A Modal Pushover Analysis Procedure for Estimating Seismic Demands for Buildings - Chopra

EARTHQUAKE ENGINEERING ENGINEERING AND STRUCTURAL STRUCTURAL DYNAMICS :561–582 (DOI: (DOI: 10.1002/ 10.1002/eqe.1 eqe.144) 44) Earthquake Engng Struct. Dyn. 2002; 31:561–582

A modal pushover analysis procedure for estimating seismic demands for buildings Anil K. Chopra1;∗;† and Rakesh K. Goel 2 1 Department

of Civil and Environmental Engineering; Engineering ; University of California at Berkeley; Berkeley ; Berkeley; Berkeley; CA, 94720-1710; 94720-1710 ; U.S.A. 2 Department of Civil and Environmental Engineering; Engineering ; California Polytechnic State University; University ; San Luis Obispo, CA; CA; U.S.A.

SUMMARY Develope Developed d herein herein is an improved improved pushover analysis procedure procedure based on structural structural dynamics dynamics theory, theory, which retains the conceptual simplicity and computational attractiveness of current procedures with invariant force distribution. In this modal pushover analysis (MPA), the seismic demand due to individual terms in the modal expansion of the eective earthquake forces is determined by a pushover analysis using the inertia force distribution for each mode. Combining these ‘modal’ demands due to the ÿrst two or three terms of the expansion provides an estimate of the total seismic demand on inelastic systems. When applied to elastic systems, the MPA procedure is shown to be equivalent to standard response spectrum analysis (RSA). When the peak inelastic response of a 9-storey steel building determined by the approximate approximate MPA procedure procedure is compared compared with rigorous rigorous non-linear non-linear response response history analysis, it is demonstrated that MPA estimates the response of buildings responding well into the inelastic range to a similar degree of accuracy as RSA in estimating peak response of elastic systems. Thus, the MPA procedur proceduree is accurate accurate enough for practical practical application application in building building evaluation evaluation and design. design. Copyright Copyright ? 2001 John Wiley & Sons, Ltd. KEY WORDS:

building building evaluation evaluation and retroÿt; modal analysis; analysis; pushover; pushover; seismic seismic demands demands

INTRODUCTION Estima Estimatin ting g seismic seismic demands demands at low performa performance nce levels, levels, such such as life life safety safety and collap collapse se preprevention, vention, requires requires explicit explicit considerati consideration on of inelastic inelastic behaviour behaviour of the structure. structure. While non-linear non-linear response history analysis (RHA) is the most rigorous procedure to compute seismic demands, curren currentt civil civil engine engineeri ering ng practic practicee prefer preferss to use the non-li non-linea nearr static static proced procedure ure (NSP) (NSP) or pushover analysis in FEMA-273 [1]. The seismic demands are computed by non-linear static

∗

Correspondence to: Anil K. Chopra, Department of Civil and Environmental Engineering, University of California at Berkeley, Berkeley, CA 94720-1710, U.S.A. † E-mail: [email protected]

Copyright ? 2001 John Wiley & Sons, Ltd.

Received 15 January 2001 Revised 31 August 2001 Accepted 31 August 2001

562

A. K. CHOPRA AND R. K. GOEL

analysis of the structure subjected to monotonically increasing lateral forces with an invariant height-wise distribution until a predetermined target displacement is reached. Both the force distribution and target displacement are based on the assumption that the response is controlled by the fundamental mode and that the mode shape remains unchanged after the structure yields. Obviously, after the structure yields, both assumptions are approximate, but investigations [2–9] have led to good estimates of seismic demands. However, such satisfactory predictions of seismic seismic demand demandss are mos mostly tly restri restricte cted d to lowlow- and medium medium-ri -rise se structu structures res provid provided ed the inelastic action is distributed throughout the height of the structure [7 ; 10]. None of the invariant force distributions can account for the contributions of higher modes to respon response, se, or for a redist redistrib ributi ution on of inerti inertiaa forces forces becaus becausee of structu structural ral yieldi yielding ng and the associated changes in the vibration properties of the structure. To overcome these limitations, severa severall resear researche chers rs have have propos proposed ed adapti adaptive ve force force distri distribut bution ionss that that attemp attemptt to follow follow more more closely the time-variant distributions of inertia forces [5 ; 11; 11; 12]. While these adaptive force distri distribut bution ionss may provid providee better better estimat estimates es of seismi seismicc demand demandss [12], [12], they they are concep conceptua tually lly complicated complicated and computation computationally ally demanding demanding for routine routine application application in structural structural engineerin engineering g practice. Attempts have also been made to consider more than the fundamental vibration mode in pushover analysis [12–16]. The princi principal pal object objective ive of this this investi investigat gation ion is to develo develop p an improv improved ed pus pushov hover er analyanalysis procedure based on structural dynamics theory that retains the conceptual simplicity and computational attractiveness of the procedure with invariant force distribution—now common in structural engineering practice. First, we develop a modal pushover analysis (MPA) procedure for linearly elastic buildings and demonstrate that it is equivalent to the well-known response spectrum analysis (RSA) procedure. The MPA procedure is then extended to inelastic buildings, the underlying assumptions and approximations are identiÿed, and the errors in the procedure relative to a rigorous non-linear RHA are documented.

DYNAMIC AND PUSHOVER ANALYSIS PROCEDURES: ELASTIC BUILDINGS Modal response history analysis The dierential equations governing the response of a multistorey building to horizontal earthquake quake ground ground motion motion u g (t ) are as follows: mu + cu˙ + ku = −mÃu g (t )

(1)

where u is the vector of N lateral oor displacements relative to the ground, m; c; and k are the mass, classical damping, and lateral stiness matrices of the systems; each element of the inuence vector Ã is equal to unity. The right-hand side of Equation (1) can be interpreted as eective earthquake forces: pe (t ) = −mÃu g (t )

(2)

The spatial distribution of these eective forces over the height of the building is deÿned by the vector s = mÃ and their their time time variation variation by u g (t ). ). This force distribution can be expanded Copyright ? 2001 John Wiley & Sons, Ltd.

Earthquake Engng Struct. Dyn. 2002; 31:561–582

ESTIMATING SEISMIC DEMANDS FOR BUILDINGS

563

as a summation of modal inertia force distribution s n [17, Section 13: 13 :2]: N

mÃ =

N

 = sn

n=1

n m Mn

(3)

n=1

where Mn is the nth natural vibration mode of the structure, and n =

Ln ; M n

Ln = MT n mÃ;

M n = MT n mMn

(4)

The eective earthquake forces can then be expressed as N



pe (t ) =

n=1

N

pe ; e ; n (t ) =



−

s n u g (t )

(5)

n=1

The contribution of the nth mode to s and to pe (t ) are: s n = n mMn

 g (t ) pe ; e ; n (t ) = −s n u

(6a) (6b)

The response of the MDF system to pe ; e ; n (t ) is entirely in the nth-mode, with no contributions from other modes. Then the oor displacements are u n (t ) = Mn qn (t )

(7)

where the modal co-ordinate qn (t ) is governed by qn + 2 2n !n q˙n + !2n qn = −n u g (t )

(8)

in which !n is the natural vibration frequency and n is the damping ratio for the nth mode. The solution qn of Equation (8) is given by qn (t ) = n Dn (t )

(9)

where Dn (t ) is governed by the equation of motion for the nth-mode linear SDF system, an SDF system with vibration vibration properties— properties—natur natural al frequency frequency !n and dampin damping g ratio ratio n —of —of the nth-mod th-modee of the MDF system, system, subjecte subjected d to u g (t ): ): 2n !n D˙ n + !n2 Dn = −u g (t ) D n + 2

(10)

Substituting Equation (9) into Equation (7) gives the oor displacements u n (t ) = n Mn Dn (t )

(11)

Any response response quantity quantity r (t )—storey )—storey drifts, internal element forces, etc.—can be expressed as r n (t ) = r nst An (t )

(12)

where r nst denotes the modal static response, the static value of r due to external forces s n , and An (t ) = !n2 Dn (t ) Copyright ? 2001 John Wiley & Sons, Ltd.

