windward side,

outward or reduced inward pressure on the leeward side.

FIGURE 9.2 Roof overhang restricts airflow, creates large local forces onthe structure.

Although gusting provides only a short-duration dynamic loading to the structure, a majorconcern may be the vibration, rocking, or buffeting caused by the dynamic effect Thepressure distribution caused by these combined effects must be applied to the building as awind load.

9.2DETERMINATION OF WIND LOADS

Wind loading as described in Art 9.1 is the basis for design wind loads specified in imum Design Loads for Buildings and Other Structures,’’ ASCE 7-88, American Society ofCivil Engineers Model building codes specify simplified methods based on these provisionsfor determining wind loads These methods can be used for most structures One such methodis incorporated in the ‘‘Uniform Building Code’’ (UBC) of the International Conference ofBuilding Officials, Inc (See Art 6.6 for ASCE 7-95.)

‘‘Min-9.2.1Wind-Load Provisions in the UBC

The basic wind speeds specified by the UBC for the continental United States and Alaskaare shown in Fig 9.3 The contours on the map indicate wind speeds that have a 2%probability of being exceeded in a year at a height 10 m above ground on open sites (Theseare wind speeds that are expected to occur once in 50 years.) The effects of extreme con-ditions, such as tornadoes, hurricanes, or local wind currents in mountainous regions are notcovered by this map Further, special wind regions are identified in the map where localwind velocity may significantly exceed the indicated values for the location The possibilityof occurrence of these local variations should be considered in design.

FIGURE 9.3 Contours indicate for regions of the continental United States and Alaska the basic wind speeds, mph,

the fastest-mile speeds 10 m above ground in open terrain with a 2% annual probability of occurrence (Based on data

in ‘‘Minimum Design Loads for Buildings and Other Structures,’’ ASCE 7-88, American Society of Civil Engineers andthe ‘‘Uniform Building Code,’’ International Conference of Building Officials.)

The stagnation pressures qs[Eq (9.3)] at a height of 10 m above ground are provided intabular form in the UBC:

The UBC integrates the combined effects of gusting, changes of wind velocity with height

above ground, and the local terrain or surface roughness of the earth in a coefficient, Ce.

Values of Ce are given in the UBC for specific exposure conditions as a stepwise function

of height (Table 9.1) The UBC defines three exposure conditions, B to D Exposure Crepresents open terrain (assumed in Fig 9.3) Exposure B applies to protected sites ExposureD is an extreme exposure primarily intended for open shorelines and coastal regions Co-efficient Ceas well as stagnation pressure qsare factors used in determination of design windpressures.

The UBC also specifies an importance factor I to be assigned to a building so that more

important structures are designed for larger forces to assure their serviceability after an

TABLE 9.1 Coefficient CeforEq (9.4)

Height, ft*

* Height above average level ofadjoining ground.

extreme windstorm For most buildings, I⫽ 1.0 For such buildings as hospitals, fire andpolice stations, and communications centers, and where the primary occupancy is for assem-

bly of 300 or more persons, I⫽1.15.

A final factor Cqdepends on the geometry of the structure and its appendages and on thecomponent or portion of the structure to be loaded It is intended to account for the pressuredistribution on buildings, which may affect the major load elements.

The design pressure p, psf, is then given by

The UBC presents two methods of distributing the pressures to the primary load-resisting

system Method 1 (Fig 9.4b) is a normal-force method, which distributes pressures normal

to the various parts of the building The pressures act simultaneously in a direction normal

to the plane of roofs or walls In this method, Cq⫽ 0.8 inward for all windward walls and0.5 outward for all leeward walls For winds parallel to the ridge line of sloped roofs and

for flat roofs, Cq⫽0.7 outward For winds perpendicular to the ridge line, C⫽0 7 outwardon the leeward side.

On the windward side:

Cq⫽ 0.7 outward with roof slope less than 2:12

⫽ 0.9 outward or 0.3 inward with roof slope between 2:12 and 9:12

⫽ 0.4 inward with roof slope between 9:12 and 12:12

⫽ 0.7 inward with roof slope greater than 12:12.