(13)


564


Forces sn

An (t ) t )

ω n , ζ n

r n st

¨ u g (t ) t )

(a) Static Analysis of

(b) Dynamic Analysis of

Structure

SDF System

Figure 1. Conceptual explanation of modal RHA of elastic MDF systems.

is the pseudo pseudo-ac -accel celera eratio tion n respon response se of the nth-m th-mod odee SDF SDF syste system m [17, [17, Sect Sectio ion n 13: 13:1]. 1]. The The two analyses that lead to r nst and An (t ) are shown schematically in Figure 1. Equations (11) and (12) represent the response of the MDF system to pe ; [Equation (6b)]. Therefore, Therefore, e ; n (t ) [Equation the response of the system to the total excitation pe (t ) is N

N

 (t ) =   D (t )  r (t ) =  r A (t ) r (t ) =

u(t ) =

un

n Mn

n=1

n

(14)

n=1

N

N

st n

n

n=1

n=1

n

(15)

This is the classical modal RHA procedure: Equation (8) is the standard modal equation governing qn (t ), ) , Equa Equati tion onss (11) (11) and and (12) (12) deÿn deÿnee the the cont contri ribu buti tion on of the the nth-m th-mod odee to the the respon response, se, and Equati Equations ons (14) (14) and (15) (15) reect reect combin combining ing the respon response se contri contribut bution ionss of all modes. However, these standard equations have been derived in an unconventional way. In contra contrast st to the classic classical al deriva derivatio tion n found found in textbo textbooks oks (e.g. (e.g. Refere Reference nce [17, [17, Sectio Sections ns 12 :4 and 13: 13:1:3]), 3]), we have have used used the modal expansio expansion n of the spatial spatial distri distribut bution ion of the eective eective earthquake forces. This concept will provide a rational basis for the MPA procedure developed later. Modal response spectrum analysis The peak value r no no of the nth-mode contribution r n (t ) to response r (t ) is determined from r no = r nst An

(16)

where An is the ordinate A(T n ; n ) of the pseudo-acc pseudo-accelerat eleration ion response (or design) design) spectrum spectrum for the nth-mode SDF system, and T n = 2=!n is the nth natural vibration period of the MDF system. The peak peak mod modal al respon responses ses are combin combined ed accord according ing to the squ square are-ro -rootot-ofof-sum sum-of -of-sq -squar uares es (SRSS) or the complete quadratic combination (CQC) rules. The SRSS rule, which is valid for structures with well-separated natural frequencies such as multistorey buildings with symmetric plan, provides an estimate of the peak value of the total response: N

r o ≈

n=1


1= 2

  2 r no

(17) Earthquake Engng Struct. Dyn. 2002; 31:561–582


565

Modal pushover analysis To develop a pushover analysis procedure consistent with RSA, we observe that static analysis of the structure subjected to lateral forces f no no = n m Mn An

(18)

will will prov provid idee the the same same valu valuee of r no the peak peak nth-mod th-modee respon response se as in Equatio Equation n (16) (16) [17, [17, no , the Section Section 13: 13:8:1]. Altern Alternati ativel vely, y, this this respons responsee value value can be obtain obtained ed by static static analys analysis is of the structure subjected to lateral forces distributed over the building height according to s∗n = mMn

(19)

with the structure pushed to the roof displacement, urno , the peak value of the roof displacement due to the nth-mode, which from Equation (11) is urno = n rn Dn

(20)

where Dn = An =!n2 ; obviously obviously Dn or An are readil readily y availa available ble from from the respon response se (or design design)) spectrum. The peak modal responses r no no , each determined by one pushover analysis, can be combined according to Equation (17) to obtain an estimate of the peak value r o of the total response. This MPA for linearly elastic systems is equivalent to the well-known RSA procedure.

DYNAMIC AND PUSHOVER ANALYSIS PROCEDURES: INELASTIC BUILDINGS Response history analysis For each structural element of a building, the initial loading curve can be idealized appropriately (e.g. bilinear with or without degradation) and the unloading and reloading curves dier from from the initia initiall loadin loading g branch branch.. Thus, Thus, the relati relations ons betwee between n latera laterall forces forces f s at the N oor levels and the lateral displacements u are not single-valued, but depend on the history of the displacements: f s = f s (u; sign u˙ )

(21)

With this generalization for inelastic systems, Equation (1) becomes mu + cu˙ + f s (u; sign u˙ ) = −mÃu g (t )

(22)

The standa standard rd approa approach ch is to directl directly y solve solve these these couple coupled d equati equations ons,, leadin leading g to the ‘exact ‘exact’’ non-linear RHA. Althou Although gh classic classical al mod modal al analys analysis is is not valid for inelastic inelastic systems, systems, it will be used used next next to transf transform orm Equati Equation on (22) (22) to the mod modal al co-ord co-ordina inates tes of the corres correspon pondin ding g linear linear sys system tem.. Each Each stru struct ctur ural al eleme element nt of this this elas elastic tic syste system m is deÿn deÿned ed to have have the the same same stin stiness ess as the the initial stiness of the structural element of the inelastic system. Both systems have the same mass and damping. Therefore, the natural vibration periods and modes of the corresponding linear system are the same as the vibration properties of the inelastic system undergoing small oscillations (within the linear range). Copyright ? 2001 John Wiley & Sons, Ltd.


566


Expanding the displacements of the inelastic system in terms of the natural vibration modes of the corresponding linear system, we get N

u

 (t ) =

Mn qn (t )

(23)

n=1

Substituting Equation (23) into Equation (22), premultiplying by MT n , and using the mass- and classical damping-orthogonality property of modes gives 2n !n q˙n + qn + 2

F sn = −n ug (t ); M n

n = 1 ; 2; : : : ; N

(24)

where the only term that diers from Equation (8) involves ˙) = MT n f s (u; sign u˙ ) F sn = F sn (q; sign q

(25)

This This resisti resisting ng force force depend dependss on all mod modal al co-ord co-ordina inates tes qn (t ), ), implyi implying ng coupli coupling ng of mod modal al co-ordinates because of yielding of the structure. Equation Equation (24) represents represents N equations equations in the modal co-ordinates co-ordinates q n . Unlike Equation (8) for linearly elastic systems, these equations are coupled for inelastic systems. Simultaneously solvin solving g these these couple coupled d equati equations ons and using Equati Equation on (23) (23) will, will, in princi principle ple,, give give the same same results results for u(t ) as obtain obtained ed direct directly ly from from Equati Equation on (22). (22). Howeve However, r, Equati Equation on (24) (24) is rarely rarely used because it oers no particular advantage over Equation (22). Uncoupled modal response history analysis Neglecting the coupling of the N equati equations ons in mod modal al co-ord co-ordina inates tes [Equat [Equation ion (24)] (24)] leads leads to the uncoupled modal response history analysis (UMRHA) procedure. This approximate RHA procedure was used as a basis for developing an MPA procedure for inelastic systems. The spatial spatial distributio distribution n s of the eecti eective ve earthq earthquak uakee forces forces is expand expanded ed into into the mod modal al contributions s n according to Equation (3), where Mn are now the modes of the corresponding linear system. The equations governing the response of the inelastic system to pe ; e ; n (t ) given by Equation (6b) are mu + cu˙ + f s (u; sign u˙ ) = −s n u g (t )

(26)

The solution of Equation (26) for inelastic systems will no longer be described by Equation (7) becau because se ‘modes ‘modes’’ other other than than the nth-‘mo th-‘mode’ de’ will will also also contri contribut butee to the soluti solution. on. Howeve However, r, because for linear systems qr (t )=0 ) =0 for all modes other than the nth-mode, it is reasonable to expect that the nth-‘mode’ should be dominant even for inelastic systems. This This asser assertio tion n is illus illustr trat ated ed nume numeri rica call lly y in Figu Figure re 2 for for a 9-st 9-stor orey ey SAC steel steel buil buildi ding ng described in Appendix A. Equation (26) was solved by non-linear RHA, and the resulting roof displacement history was decomposed into its ‘modal’ components. The beams in all storeys except two yield when subjected to the strong excitation of 1 :5 × El Centro ground motion, and and the the mo mode dess othe otherr than than the the nth-mod th-modee contri contribut butee to the respon response. se. The second second and third third modes start responding to excitation pe ; e ; 1 (t ) the instant the structure ÿrst yields at about 5 :2 s; however, their contributions to the roof displacement are only 7 and 1 per cent, respectively, of the ÿrst mode response [Figure 2(a)]. The ÿrst and third modes start responding to excitation the inst instan antt the the struc structu ture re ÿrst ÿrst yiel yields ds at abou aboutt 4 :2 ss;; howeve however, r, their their contri contribut bution ionss to pe ; e ; 2 (t ) the Copyright ? 2001 John Wiley & Sons, Ltd.