Method 2 (Fig 9.4c) uses a projected-area approach with horizontal and vertical pressures

applied simultaneously to the vertical and horizontal projections of the building, respectively.

For this case, Cq⫽1.4 on the vertical projected area of any structure over 40 ft tall, 1.3 onthe vertical projected area of any shorter structure, and 0.7 upward (uplift) on any horizontalprojection.

FIGURE 9.4 Distribution of wind pressure on a single-story building with sloping roof (a)Buildmg in open terrain subjected to a 70-mph wind; (b) pressures computed by the normal-forcemethod; (c) pressures computed by the projected-area method.

Individual components and local areas may have local pressure concentrations due tolocal disturbance of the airflow (Fig 9.2) These normally do not affect the design of loadframes and major load-carrying elements, but they may require increased resistance for ar-chitectural elements, local structural members supporting these elements, and attachment

details The UBC also contains values of Cqfor these local conditions Some of these

com-ponent requirements for Cqfor wall elements include:1.2 inward for all wall elements

1.2 outward for wall elements of enclosed and unenclosed structures

1.6 outward for wall elements of open structures1.3 inward and outward for all parapet walls

An unenclosed structure is a structure with openings in one or more walls, but the sumsof the openings on each side are within 15% of each other An open structure has similarwall openings but the sum of the openings on one wall is more than 15% greater that thesum of the openings of other walls Open structures may accumulate larger internal pressuresthan enclosed or unenclosed structures (Fig 9.1) and must be designed for larger outwardpressures.

There are similar component requirements for Cqfor roof elements These include:

Cq⫽ 1.7 outward for roof elements of open structures with slope less than 2:12

⫽1.6 outward or 0.8 inward for roof elements of open structures with slope greaterthan 2:12 but less than 7:12

⫽1.7 inward and outward for roof elements of open structures with slope greater than7:12

⫽1.3 outward for roof elements of enclosed and unenclosed structures with roof slopeless than 7:12

⫽1.3 outward or inward for roof elements of enclosed and unenclosed structures withroof slope greater than 7:12

Corners of wall elements must also be subjected to Cq⫽1.5 outward or 1.2 inward for thelesser of 10 ft or 10% of the least width of the structure Roof eaves and other projectionsare also collectors of concentrated wind pressure (Fig 9.2) Building codes require consid-erations of these local pressure distributions with

Cq⫽2.3 upward of roof rakes, ridges, and eaves without overhang and slope less than2:12

⫽ 2.6 upward of roof rakes, ridges, and eaves without overhang and slope greaterthan 2:12 but less than 7:12

⫽ 1.6 upward of roof rakes, ridges, and eaves without overhang and slope greaterthan 7:12

⫽ 0.5 greater coefficient for overhanging elements and canopies.

These factors combine to produce a complex distribution of design pressures Some of thedistributions are illustrated in Fig 9.5.

These localized distributions affect the strength of local elements and the strength ofattachment details of local elements, but they do not affect the global strength requirementsof the structure.

9.2.2Other Provisions for Wind Loads

Alternative methods for determining wind loads, such as that in ASCE Standard 7-88, areavailable, and give more detailed provisions than those in the UBC (Art 9.2.1) for definingand distributing wind loads Tabulated data may be more detailed in these other methods,and more equations may be required However the pressure distributions are similar to thatprovided by the UBC.

These methods provide basic wind loads for buildings, but they do not specify how toestimate or control aerodynamic effects Aerodynamic effects may result in interaction be-tween the dynamic response of a structure and the wind flow around it This interaction mayamplify the dynamic response and cause considerable occupant discomfort during somewindstorms.

FIGURE 9.5 Typical distributions of local wind pressures.