(a) peff,1 = _ s1 × 1.5 × El Centro 80 ) m c (

1

(b)

p eff,2

) m c (

0

1

r

0

1.437

u

48.24

_ 20

80 ) m c (

2

20

Mode 2

) m c (

3.37

0

2

Mode 1

r

u

_ 80

= _ s × 1.5 × El Centro

20

Mode 1

567

2

r

Mode 2

0

r

u

u

_ 80 0

5

10

15

20

80 ) m c (

3

25

30

Mode 3

) m c (

0.4931

0

_ 20 0 20

3

r

11.62 5

0

5

10

15

20

Time (sec)

25

30

20

25

30

0.7783

u

_ 80 0

15

Mode 3

r

u

10

_ 20 0

5

10

15

20

25

30

Time (sec)

Figure 2. Modal decomposition of the roof displacement for ÿrst three modes due to 1 :5 × El Centro ground motion: (a) pe ; 1 (t ) = − s1 × 1:5 × El Centro; (b) pe ; 2 (t ) = − s 2 × 1:5 × El Centro ground motion.

the the roof roof disp displa lace ceme ment nt of the the seco second nd mo mode de resp respon onse se (Fig (Figur uree 2(b) 2(b))) are are 12 and and 7 per per cent cent,, respectively, of the second mode response [Figure 2(b)]. Approximating the response of the structure to excitation pe ; e ; n (t ) by Equation (7), substituting Equation (7) into Equation (26), and premultiplying by MT n gives Equation (24) except for the important approximation that F sn now depends only on one modal co-ordinate, qn : F ˙n ) = MT n f s (qn ; sign q˙n ) sn = F sn (qn ; sign q

(27)

With With this this approx approxima imatio tion, n, the soluti solution on of Equatio Equation n (24) (24) can be expres expressed sed by Equati Equation on (9), (9), where Dn (t ) is governed by F sn D n + 2 2n !n D˙ n + = −u g (t ) Ln

(28)

T ˙ ˙ F sn = F sn ( Dn ; sign Dn ) = Mn f s ( Dn ; sign Dn )

(29)

and

is related to F ˙n ) because of Equation (9). sn (qn ; sign q Equati Equation on (28) (28) may be interp interpret reted ed as the govern governing ing equation equation for the nth-‘mode’ th-‘mode’ inelastic inelastic SDF system, an SDF system with (1) small amplitude vibration properties—natural frequency !n and damping ratio n —of the nth-mode of the corresponding linear MDF system; and (2) F sn =Ln – Dn relation between resisting force F sn =Ln and modal co-ordinate Dn deÿned by Equation (29). Although Equation (24) can be solved in its original form, Equation (28) can be solved conven convenien iently tly by standa standard rd softwa software re becaus becausee it is of the same same form form as the standa standard rd equati equation on for an SDF system, and the peak value of Dn (t ) can be estimated from the inelastic response (or design) spectrum [17, Sections 7: 7 :6 and 7: 7 :12: 12:1]. Introducing the nth-‘mode’ inelastic SDF system also permitted extension of the well-established concepts for elastic systems to inelastic Copyright ? 2001 John Wiley & Sons, Ltd.


568


Forces sn

Unit mass An (t ) st r n

ω n , ζ n , F sn / Ln ¨ u g (t )

(a) Static Analysis of Structure

(b) Dynamic Analysis of Inelastic SDF System

Figure 3. Conceptual explanation of uncoupled modal RHA of inelastic MDF systems.

systems; compare Equations (24) and (28) to Equations (8) and (10), and note that Equation (9) applies to both systems. ‡ Solution of the non-linear Equation (28) formulated in this manner provides Dn (t ), ), which when substituted into Equation (11) gives the oor displacements of the structure associated with with the nth-‘mo th-‘mode’ de’ inelas inelastic tic SDF system. system. Any oor oor displa displacem cement ent,, storey storey drift, drift, or anothe anotherr deformation response quantity r n (t ) is given by Equations (12) and (13), where An (t ) is now the pseudo pseudo-ac -accel celera eratio tion n respon response se of the nth-‘mo th-‘mode’ de’ inelas inelastic tic SDF sys system tem.. The two analyanalyses that lead to r nst and An (t ) for the inelastic system are shown schematically in Figure 3. Equations (12) and (13) now represent the response of the inelastic MDF system to pe ; ), e ; n (t ), the nth-mode contribution to pe (t ). ). Therefore, the response of the system to the total excitation pe (t ) is given by Equations (14) and (15). This is the UMRHA procedure. Underlyi Underlying ng assumptio assumptions ns and accuracy. accuracy. Using Using the the 3:0 × El Centro Centro ground ground motion motion for both both analyses, the approximate solution of Equation (26) by UMRHA is compared with the ‘exact’ solution by non-linear RHA. This intense excitation was chosen to ensure that the structure is excited excited well beyond beyond its linear linear elasti elasticc limit. limit. Such Such compar compariso ison n for roof roof displa displacem cement ent and top-st top-store orey y drift drift is presen presented ted in Figure Figuress 4 and 5, respec respectiv tively ely.. The errors errors are slightl slightly y larger larger in drift drift than than in displa displacem cement ent,, but even for this very very intens intensee excita excitatio tion, n, the errors errors in either either response quantity are only a few per cent. These These errors errors arise arise from from the follow following ing assump assumptio tions ns and approx approxima imatio tions: ns: (i) the coupli coupling ng between modal co-ordinates qn (t ) arising from yielding of the system [recall Equations (24) and (25)] is neglected; (ii) the superposition of responses to pe ; according e ; n (t ) (n = 1; 2; : : : ; N ) to Equati Equation on (15) (15) is strict strictly ly valid valid only only for linearly linearly elastic elastic sys system tems; s; and (iii) the F sn =Ln – Dn relation is approximated by a bilinear curve to facilitate solution of Equation (28) in UMRHA. Although Although approximations approximations are inherent inherent in this UMRHA procedure procedure,, when specialized specialized for linearly elastic systems it is identical to the RHA procedure described earlier for such systems. The overall errors in the UMRHA procedure are documented in the examples presented in a later section. How is the the F relatio tion n to be Proper Propertie tiess of the nth-mod th-modee inelas inelastic tic SDF sys system tem.. How sn =Ln – Dn rela determined in Equation (28) before it can be solved? Because Equation (28) governing Dn (t ) is based based on Equati Equation on (7) for oor oor displac displaceme ements nts,, the relatio relationsh nship ip betwee between n latera laterall forces forces f s ‡

Equivalent inelastic SDF systems have been deÿned dierently by other researchers [18; 19].




(a) Nonlinear RHA 150 ) m c (

(b) UMRHA 150

n = 1

) m c (

0

1

569

0

1

r

n = 1

r

u

u

75.89

_150 50 ) m c (

50

n = 2

14.9

2

78.02

_150

) m c (

0

2

r

n = 2

14.51 0

r

u

u

_ 50

_ 50 10 ) m c (

3

10

n = 3

) m c (

0

3

r

n = 3

0

r

u

u

5.575

_10 0

5

10 15 20 Time (sec)

25

30

5.112

_10

0

5

10 15 20 Time (sec)

25

30

Figure Figure 4. Compariso Comparison n of an approxima approximate te roof displacemen displacementt from UMRHA with exact exact soluti solution on by non-li non-linea nearr RHA for pe ; n (t ) = − s n u g (t ); n = 1; 2 and 3, whe where u g (t ) = 3:0 × El Centro ground motion. (a) Nonlinear RHA 20 ) m c (

1

(b) UMRHA 20

n = 1

) m c (

0

r

∆

1

_ 20

20

n = 2

7.008

) m c (

0

0

r

∆

_ 20

_ 20

10

3

n = 2

6.744

2

r

∆

) m c (

5.855

_ 20

20

2

0

r

∆

6.33

) m c (

n = 1

10

n = 3

) m c (

0

3

r

0

r

∆

∆

_10

n = 3

5.956 0

5

10

15

20

Time (sec)

25

30

_10

5.817 0

5

10

15

20

25

30

Time (sec)

Figure Figure 5. Compar Compariso ison n of approx approxima imate te storey storey drift drift from from UMRHA UMRHA with with exact exact soluti solution on by non-li non-linea nearr RHA for pe ; n (t ) = − s n u g (t ), ), n = 1; 2 and 3, wher wheree u g (t ) = 3:0 × El Centro motion.

and Dn in Equation (29) should be determined by non-linear static analysis of the structure as the structure undergoes displacements u = Dn Mn with increasing increasing Dn . Although most commercially mercially available available software software cannot cannot implement implement such displacemen displacement-con t-controll trolled ed analysis, analysis, it can conduct conduct a force-cont force-controlle rolled d non-linear non-linear static analysis analysis with an invariant invariant distribution distribution of lateral lateral forces forces.. Theref Therefore ore,, we impose impose this this constra constraint int in develo developin ping g the UMRHA UMRHA proced procedure ure in this this section and MPA in the next section. What What is an approp appropria riate te invari invariant ant distri distribut bution ion of lateral lateral forces forces to determ determine ine F sn ? For an inelas inelastic tic sys system tem no invari invariant ant distri distribut bution ion of forces forces can produc producee displa displacem cement entss propor proportio tional nal Copyright ? 2001 John Wiley & Sons, Ltd.


570


V

bn

(b) F sn / Ln _ D n Relationship

F sn / Ln

(a) Idealized Pushover Curve

Idealized V bny

1

αn k n

V

bny

* n

/ M

1

2 αn ω n

Actual

k n

2 n

ω

1

1

u r n

u

D D = u

r n y

ny

r n y

/ Γ φ n

n

r n

Figure 6. Properties of the nth-‘mode’ inelastic SDF system from the pushover curve.

to M n at all displacements or force levels. However, before any part of the structure yields, the only force distribution that produces displacements proportional to Mn is given by Equation (19). Therefore, this distribution seems to be a rational choice—even after the structure yields—to determine F sn in Equation (29). When implemented by commercially available software, such non-linear static analysis provides the so-called pushover curve, which is dierent than than the F – Dn curve. curve. The structu structure re is pus pushed hed using using the force force distri distribut bution ion of Equati Equation on sn =Ln – D (19) to some predetermined roof displacement, and the base shear V bn is plotted against roof displacement urn . A bilinear idealization of this pushover curve for the nth-‘mode’ is shown in Figure 6(a). At the yield point, the base shear is V bny and roof displacement is u rny rny . How How to conv conver ertt this this V bn –u –u rn pus pushov hover er curve curve to the F relatio tion? n? The The two two sets sets of sn =Ln – Dn rela forces and displacements are related as follows: F sn =

V bn ; n

Dn =

u rn n rn

(30)

Equati Equation on (30) (30) enable enabless conver conversio sion n of the pus pushov hover er curve curve to the desire desired d F sn =Ln – Dn relation shown in Figure 5(b), where the yield values of F sn =Ln and Dn are F V bny sny = ∗; Ln M n

Dny =

u rny rny n rn

(31)

in which which M n∗ = Ln n is the the eec eecti tive ve mo moda dall mass mass [17, [17, Sect Sectio ion n 13: 13:2:5]. 5]. The The two two are are rela relate ted d through F sny = !n2 Dny Ln

(32)

implyi implying ng that that the initial initial slope slope of the bilinear bilinear curve curve in Figure Figure 6(b) is !n2 . Knowin Knowing g F sny =Ln and Dny from Equation (31), the elastic vibration period T n of the nth-‘mode’ inelastic SDF system is computed from T n = 2 Copyright ? 2001 John Wiley & Sons, Ltd.

1= 2

 L D  n

ny

F sny

(33)



571

This value of T n , which may dier from the period of the corresponding linear system, should be used in Equation (28). In contrast, the initial slope of the pushover curve in Figure 6(a) is k n = !n2 Ln , which is not a meaningful quantity. Modal pushover analysis Next Next a pus pushov hover er analys analysis is proced procedure ure is presen presented ted to estima estimate te the peak peak respon response se r no the no of the inelas inelastic tic MDF sys system tem to eectiv eectivee earthq earthquak uakee forces forces pe ; ). Consid Consider er a non-li non-linea nearr static static e ; n (t ). analys analysis is of the struct structure ure sub subjec jected ted to latera laterall forces forces distrib distribute uted d over over the buildi building ng height height ac∗ cording to s n [Equation (19)] with structure pushed to the roof displacement u rno . This value of the roof displacement is given by Equation (20) where Dn , the peak value of Dn (t ), ), is now determined by solving Equation (28), as described earlier; alternatively, it can be determined from the inelastic response (or design) spectrum [17, Sections 7 :6 and 7: 7 :12]. At this roof dis placement, the pushover analysis provides an estimate of the peak value r no of any response ): oor displacements, storey drifts, joint rotations, plastic hinge rotations, etc. r n (t ): This pushover analysis, although somewhat intuitive for inelastic buildings, seems rational for two reasons. First, pushover analysis for each ‘mode’ provides the exact modal response for elastic buildings and the overall procedure, as demonstrated earlier, provides results that are identi identical cal to the well-k well-know nown n RSA proced procedure ure.. Second Second,, the latera laterall force force distri distribut bution ion used used appears to be the most rational choice among all invariant distribution of forces. The response value r no is an estimate of the peak value of the response of the inelastic system to pe ; ), governed by Equation (26). As shown earlier for elastic systems, r no no also e ; n (t ), represents the exact peak value of the nth-mode contribution r n (t ) to response r (t ). ). Thus, we will refer to r no as the peak ‘modal’ response even in the case of inelastic systems. The peak ‘modal’ ‘modal’ responses responses r no each determ determine ined d by one pushover pushover analys analysis, is, is combin combined ed no , each using an appropriate modal combination rule, e.g. Equation (17), to obtain an estimate of the peak value r o of the total response. This application of modal combination rules to inelastic systems obviously lacks a theoretical basis. However, it provides results for elastic buildings that are identical to the well-known RSA procedure described earlier. COMPARATIVE EVALUATION OF ANALYSIS PROCEDURES The ‘exact ‘exact’’ respon response se of the 9-stor 9-storey ey SAC buildi building ng descri described bed earlier earlier is determ determine ined d by the two approx approxima imate te method methods, s, UMRHA UMRHA and MPA, and compared compared with the ‘exact ‘exact’’ result resultss of a rigorous rigorous non-linear RHA using the DRAIN-2DX DRAIN-2DX computer program [20]. Gravity-loa Gravity-load d (and P-delta) eects are excluded from all analyses presented in this paper. However, these eects were included in Chopra and Goel [21]. To ensure that this structure responds well into the inelastic range, the El Centro ground motion is scaled up a factor varying from 1.0 to 3.0. The ÿrst three vibration modes and periods of the building for linearly elastic vibration are shown sho wn in Figure Figure 7. The vibrat vibration ion periods periods for the ÿrst three three mod modes es are 2.27, 2.27, 0.85, and 0.49 s, ∗ respectively. The force distribution s n for the ÿrst three modes are shown in Figure 8. These force distributions will be used in the pushover analysis to be presented later. Uncoupled modal response history analysis The The stru struct ctur ural al resp respon onse se to 1 :5 × the the El Cent Centro ro grou ground nd mo moti tion on incl includ udin ing g the the resp respon onse se contributions associated with three ‘modal’ inelastic SDF systems, determined by the UMRHA Copyright ? 2001 John Wiley & Sons, Ltd.