Furthermore, local variations in wind velocity can be caused by adjacent buildings Thewind may be funneled onto the structure, or the structure may be protected by surroundingstructures Wind tunnel testing is often required for designing for these effects Local windvariations are most likely to be significant for tall, slender structures As a general rule,buildings with unusual geometry or a height more than 5 times the base dimension are logicalcandidates for a wind tunnel test Such a test can reveal the predominant wind speeds anddirections for the site, local effects such as channeling of the wind by surrounding buildings,effects of the new building on existing surrounding structures, the dynamic response of thebuilding, and the interaction of the response with the wind velocity The model used for thetest can include the stiffness of the building, and wind pressures can be measured at criticallocations Major structures often are based on wind-tunnel-test results, since greater economyand more predictable structural performance are possible.

Special structures, such as antennas, transmission lines, and supports for signs and ing, may also be susceptible to aerodynamic effects and require special analysis Aerody-namic effects are beyond the scope of this section, but analytical methods of dealing withthese are available Wind tunnel testing may also be required for these systems.

light-(E Simu and R H Scanlan, Wind Effects on Structures, Wiley-lnterscience, New York.)

9.3SEISMIC LOADS IN MODEL CODES

The ‘‘Uniform Building Code’’ (UBC) of the International Conference of Building Officialshas been the primary source of seismic design provisions for the United States It adopts

provisions based on recommendations of the Structural Engineers Association of California(SEAOC) The UBC and SEAOC define design forces and establish detailed requirementsfor seismic design of many structural types Another model code is the ‘‘National EarthquakeHazard Reduction Program (NEHRP) Recommended Provisions for the Development ofSeismic Regulations for New Buildings,’’ of the Building Seismic Safety Council (BSSC),Federal Emergency Management Agency (FEMA), Washington, D.C There have historicallybeen considerable similarities between the UBC and NEHRP recommendations, since therationale is similar for both documents and many engineers participate in the developmentof both documents However, there have also been differences in the detailed approach usedby the UBC and NEHRP provisions, and in recent years efforts have been made to resolvethese differences, because of the move toward an International Building Code (IBC) As aresult, the 1997 edition of the UBC has much greater similarity with NEHRP than has pasteditions ‘‘Minimum Design Loads for Buildings and Other Structures,’’ ASCE 7-95, Amer-ican Society of Civil Engineers adopts seismic force requirements similar to those includedin the NEHRP provisions.

The American Institute of Steel Construction (AISC) promulgates ‘‘Seismic Design visions for Structural Steel Buildings.’’ This document does not establish design forces, butit provides detailed design requirement for steel structures The detailed seismic design pro-visions provided in the UBC and NEHRP provisions include most of the AISC seismicprovisions, since the AISC seismic provisions are often directly inserted into the model codesor are referenced in the seismic design provisions As a result, there is great similaritybetween the UBC, NEHRP, ASCE and AISC LRFD provisions The UBC and AISC pro-visions are emphasized in the following, but several issues that are better understood byexamining NEHRP provisions are noted.

Pro-9.4EQUIVALENT STATIC FORCES FOR SEISMIC DESIGN

The UBC offers two methods for determining and distributing seismic design loads One isthe dynamic method, which is required to be used for a structure that is irregular or ofunusual proportions (Art 9.5) The other specifies equivalent static forces and is the mostwidely used, because of its relative simplicity.

The equivalent-static-force method defines the static shear at the base of the building asbeing the smaller of

FIGURE 9.6 Zones of probable seismic intensity (Adapted from the ‘‘Uniform Building Code,’’ International

Conference of Building Officials.)

of floor live loads in storage and warehouse occupancies The base is the level at whichseismic motions are imparted to the building.

These equations satisfy the multiple goals of narrowing historic differences between theNEHRP and UBC seismic forces provisions while addressing specific concerns of the regions

with greater seismic risk Equation (9.5a) provides a minimum base shear for longer periodstructures, while Eq (9.5b) establishes the minimum base shear for short period structures.Caand CVare seismic coefficients for the acceleration dependent (short period) and velocitydependent (intermediate and long period) structures, respectively In the NEHRP provisions,these seismic coefficients are determined from a detailed mapping of the United States com-bined with consideration of the soil conditions at the site The maps were developed by theUnited States Geological Survey based upon a statistical evaluation of expected earthquakeaccelerations The UBC includes a much simplified mapping as shown in Fig 9.6 From this

map, a Z coefficient is determined for a given location, and this coefficient can be translatedinto CVand Ca terms through tables which define the seismic coefficients based upon theseismic zone and the soil conditions at the site In general, soft soils translate into larger

seismic coefficients In seismic zone 4, CVand Caalso depend upon the seismic source typeand the distance from the earthquake fault The seismic source type is either A, B, or C.The A designation is given to faults with more rapid slip rates (greater than 5 mm per year)and maximum possible moment magnitudes of 7.0 or greater, and the source type C is givento all faults with possible moment magnitudes less than 6.5 and with slower slip rates (lessthan 2 mm per year).

Equation (9.6a) provides a lower bound limit on all seismic forces, and Eq (9.6b)

pro-vides a lower bound seismic base shear which accounts for the increased damage expected

with buildings constructed very near the earthquake fault The NVterm varies between 1.0

and 2.0, and it depends upon the seismic source type and the distance from this source NV

is always 1.0 for structures more than 15 km from the source and for source type C.

T is the fundamental period of the structure It may be computed by dynamic analysis or

by an approximate equation such as

3 / 4

where hnis the height, ft, from the base of the building to level n, which is the uppermostlevel in the main portion of the structure The factor Ct⫽ 0.035 for steel moment-resisting

frames, which are relatively flexible structures, Ct⫽0.030 for eccentric-braced frames and

reinforced concrete moment-resisting frames Ct⫽0.02 for braced frames and other relativelystiff structures.

Equation (9.7) yields periods that are shorter than those computed for some steel

struc-tures Hence, when T is computed from the structural properties and deformation istics of the resisting elements, the UBC requires that the period used in Eq (9.5a) be not

character-more than 30% larger than computed by Eq (9.7) for seismic zone 4 and not character-more than40% larger than that computed for the other seismic zones This limitation is particularlyimportant for steel moment-resisting frames because it frequently controls their design.

The seismic design shear V depends on regional seismicity (Fig 9.6), which is quantifiedby a zone factor Z, which approximates an effective peak ground acceleration (on firms soilfor the region) Z⫽0.075 for zone 1 in Fig 9.6, 0.15 for zone 2, 0.30 for zone 3, and 0.40for zone 4 Seismic zone 2 is divided into two regions (2A and 2B) which have the samedesign base shear, but different detailing requirements.

The importance factor I in Eqs (9.5) and (9.6) depends on the importance of the building.I ⫽ 1.25 for essential or hazardous facilities and 1.0 for standard or special-occupancystructures.

The coefficient R in Eq (9.5) reduces the seismic design forces in recognition of the

ductility achieved by the structural system during a major earthquake A measure of the

ductility and inelastic behavior of the structure, R ranges from 2.2 to 8.5 The largest valuesof R are used for ductile structural systems that can dissipate large amounts of energy and

can sustain large inelastic deformations The smallest values are intended to assure nearlyelastic behavior when the overstrength normally achieved in design is considered Specialsteel moment resisting frames have historically been regarded as one of the most ductile

structural systems and are assigned R ⫽ 8.5 Moment resisting steel frames are dimensional frames, where the members and joints are capable of resisting lateral forces onthe structure primarily by flexure While this structural system is still highly regarded, theperformance of special moment frames during the January 17, 1994, Northridge earthquakeraised serious questions as to the performance of these structures, and this will be discussedin some detail in Art 9.6 Ordinary moment frames are designed to lesser ductility criteria,

three-and R⫽4.5.

For steel eccentric braced frames (Fig 9.13), at least one end of each diagonal braceintersects a beam at a point away from the column-girder joint or from an adjacent brace-girder joint This eccentric intersection forms a link beam, which must be designed to yieldin shear or bending to prevent buckling of the brace This system is also quite ductile, and

Concentrically braced frames (Fig 9.12) have concentric joints for brace, beam and umn and the inelastic seismic behavior is dominated by buckling of the brace The ductilityachieved with buckling systems is limited because brace fracture may occur during the

col-inelastic deformation, and as a result R ⫽5.6 for these ordinary concentrically braced tems Fracture of the brace is less likely to occur if the connections of the system are designedto avoid deterioration and fracture during brace buckling As a result, special concentrically

sys-braced frames are designed for these enhanced ductility conditions and R⫽6.4.