572


9th T 3 = 0.49 sec

8th

T 2 = 0.85 sec

7th 6th

T 1= 2.27 sec

r 5th o o l F 4th

3rd 2nd 1st Ground _1.5

_1

_ 0.5

0 0.5 Mode Shape Component

1

1.5

Figure 7. First three natural-vibration periods and modes of the 9-storey building. 3.05

3.05

2.61

3.05

_ 0.39

1.51

0.0272 _ 2.72

2.33 2.04 _ 1.13

_ 2.93 _ 1.38

1.75 _ 1.8 1.44 _ 2.1

0.728

1.12 _ 2.03 0.796 0.487

2.37

_ 1.67

2.94

_1.1

* 1

s

2.31

* 2

s

* 3

s

Figure 8. Force distributions s∗n = mMn ; n = 1; 2 and 3.

procedure, is presented next. Figure 9 shows the individual ‘modal’ responses, the combined response due to three ‘modes’, and the ‘exact’ values from non-linear RHA for the roof dis placement and top-storey drift. The peak values of response are as noted; in particular, the peak peak roof roof displa displacem cement ent due to each each of the three ‘modes ‘modes’, ’, is u r10 = 48: 48:3 cm, u r20 = 11: 11:7 cm and u r30 = 2:53 cm. The peak peak values values of oor oor displa displacem cement entss and storey storey drifts drifts includ including ing one, one, two, and three modes are compared with the ‘exact’ values in Figure 10 and the errors in the approximate results are shown in Figure 11. Observe that errors tend to decrease as response contributions of more ‘modes’ are included, although the trends are not as systematic as when the system remained elastic [22]. This is to be expected because in contrast to classical modal analysis, the UMRHA procedure lacks a rigoro rigorous us theory theory.. This This deÿcie deÿciency ncy also implies implies that, that, with, with, say, say, three three ‘modes ‘modes’’ includ included, ed, the response is much less accurate if the system yields signiÿcantly versus if the system remains within the elastic range [22]. However, for a ÿxed number of ‘modes’ included, the errors in storey drifts are larger compared to oor displacements, just as for elastic systems. Next Next we invest investiga igate te how the errors errors in the UMRHA vary vary with with the deformat deformation ion demands demands imposed by the ground motion, in particular, the degree to which the system deforms beyond its elastic limit. For this purpose the UMRHA and exact analyses were repeated for ground Copyright ? 2001 John Wiley & Sons, Ltd.


573


(a) Roof Displacement ) m c (

1

_

) m c (

2

) m c (

0

r

u

(b) Top Story Drift

80

80

_ 12

) m c (

11.7 •

) m c ( r

) m c ( r

0

) m c (

• 2.53

• 2.88

r

_ 12

) m c (

0

12

UMRHA 3 "Modes"

0

r

∆

•48.1

• 7.38

_

12

80

) m c (

0 80 0

"Mode" 3

0

3

80

80

12

∆

80

u

_

_ 12

80

u

_

0

r

r

_

"Mode" 2

5.44 •

∆

_ 80

3

12

2

r

u

• 3.62

r

∆

u

) m c (

"Mode" 1

0

1

•48.3

80 0

12

•44.6 5 10

r

12

•6.24

NL RHA

0

∆

15

20

25

30

_12

0

5

10

15

20

25

30

Time, sec

Time, sec

Figure Figure 9. Respo Response nse histor histories ies of roof roof displa displacem cement ent and top-store top-storey y drift drift due to 1:5 × El Centro Centro ground ground motion: motion: individual individual ‘modal’ responses and combined combined response from UMRHA, UMRHA, and total response from non-linear RHA. (a) Floor Displacements

(b) Story Drift Ratios

9th

9th

8th

8th

7th

7th

6th

6th

r 5th o o l F 4th

3rd 2nd

r 5th o o l F 4th

NL_ RHA UMRHA 1 "Mode" 2 "Modes" 3 "Modes"

3rd 2nd

1st Ground 0

NL_ RHA UMRHA 1 "Mode" 2 "Modes" 3 "Modes"

1st 0.5 1 1.5 Displacement/Height Displacement/Height (%)

2

Ground 0

0.5

1 1.5 Story Drift Ratio (%)

2

2.5

Figure Figure 10. Height Height-wi -wise se variat variation ion of oor oor displa displacem cement entss and storey storey drift drift ratios ratios from from UMRHA UMRHA and non-linear RHA for 1: 1:5 × El Centro ground motion.

motions of varying intensity. These excitations were deÿned by the El Centro ground motion multiplied by 0.25, 0.5, 0.75, 0.85, 1.0, 1.5, 2.0, and 3.0. For each excitation, the errors in responses computed by UMRHA including three ‘modes’ relative to the ‘exact’ response were determined. Copyright ? 2001 John Wiley & Sons, Ltd.


574


(a) Floor Displacements


9th

9th

8th

8th

7th

7th

6th

6th

r o 5th o l F 4th

r 5th o o l F 4th

UMRHA 1 "Mode" 2 "Modes" 3 "Modes"

3rd 2nd

UMRHA 1 "Mode" 2 "Modes" 3 "Modes"

3rd 2nd

1st

1st

Ground _ 60

_ 40

_ 20

0 Error (%)

20

40

60

Ground _ 60

_

40

_

20

0 20 Error (%)

40

60

Figure 11. Height-wise variation of error in oor displacements and storey drifts estimated by UMRHA including one, two or three ‘modes’ for 1: 1:5 × El Centro ground motion.

(a) Floor Displacements 60

(b) Story Drifts 60

Error Envelope Error for Floor No. Noted

50

Error Envelope Error for Story No. Noted

50

7

40

40

) % ( r 30 o r r E 20

6

) % ( r 30 o r r E 20

1

5

2

6

2

5

10

8

9

10

4

3 1

9

4

0

3

0

0 .5

1

8

1 .5 2 GM Multiplier

7

2 .5

3

0

0

0. 5

1

1. 5 2 GM Multiplier

2 .5

3

Figure Figure 12. Errors in UMRHA UMRHA as a functi function on of ground ground motion motion intens intensity ity:: (a) oor displacements; and (b) storey drifts.

Figu Figure re 12 summ summar ariz izes es the the erro errorr in UMRH UMRHA A as a func functi tion on of grou ground nd mo motio tion n inte intens nsity ity,, indicated by a ground motion multiplier. Shown is the error in each oor displacement [Figure 12(a)], in each storey drift [Figure 12(b)], and the error envelope for each case. To interpret thes thesee resu result lts, s, it will will be usef useful ul to know know the the defo deform rmat atio ion n of the the syst system em rela relati tive ve to its its yiel yield d ∗ deformation deformation.. For this purpose, pushover curves using force distributio distributions ns s n [Equation [Equation (19)] for the ÿrst three modes of the system are shown in Figure 13, with the peak displacement of each ‘modal’ SDF system noted for each ground motion multiplier. Two versions of the pushover curve are included: the actual curve and its idealized bilinear version. The location of plasti plasticc hinges hinges and their their rotati rotations ons,, determ determine ined d from from ‘exact ‘exact’’ analys analyses, es, were were noted noted but not shown here. Figu Figure re 12 perm permit itss the the foll follow owin ing g obse observ rvat atio ions ns rega regard rdin ing g the the accu accura racy cy of the the UMRH UMRHA A proce procedur dure: e: the errors errors (i) are small (less (less than than 5 per cent) for ground ground motion motion multipli multipliers ers up Copyright ? 2001 John Wiley & Sons, Ltd.


575


(a) "Mode" 1 Pushover Curve

(b) "Mode" 2 Pushover Curve

0.25

0.25 /W = 0.1684; α = 0.19 u y = 36.3 cm; V by

0.2 t h g i e 0.15 W / r a e h S 0.1 e s a B 0.05

u y = 9.9 cm; V by /W = 0.1122; α = 0.13

3

0.2 t h g i e 0.15 W / r a e h S 0.1 e s a B 0.05

2 1.5 1 0.85 0.75

0.5

Actual Idealized

0.25

1.5

2

3

1 0.85 0.75

Actual Idealized

0.5 0.25

0 0

20 40 60 Roof Displacement (cm)

0

80

0

5


25

(c) "Mode" 3 Pushover Curve 0.25 u y = 4.6 cm; V by /W = 0.1181;α = 0.14

0.2 t h g i e W0.15 / r a e h S 0.1 e s a B 0.05

3 2 1.5 0.85 0.5

0

1

Actual Idealized

0.75

0.25

0

2


10

Figure 13. ‘Modal’ pushover curves with peak roof displacements identiÿed for 0.25, 0.5, 0.75, 1.0, 1.5, 2.0, and 3: 3 :0 × El Centro ground motion.

to 0.75; (ii) increase rapidly as the ground motion multiplier increases to 1.0; (iii) maintain roughly similar values for more intense ground motions; and (iv) are larger in storey drifts compar compared ed to oor oor displa displacem cement ents. s. Up to ground ground motion motion multip multiplier lier 0.75, 0.75, the sys system tem remain remainss elastic elastic and the errors errors in trunca truncatin ting g the higher higher mod modee contri contribut bution ionss are neglig negligibl ible. e. Additio Additional nal errors are introduced in UMRHA of systems responding beyond the linearly elastic limit for at least two reasons. First, as mentioned earlier, UMRHA lacks a rigorous theory and is based on several approximations. Second, the pushover curve for each ‘mode’ is idealized by a bilinear curve in solving Equation (28) for each ‘modal’ inelastic SDF system (Figures 6 and 13). 13). The idealize idealized d curve curve for the ÿrst ÿrst ‘mode’ ‘mode’ deviates deviates mos mostt from from the actual actual curve curve near near the peak peak displa displacem cement ent corres correspon pondin ding g to ground ground motion motion multip multiplie lierr 1.0. 1.0. This This would would explai explain n why the errors are large at this excitation intensity; although the system remains essentially elastic; the ductility factor for the ÿrst mode system is only 1.01 [Figure 13(a)]. For more intense excitations, the ÿrst reason mentioned above seems to be the primary source for the errors. Modal pushover analysis The MPA proced procedure ure,, consid consideri ering ng the respon response se due to the ÿrst ÿrst three three ‘modes ‘modes’, ’, was impleimplemented for the selected building subjected to 1 :5 × the El Centro ground motion. The strucCopyright ? 2001 John Wiley & Sons, Ltd.