Finally dual systems, such as special steel moment-resisting frames capable of resistingat least 0.25 V, combined with steel eccentric braced frames, special concentrically braced

frames, or ordinary braced frames have improved inelastic performance over the individual

systems acting alone, and R⫽ 8.5, 7.5 and 6.5, respectively.

Force Distribution.The seismic base shear V (Eqs (9.5) and (9.6)) is distributed out the structure in accordance with its mass and stiffness A concentrated force Ft, however,

through-should be applied to the top of the structure if the period T is greater than 0.7 sec.

where Fiand wiare the seismic force and floor weight at the ith level and xiis the height

from the base to the ith floor The force Fiat each floor is distributed horizontally in portion to the distribution of the mass of the floor The stiffness of the floor diaphragm mustbe evaluated to determine whether the diaphragm satisfies the rigid or flexible diaphragmstiffness requirements With rigid diaphragms, the horizontal forces are distributed to verticalframes with consideration of the horizontal mass distribution (including minimum torsion)and the relative stiffness of the frames With flexible diaphragms, the horizontal forces aredistributed to vertical frames with consideration of the mass distribution and the tributaryarea of each frame Floor slabs and their attachments between floor diaphragms and lateralload frames must have adequate strength to distribute these inertial forces The frames mustbe designed for a minimum torsion, which is produced by a mass eccentricity of 5% of thenormal maximum base dimension This minimum is in addition to torsion due to the com-puted eccentricity between the center of mass and gravity.

pro-Deflections and Element Design Forces. The UBC permits both the Allowable Stress andStrength Design methods The strength design methods use load factors which are derivedbased upon a statistical evaluation of the probability of the various load occurrences and arequite similar to those used in the AISC LRFD provisions Allowable Stress Design uses theseismic design loads as service loads A separate set of Allowable Stress load factors areused to translate the seismic design forces to service load forces which can be used withallowable stresses and stress increases where appropriate The horizontal earthquake loadsare applied to the structure as described earlier They are applied separately in all horizontaldirections, but they are combined with vertical acceleration effects in element evaluations.This is done as follows:

the maximum story shear carried by a single element compared to the total story shear This

definition varies somewhat for different structural systems ABis the ground floor area of thestructure This redundancy factor tends to increase the design forces for elements in structureswith little redundancy in the lateral load system.

Element forces and structural deflections are determined by computer analysis or theymay be estimated by approximate analysis methods described later in this section The storydrift and critical deflections of the structure can be estimated by a nonlinear analysis in-

cluding P⫺ ⌬effects Most structural engineers are unprepared to do a nonlinear analysis,and so an alternate method is provided With this alternate method, a linear elastic analysisis performed with the factored loads applied The maximum story drifts and deflections,⌬S,from this elastic analysis are then corrected by the equation

to account for inelastic behavior and to estimate the maximum inelastic response,⌬M P⫺⌬ effects are intended to be included in this estimated deflection The maximum inelasticstory drift,⌬M, must not exceed 0.025 times the story height for structures with periods lessthan 0.7 sec and 0.020 times story height for structures with longer periods.

9.5DYNAMIC METHOD OF SEISMIC LOAD DISTRIBUTION

The ‘‘Uniform Building Code’’ static-force method (Art 9.4) is based on a single-moderesponse with approximate load distributions and corrections for higher-mode response.These simplifications are appropriate for simple regular structures However, they do notconsider the full range of seismic behavior in complex structures The dynamic method ofseismic analysis is required for many structures with unusual or irregular geometry, since itresults in distributions of seismic design forces that are consistent with the distribution ofmass and stiffness of the frames, rather than arbitrary and empirical rules Irregular structuresinclude frames with any of the following characteristics:

The lateral stiffness of any story is less than 70% of that of the story above or less than80% of the average stiffness for the three stories above

The mass of any story is more than 150% of the effective mass for an adjacent story,except for a light roof above

The horizontal dimension of the lateral-force-resisting system in any story is more than130% of that of an adjacent story

The story strength is less than 80% of the story above

Frames with a story strength that is less than 80% of that of the story above must be

designed with consideration of the P⫺ ⌬effects caused by gravity loading combined withthe seismic loading.

Frames with horizontal irregularities place great demands on floors acting as diaphragmsand the horizontal load-distribution system Special care is required in their design when anyof the following conditions exist:

The maximum story drift due to torsional irregularity is more than 1.2 times the averagestory drift for the two ends of the structure.

There are reentrant corners in the plan of the structure with projections more than 15%of the plan dimension

The diaphragms are discontinuous or have cutouts or openings totaling more than 50%of the enclosed area or changes of stiffness of more than 50%

There are discontinuities in the lateral-force load path

Irregular structures commonly require use of a variation of the dynamic method of seismicanalysis, since it provides a more appropriate distribution of design loads Many of thesestructures should also be subjected to a step-by-step dynamic analysis (linear or nonlinear)for specific accelerations to check the design further.

The dynamic method is based on equations of motion for linear-elastic seismic response.The equation of motion for a single-degree-of-freedom system subjected to a seismic ground

acceleration agmay be expressed as2

where d2x / dt2is the acceleration of the structure, dx / dt is the velocity relative to the groundmotion, and x is the displacement from an equilibrium position The coefficients m, c, andk are the mass, damping, and stiffness of the system, respectively Equation (9.12) can be

solved by a number of methods.

The maximum acceleration is often expressed as a function of the fundamental period ofvibration of the structure in a response spectrum The response spectrum depends on theacceleration record Since response varies considerably with acceleration records and struc-tural period, smoothed response spectra are commonly used in design to account for themany uncertainties in future earthquakes and actual structural characteristics.

Most structures are multidegree-of-freedom systems The n equations of motion for asystem with n degrees of freedom are commonly written in matrix form as

[M]{x¨}⫹[C]{x¨}⫹[K]{x}⫽ ⫺[M]{B} ag (9.13)

where [M], [C], and [K] are n⫻n square matrices of the mass, damping, and stiffness, and

{ }, { }, and {x} are column vectors of the acceleration, relative velocity, and relativex¨x˙

displacement The column vector {B} defines the direction of the ground acceleration relative

to the orientation of the mass matrix The multidegree-of-freedom equations are coupled.They can be solved simultaneously by a number of methods However, the single-degree-of-freedom response spectrum method is also commonly used for multidegree-of-freedom

systems The solution is assumed to be separable and the n eigenvalues (natural frequencies)

␻iand eigenvectors (mode shapes) {⌽i} are found The solutions for the relative

displace-ments, relative velocities and accelerations are then for i equal 1 to n:n

The response-spectrum technique can then be used to find the maximum values of ƒj(t)

for each mode of vibration Figure 9.7 shows the design response spectra recommended bythe UBC unless site-specific spectra are employed The response is based on calculations ofthe single-degree-of-freedom elastic response for a range of earthquake acceleration records

FIGURE 9.7 Response spectra recommended for the dynamic method for determining seismic behavior of

structures (After the ‘‘Uniform Building Code,’’ 1997 International Conference of Building Officials.)

and is normalized by the zone factor Z used in the static-force method Given the modes ofvibration for a multidegree-of-freedom system, a spectral acceleration for each mode, Sai,

can be determined from the response spectra The base shear Viacting in each mode canthen be determined from

Earthquake Engineering and Structural Dynamics, vol 9, pp 187–194, 1981.) The method

degenerates into a variation of the square root of the sum of the squares (SRSS) method

when the modes of vibration are well-separated The summation must include an adequatenumber of modes to assure that at least 90% of the mass of the structure is participating inthe seismic loading.