576



(a) Floor Displacements 9th

9th

8th

8th

7th

7th

6th

6th

r 5th o o l F 4th

r 5th o o l F 4th

_

NL RHA MPA 1 "Mode" 2 "Modes" 3 "Modes"

3rd 2nd

MPA 1 "Mode" 2 "Modes" 3 "Modes"

3rd 2nd

1st Ground

_

NL RHA

1st 0

0.5

1

1.5

Displacement/Height Displacement/Height (%)

2

Ground

0

0.5

1 1.5 Story Drift Ratio (%)

2

2.5

Figu Figure re 14. 14. Heig Height ht-w -wis isee vari variat atio ion n of oor oor disp displa lace ceme ment ntss and and stor storey ey drif driftt rati ratios os from from MPA MPA and and nonnon-li line near ar RHA RHA for for 1:5 × El Centr Centro o ground ground motion motion;; shadin shading g indica indicates tes errors errors in MPA including three ‘modes’.

ture ture is pus pushed hed using using the force force distri distribut bution ion of Equati Equation on (19) (19) with with n = 1; 2 and 3 (Figure 8) to roof displacements displacements u rno = 48: 48:3, 11.7 and 2: 2 :53 cm, respectivel respectively, y, the values values determined determined by RHA of the nth-mode inelastic SDF system (Figure 9). Each of these three pushover analyses provi provides des the pus pushov hover er curve curve (Figur (Figuree 13) and the peak peak values values of mod modal al respon responses. ses. Because Because this this buildi building ng is unusua unusually lly strong—i strong—its ts yield yield base base shear shear = 16: 16 :8 per cent of the building weight [Figur [Figuree 13(a)] 13(a)]—th —thee displa displacem cement ent ductili ductility ty demand demand impose imposed d by three three times times the El Centro Centro ground motion is only slightly larger than 2. Figure 14 presents estimates of the combined response according to Equation (17), considering one, two, and three ‘modes’, respectively, and Figure 15 shows the errors in these estimates relative to the exact response from non-linear RHA. The errors in the modal pushover results for two or three modes included are generally signiÿcantly smaller than in UMRHA (compare Figures 15 and 11). Obviously, the additional errors due to the approximation inherent in modal combination rules tend to cancel out the errors due to the various approximation contained in the UMRHA. The ÿrst ‘mode’ alone is inadequate, especially in estimating the storey drifts (Figure 14). Signiÿcant improvement is achiev achieved ed by includ including ing respon response se contri contribut bution ionss due to the second second ‘mode’ ‘mode’,, howeve however, r, the third third ‘mode’ contributions do not seem especially important (Figure 14). As shown in Figure 15(a), MPA including three ‘modes’ underestimates the displacements of the lower oors by up to 8 per cent and overestimates the upper oor displacements by up to 14 per cent. The drifts are underestimated by up to 13 per cent in the lower storeys, overestimated by up to 18 per cent in the middle storeys, and are within a few per cent of the exact values for the upper storeys [Figure 15(b)]. The errors errors are especi especially ally large in the hinge plasti plasticc rotatio rotations ns estima estimated ted by the MPA proprocedures, even if three ‘modes’ are included [Figure 15(c)]; although the error is recorded as 100 per cent if MPA estimates zero rotation when the non-linear RHA computes a non-zero value, this error is not especially signiÿcant because the hinge plastic rotation is very small. Observe that the primary contributor to plastic rotations of hinges in the lower storeys is the ÿrst ‘mode’, in the upper storeys it is the second ‘mode’; the third ‘mode’ does not contribute because this SDF system remains elastic [Figure 13(c)]. Copyright ? 2001 John Wiley & Sons, Ltd.


577


(a) Floor Displacements


9th

9th

8th

8th

7th

7th

6th

6th

r 5th o o l F 4th

r 5th o o l F 4th


3rd 2nd

2nd

1st Ground _ 60


3rd

1st

_

40

_ 20

0 Error (%)

20

40

60

Ground _ 60

_ 40

_ 20

0 Error (%)

20

40

60

(c) (c) Hinge Hinge Plastic Plastic Rotations Rotations 9th 8th 7th 6th 5th

r r o o o o l l F F

4th MPA 1 "Mode" 2 "Modes" 3 "Modes"

3rd 2nd 1st Ground Ground _ 12 0

_ 80

_

40

0 Error Error (%) %

40

80

1 20

Figure Figure 15. Errors in oor displacemen displacements, ts, storey drifts, and hinge plastic rotations rotations estimated estimated by MPA including one, two and three ‘modes’ for 1 :5 × El Centro ground motion.

The The loca locati tion onss of plas plasti ticc hing hinges es show shown n in Figur Figuree 16 were were dete determ rmin ined ed by four four anal analys yses es:: MPA consid consideri ering ng one ‘mode’ ‘mode’,, two ‘modes ‘modes’, ’, and three three ‘modes ‘modes’; ’; and non-li non-linea nearr RHA. RHA. One ‘mode’ pushover analysis is unable to identify the plastic hinges in the upper storeys where higher mode contributions to response are known to be more signiÿcant. The second ‘mode’ is nece necessa ssary ry to iden identi tify fy hing hinges es in the the uppe upperr stor storey eys, s, howe howeve ver, r, the the resu result ltss are are not not alwa always ys accurate. For example, the hinges identiÿed in beams at the sixth oor are at variance with the ‘exact’ results. Furthermore, MPA failed to identify the plastic hinges at the column bases in Figure 16, but was successful when the excitation was more intense. Figure 17 summarizes the error in MPA considering three ‘modes’ as a function of ground motion intensity, indicated by a ground motion multiplier. Shown is the error in each oor displacement [Figure 17(a)], each storey drift [Figure 17(b)], and the error envelope for each case case.. Whil Whilee vari variou ouss sour source cess of erro errors rs in UMRH UMRHA A also also appl apply y to MPA, MPA, the the erro errors rs in MPA MPA are fortui fortuitou tously sly smaller smaller than than in UMRHA UMRHA (compa (compare re Figure Figuress 17 and 12) for ground ground motion motion multipliers larger than 1.0, implying excitations intense enough to cause signiÿcant yielding of the structure. However, errors in MPA are larger for ground motion multipliers less than Copyright ? 2001 John Wiley & Sons, Ltd.


578

A. K. CHOPRA AND R. K. GOEL (b) MPA, 2 _ "Modes"

(a) MPA, 1 _"Mode"

• • • • •

• •• •• •• ••

•• •• •• •• ••

•• •• •• •• ••

• •

•• ••

•• ••

•• ••

• •

• • •• • ••

• • • • •

• •• •• •• ••

•• •• •• •• ••

•• •• •• •• ••

• • •• • ••

•• ••

•• ••

•• ••

• • •

• •• •• •• ••

•• •• •• •• ••

• •• •• •• ••

• • •• •• ••

•

•

•

(c) 3 _"Modes"

• •

•• ••

•• ••

•• ••

• •

• • • •

• • • • •

• •• •• •• ••

•• •• •• •• ••

•• •• •• •• ••

• • •• • ••

• • • • •

(d) Nonlinear RHA

Figure 16. Locations of plastic hinges determined by MPA considering one, two and three ‘modes’ and by non-linear RHA for 1: 1:5 × El Centro ground motion.