The total seismic design force and the force distribution over the height and width of thestructure for each mode can be determined by this method The combined force distributiontakes into account the variation of mass and stiffness of the structure, unusual aspects of thestructure, and the dynamic response in the full range of modes of vibration, rather than thesingle mode used in the static-force method The combined forces are used to design the

structure, often reduced by R in accordance with the ductility of the structural system In

many respects, the dynamic method is much more rational than the static-force method,which involves many more assumptions for computing and distributing design forces Thedynamic method sometimes permits smaller seismic design forces than the static-forcemethod However, while it offers many rational advantages, the dynamic method is still alinear-elastic approximation to an inelastic-design method As a result, it assumes that theinelastic response is distributed throughout the structure in the same manner as predicted bythe elastic-mode shapes This assumption may be inadequate if there is a brittle link in thesystem.

9.6STRUCTURAL STEEL SYSTEMS FOR SEISMIC DESIGN

Since seismic loading is an inertial loading, the forces are dependent on the dynamic acteristics of the acceleration record and the structure Seismic design codes use a responsespectrum as shown in Fig 9.7 to model these dynamic characteristics These forces areusually reduced in accordance with the ductility of the structure This reduction is accom-

char-plished by the R factor in the static-force method, and the reduction may be quite large (Art.

9.5) The designer must ensure that the structure is capable of developing the required tility, as it is well-known that the available ductility varies with different structural systems.Therefore, the structural engineer must ensure that the structural system selected for a given

duc-application is capable of achieving the ductility required for the R value used in the design.

The engineer also must complete the details of the design of members and connections sothat the structure lives up to these expectations.

Evaluation of Ductility. Two major factors may affect evaluation of the ductility of tural systems First, the ductility is often measured by the hysteretic behavior of the criticalcomponents The hysteretic behavior is usually examined by observing the cyclic force-deflection (or moment-rotation) behavior as shown in Fig 9.9 The slope of the curvesrepresents the stiffness of the structure or component The enclosed areas represent the energythat is dissipated, and this can be large, because of the repeated cycles of vibration These

struc-enclosed areas are sometimes full and fat (Fig 9.9a), or they may be pinched or distorted(Fig 9.9b) The hysteretic curves also show the inelastic deformation that can be tolerated

at various resistance levels Structural framing with curves enclosing a large area representinglarge dissipated energy, and structural framing which can tolerate large inelastic deformationswithout excessive loss in resistance, are regarded as superior systems for resisting seismic

loading As a result, these systems are commonly designed with larger R values and smaller

seismic loads.

Special steel moment-resisting frames and eccentric braced frames, defined in Art 9.4,are capable of developing large plastic deformations and large hysteretic areas As a result,

they are designed for larger values of R, thus smaller seismic forces and greater inelastic

deformation This hysteretic behavior is important, since it dampens the inelastic responseand improves the seismic performance of the structure without requiring excessive strengthor deformation in the structure This is illustrated in Fig 9.8, which shows the inelastic

FIGURE 9.8 Curves show inelastic dynamic response of steel frames designed for different values of R,

plotted for eight-story, weak-column, strong-beam framing with 2% damping (Imperial Valley College record).

dynamic response of two steel moment-resisting frames, which had identical mass, stiffnessand seismic excitation (1979 Imperial Valley College), but different seismic resistance Thestory drift and inelastic deformation cycles are larger for the structure with the smaller

resistance This shows that structures with smaller design force (larger R) require the structure

to have the ability to maintain its integrity through larger inelastic deformations than if a

larger design force (smaller R) were employed.

While some steel structures are very ductile, not all structures have this great ductility.Fracture of the connections has a very detrimental effect on the structural performance, sinceit may cause a significant loss in both resistance and deformational capacity Local and global

buckling may also change the hysteretic behavior from that of Fig 9.9a to Fig 9.9b The

combined effects of these potential problems means that the structural engineer must payparticular attention to the design details in the seismic design of buildings, since those detailsare essential to ensuring good seismic performance.

Effects of Inelastic Deformations. The distribution of inelastic deformation is a secondfactor that can effect the inelastic seismic performance of a structural system Some structuralsystems concentrate the inelastic deformation (ductility demand) into a small portion of thestructure This can dramatically increase the ductility demand for that portion of the structure.This concentration of damage is sometimes related to factors that cause pinched hystereticbehavior, since buckling may change the stiffness distribution as well as affect the energydissipation.

Ductility demand, however, can also be related to other factors Figure 9.10 shows thecomputed inelastic response of two steel moment-resisting frames that have identical massand nearly identical strength and stiffness and are subject to the same acceleration record asthat in Fig 9.8 The frames differ, however, in that one is designed to yield in the beamswhile the other is designed to yield in the columns This difference in design concept resultsin a significant difference in seismic response and ductility demand Design codes attemptto assure greater ductility from structures designed for smaller seismic forces, but attaining

FIGURE 9.9 Hysteretic behavior of three steel frames (a) Moment-resisting frame: (b) centric braced frame: (c) eccentric braced frame.

con-FIGURE 9.9 Continued.

FIGURE 9.10 Curves show inelastic dynamic response of two steel frames with identicalmass and nearly identical strength and stiffness but designed with two different strategies fordetermining inelastic deformations.

this objective is complicated by the fact that ductility and ductility demand are not fullyunderstood.

Steel moment-resisting frames have historically been regarded as the most ductile

struc-tural system for seismic design, but a number of special steel moment frames sustaineddamage during the January 17, 1994, Northridge earthquake In these buildings, cracks wereinitiated near the flange weld Some of the cracks penetrated into the column and panel zoneof the beam-column connections as illustrated in the photo of Fig 9.11, but others penetratedinto the beam flange or the flange welds and the heat-affected zones of these welds None

FIGURE 9.11 Photograph of crack through the column flange andinto the column web or panel zone of connection.

of these buildings collapsed and there was no loss of life, but the economic loss was siderable This unexpected damage has caused a new evaluation of the design of momentframe connections through the SAC Steel Project SAC is a joint venture of SEAOC, ATC(Applied Technology Council), and CUREE (California Universities for Research in Earth-quake Engineering), and the joint venture is funded by FEMA This work is still in progress,but it is clearly leading structural engineers in new directions in the design of special steelmoment frame buildings The work shows that great ductility is possible, but it also showsthat the engineer must exercise great care in the selection and design of members and con-nections The requirements that are evolving for special moment frames are briefly sum-marized in Art 9.7.1.

con-Concentric braced frames, defined in Art 9.4, economically provide much larger

strength and stiffness than moment-resisting frames with the same amount of steel Thereare a wide range of bracing configurations, and considerable variations in structural perform-ance may result from these different configurations Figure 9.12 shows some concentricbracing configurations The braces, which provide the bulk of the stiffness in concentricallybraced frames, attract very large compressive and tensile forces during an earthquake As aresult, compressive buckling of the braces often dominates the behavior of these frames The

pinched cyclic force-deflection behavior shown in Fig 9.9b commonly results, and failure

of braces may be quite dramatic Therefore, concentrically braced frames are regarded asstiffer, stronger but less ductile than steel moment-resisting frames In recent years, researchhas shown that concentrically braced frames can sustain relatively large inelastic deformationwithout failure if greater care is used in the design and selection of the braces and the braceconnections Concentrically braced frames, which are designed to these higher ductility stan-dards, can be designed for smaller seismic design forces and are called special concentricallybraced frames Different design provisions are required for ordinary concentrically bracedframes and special concentrically braced frames These are summarized in Art 9.7.2.

Eccentric braced frames, defined in Art 9.4, can combine the strength and stiffness of

concentrically braced frames with the good ductility of moment-resisting frames Eccentricbraced frames incorporate a deliberately controlled eccentricity in the brace connections (Fig.9.13) The eccentricity and the link beams are carefully chosen to prevent buckling of thebrace, and provide a ductile mechanism for energy dissipation If they are properly designed,

eccentric braced frames lead to good inelastic performance as depicted in Fig 9.9c, but they

require yet another set of design provisions, which are summarized in Art 9.7.3.

Dual systems, defined in Art 9.4, may combine the strength and stiffness of a braced

frame and shear wall with the good inelastic performance of special steel moment-resisting

frames Dual systems are frequently assigned an R value and seismic design force that are