(a) Floor Displacements Displacements 60

(b) Story Drifts 60

Error Envelope Error for Floor No. Noted

50

50

40 ) % ( r 30 o r r E 20

6

5

0.5

1

1.5 2 GM Multiplier

4

7

9

10

1

3

0

5

2

1

6

7 3

8

9

4

0

40 ) % ( r 30 o r r E 20

8

2

10

Error Envelope Error for Story No. Noted

2.5

3

0

0

0.5

1

1.5 GM Multiplier

2

2.5

3

Figure Figure 17. Errors Errors in MPA as a functi function on of ground ground motion motion intens intensity ity:: (a) oor displacements; and (b) storey drifts.




579

0.75 0.75,, impl implyi ying ng exci excita tatio tions ns weak weak enou enough gh to limi limitt the the resp respon onse se in the the elas elasti ticc rang rangee of the the structure. Here, UMRHA is essentially exact, whereas MPA contains errors inherent in modal combination rules. The errors are only weakly dependent on ground motion intensity (Figure 17), an observation with practical implications. As mentioned earlier, the MPA procedure for elastic systems (or weak ground motions) is equivalent to the RSA procedure—now standard in engineering practice—im practice—implyin plying g that the modal combination combination errors contained contained in these procedures procedures are acceptable. The fact that MPA is able to estimate the response of buildings responding well into the inelastic inelastic range range to a simila similarr degree degree of accura accuracy cy indica indicates tes that that this this proced procedure ure is accura accurate te enough for practical application in building retroÿt and design.

CONCLUSIONS This investigation aimed to develop an improved pushover analysis procedure based on structural dynamics theory, which retains the conceptual conceptual simplicity and computation computational al attractiven attractiveness ess of current procedures with invariant force distribution now common in structural engineering practice. It has led to the following conclusions: The standard response spectrum analysis for elastic multistorey buildings can be reformulated as a modal pushover analysis (MPA). The peak response of the elastic structure due to its nth vibration mode can be exactly determined by pushover analysis of the structure subjected to lateral forces distributed over the height of the building according to s∗n = mMn , where m is the mass matrix and Mn its nth-mode, and the structure is pushed to the roof displacement determined from the peak deformation Dn of the nth-mode elastic SDF system; Dn is available from the elastic response (or design) spectrum. Combining these peak modal responses by an appropriate modal combination rule (e.g. SRSS rule) leads to the MPA procedure. The MPA proced procedure ure develo developed ped to estimat estimatee the seismic seismic demands demands on inelas inelastic tic sys system temss is organized in two phases: First, a pushover analysis is used to determine the peak response r no no of the inelastic MDF system to individual terms, pe ;  g (t ), ), in the modal expansion of e ; n (t ) = − s n u the eective earthquakes forces, pe (t ) = −mÃu g (t ). ). The base shear–roof displacement (V ( V bn − ∗ u rn ) curve is developed from a pushover analysis for the force distribution s n . This pushover curve curve is ideali idealized zed as a biline bilinear ar force– force–def deform ormati ation on relati relation on for the nth-‘mode’ th-‘mode’ inelastic inelastic SDF system (with vibration properties in the linear range that are the same as those of the nthmode elastic SDF system), and the peak deformation of this SDF system—determined by nonlinear response history analysis (RHA) or from the inelastic response or design spectrum—is used to determine the target value of roof displacement at which the seismic response r no no is determined by the pushover analysis. Second, the total demand r o is determined by combining the r no (n = 1; 2; : : :) :) according to an appropriate modal combination rule (e.g. SRSS rule). Compar Comparing ing the peak peak inelas inelastic tic respon response se of a 9-stor 9-storey ey SAC steel steel buildi building ng determ determine ined d by the approximate MPA procedure—including only the ÿrst two or three r no terms—with nonlinear linear RHA demons demonstra trated ted that that the approx approxima imate te proced procedure ure provid provided ed good good estima estimates tes of oor oor displa displacem cement entss and storey storey drifts drifts,, and identi identiÿed ÿed locati locations ons of mos mostt plastic plastic hinges hinges;; howeve however, r, plast plastic ic hinge hinge rotatio rotations ns were were less less accura accurate. te. Based Based on result resultss presen presented ted for El Centro Centro ground ground motion scaled by factors varying from 0.25 to 3.0, MPA estimates the response of buildings respon respondin ding g well well into into the inelastic inelastic range to simila similarr degree degree of accura accuracy cy as standa standard rd RSA is capable of estimating peak response of elastic systems. Thus, the MPA procedure is accurate Copyright ? 2001 John Wiley & Sons, Ltd.


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enough enough for practi practical cal applic applicati ation on in buildi building ng evalua evaluatio tion n and design design.. That That said, said, howeve however, r, all pusho pushover ver analys analysis is proced procedure uress consid considere ered d do not seem to comput computee accura accuratel tely y local local respon response se quantities, such as hinge plastic rotations. Thus the structural engineering profession should examine the present trend of comparing comput computed ed hinge hinge plastic plastic rotati rotations ons agains againstt rotati rotation on limits limits establi establishe shed d in FEMA-2 FEMA-273 73 to judge judge structural performance. Perhaps structural performance evaluation should be based on storey drifts that are known to be closely related to damage and can be estimated to a higher degree of accuracy by pushover analyses. While pushover estimates for oor displacements are even more accurate, they are not good indicators of damage. This This pape paperr has has focu focuse sed d on deve develo lopi ping ng an MPA proc proced edur uree and and its its init initia iall eval evalua uati tion on in estimating the seismic demands on a building imposed by a selected ground motion, with the excitation scaled to cover a wide range of ground motion intensities and building response. This This new method method for estimatin estimating g seismic seismic demand demandss at low performa performance nce levels, levels, such such as life life safety and collapse prevention, should obviously be evaluated for a wide range of buildings and ground motion ensembles. Work along these lines is in progress.

APPENDIX A: SAC STEEL BUILDING The 9-storey 9-storey structure structure,, sho shown wn in Figure Figure A1, was designed designed by Brando Brandow w & Joh Johnst nston on AssoAsso§ ¶ ciates for for the the SAC SAC Phase Phase II Steel Steel Projec Project. t. Althou Although gh not actual actually ly constr construct ucted, ed, this this strucstructure ture meets meets seismic seismic code code requir requireme ements nts of the 1994 1994 UBC and repres represent entss typica typicall medium medium-ri -rise se buildings designed for the Los Angeles, California, region. A benchmark structure for the SAC project, this building is 45 :73 m (150 ft) by 45: 45 :73 m (150 ft) in plan, and 37 :19 m (122 ft) in elevation. The bays are 9 :15 m (30 ft) on centre, in both directions, with ÿve bays each in the north–south (N–S) and east–west (E–W) directions. The building’s lateral force-resisting system is composed of steel perimeter moment-resisting frames (MRF). To avoid biaxial bending in corner columns, the exterior bay of the MRF has only one moment-resisting connection. The interior bays of the structure contain frames with simple (shear) connections. The columns are 345MPa (50 ksi) steel wide-ange sections. The levels of the 9-storey building are numbered with respect to the ground level (see Figure A1) with the ninth level being the roof. The building has a basement level, denoted B-1. Typical oor-to-oor heights (for analysis purposes measured from centre-of-beam to centre-of-beam) are 3: 3:96 m (13 ft). The oor-tooor-to-oo oorr height height of the basement basement level level is 3 :65 m (12 ft) and for the ÿrst oor is 5: 5 :49 m (18 ft). The column lines employ two-tier construction, i.e. monolithic column pieces are connected every two levels beginning with the ÿrst level. Column splices, which are seismic (tension) splice splicess to carry carry bendin bending g and uplift uplift forces forces,, are located located on the ÿrst, third, third, ÿfth, and sevent seventh h levels at 1.83 m (6 ft) above the centreline of the beam to column joint. The column bases are modelled as pinned and secured to the ground (at the B-1 level). Concrete foundation walls and surrounding soil are assumed to restrain the structure at the ground level from horizontal displacement. §

Brandow & Johnston Associates, Consulting Structural Engineers, 1660 W. Third St., Los Angeles, CA 90017. SAC is a joint joint ventur venturee of three three non-pr non-proÿt oÿt organi organizat zation ions: s: The Struct Structura urall Engine Engineers ers Assoc Associat iation ion of Califo Califorrnia (SEAOC), (SEAOC), the Applied Applied Technolog Technology y Council Council (ATC), (ATC), and Californi Californiaa Universit Universities ies for Research in Earthquake Earthquake Engineering (CUREE). SAC Steel Project Technical Oce, 1301 S. 46th Street, Richmond, CA 94804-4698. ¶




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Figure A1. Nine-storey building (adapted from Reference [23]).

The The oor oor syst system em is comp compos osed ed of 248 248 MPa MPa (36 (36 ksi) ksi) steel steel wide wide- -an ange ge beam beamss in acti acting ng comp compos osit itee acti action on with with the the oor oor slab. slab. The The seis seismi micc mass mass of the the struc structu ture re is due due to vari vari-ous compon component entss of the structu structure, re, includ including ing the steel steel framin framing, g, oor oor slabs, slabs, ceilin ceiling g = ooring, mechanical= mechanical= electr electrica ical, l, partit partition ions, s, rooÿng rooÿng and a pentho penthouse use locate located d on the roof. roof. The seismic seismic 5 2 mass of the ground level is 9: 9 :65 × 10 kg (66: (66:0 kips-s = ft), for the ÿrst level is 1 :01 × 106 kg (69: (69:0 kips-s2 = ft), for the second through eighth levels is 9 :89 × 105 kg (67: (67:7 kips-s2 = ft), and for the ninth level is 1 :07 × 106 kg (73: (73:2 kips-s2 = ft). The seismic mass of the above ground levels of the entire structure is 9 :00 × 106 kg (616 kips-s 2 = ft). The two-di two-dimen mensio sional nal buildi building ng mod model el consist consistss of the perime perimeter ter N–S MRF (Figure (Figure A1), A1), repr repres esen enti ting ng half half of the the build buildin ing g in the the N–S N–S dire direct ctio ion. n. The The fram framee is assi assign gned ed half half of the the seismic mass of the building at each oor level. The model is implemented in DRAIN-2DX [20] using the M1 model developed by Gupta and Krawinkler [7]. The MI model is based on centreline dimensions of the bare frame in which beams and columns extend from centreline to centreline. The strength, dimension, and shear distortion of panel zones are neglected but large deformation deformation ( P P –) –) eects eects are includ included. ed. The simple mod model el adopte adopted d here here is sucie sucient nt for the objectives of this study; if desired, more complex models, such as those described in Reference [7], can be used. Copyright ? 2001 John Wiley & Sons, Ltd.


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ACKNOWLEDGEMENT

This research investigation is funded by the National Science Foundation under Grant CMS-9812531, a part of the U.S.–Japan Co-operative Research in Urban Earthquake Disaster Mitigation. This ÿnancial support is gratefully acknowledged. REFERENCES 1. Building Seismic Safety Council. NEHRP Guidelines for the Seismic Rehabilitation of Buildings , FEMA-273. Federal Emergency Management Agency: Washington, DC, 1997. 2. Saiid Saiidii M, Sozen Sozen MA. MA. Simple Simple non-li non-linea nearr seismi seismicc analys analysis is of R= C structures. Journal Journal of Structural Structural Division, ASCE 1981; 107(ST5):937–951. 3. Miranda Miranda E. Seismic Seismic evaluation evaluation and upgrading upgrading of existing existing buildings. buildings. Ph.D. Dissertation Dissertation, Department of Civil Engineering, University of California, Berkeley, CA, 1991. 4. Lawson RS, Vance V, Krawinkler H. Nonlinear static pushover analysis—why, when and how? Proceedings of the 5th U.S. Conference on Earthquake Engineering 1994; 1:283–292. 5. Fajfar P, Gaspersic P. The N2 method for the seismic damage analysis of RC buildings. Earthquake Engineering & Structural Dynamics 1996; 25(1):31–46. 6. Maison B, Bonowitz D. How safe are pre-Northridge WSMFs? A case study of the SAC Los Angeles nine-storey building. Earthquake Spectra 1999; 15(4):765–789. 7. Gupta A, Krawinkler H. Seismic demands for performance evaluation of steel moment resisting frame structures (SAC Task 5.4.3). Report No. 132 , John A. Blume Earthquake Engineering Center, Stanford University, CA, 1999. 8. Gupta Gupta A, Krawinkle Krawinklerr H. Estimatio Estimation n of seismic seismic drift demands demands for frame structure structures. s. Earthquake Engineering & Structural Dynamics 2000; 29:1287–1305. 9. Skokan MJ, Hart GC. Reliability of non-linear static methods for the seismic performance prediction of steel frame buildings. Proceeding Paper No. 1972, 1972, Proceedingss of the 12th World Conference Conference on Earthquake Earthquake Engineering Engineering , Paper Auckland, New Zealand, 2000. 10. Krawink Krawinkler ler H, Seneviratn Seneviratnaa GDPK. GDPK. Pros and cons of a pushover pushover analysis of seismic seismic performance performance evaluation evaluation.. Engineering Structures 1998; 20(4–6):452–464. 11. Bracci Bracci JM, Kunnath Kunnath SK, Reinhorn AM. Seismic Seismic performan performance ce and retroÿt retroÿt evaluation evaluation for reinforce reinforced d concrete concrete structures. Journal of Structural Engineering , ASCE 1997; 123(1):3–10. 12. Gupta Gupta B, Kunnat Kunnath h SK. Adapti Adaptive ve spectr spectra-b a-base ased d pus pushov hover er proced procedure ure for seism seismic ic evalua evaluatio tion n of struct structure ures. s. Earthquake Spectra 2000; 16(2):367–392. 13. Paret TF, Sasaki KK, Eilbekc DH, Freeman SA. Approximate inelastic procedures to identify failure mechanisms from higher mode eects. Proceedings of the 11th World Conference on Earthquake Engineering . Paper No. 966, Acapulco, Mexico, 1996. 14. Sasaki KK, Freeman SA, Paret TF. Multimode pushover procedure (MMP)—a method to identify the eects of higher higher modes in a pus pushov hover er analys analysis. is. Proceeding Proceedingss of the 6th U.S. National National Conferenc Conferencee on Earthquake Engineering, Seattle, Washington, 1998. 15. Kunnath SK, Gupta B. Validity of deformation demand estimates using non-linear static procedures. Proceedings U.S.–Japan U.S.–Japan Workshop Workshop on Performance Performance-Based -Based Engineerin Engineering g for Reinforced Reinforced Concrete Concrete Building Building Structures Structures , Sapporo, Hokkaido, Japan, 2000. 16. Matsu Matsumori mori T, Otani S, Shiohara Shiohara H, Kabeyasaw Kabeyasawaa T. Earthquak Earthquakee member member deformatio deformation n demands demands in reinforced reinforced concrete frame structures. Proceedings U.S.–Japan Workshop on Performance-Based Earthquake Engineering Methodology for R= R = C Building Structures. Maui, Hawaii, 1999; 79–94. 17. Chopra Chopra AK. Dynamics Prentice-Hall: all: Dynamics of Structures Structures: Theory Theory and Application Applicationss to Earthquake Earthquake Engineerin Engineering g . Prentice-H Englewood Clis, NJ, 2001. 18. Villav Villaverde erde R. Simpliÿe Simpliÿed d response response spectrum spectrum seismic seismic analysis analysis of non-linear non-linear structure structures. s. Journal Journal of Structural Structural Engineering Mechanics , ASCE 1996; 122:282–285. 19. Han Han SW, Wen YK. Method Method of reliabili reliability-ba ty-based sed seismic design. I: Equivalent Equivalent non-linear non-linear system. system. Journal Journal of Structural Engineering, ASCE 1997; 123:256–265. 20. Allahabadi R, Powell GH. DRAIN-2DX user guide. Report No. UCB = EERC-88= Earthquakee Engineeri Engineering ng EERC-88= 06 , Earthquak Research Center, University of California, Berkeley, CA, 1988. 21. Chopra AK, Goel R. Modal pushover analysis of SAC building, Proceedings SEAOC Convention , San Diego, California, 2001. 22. Chopra AK, Goel R. A modal pushover analysis procedure to estimate seismic demands for buildings: theory and preliminary evaluation. Report No. PEER 2001= 2001 = 03, Paciÿc Earthquake Engineering Research Center, University of California, Berkeley, CA, 2001. 23. Ohto Ohtori ri Y, Christ Christens enson on RE, RE, Spence Spencerr Jr BF, BF, Dyke Dyke SJ. Benchm Benchmark ark Contr Control ol Proble Problems ms for Seismi Seismical cally ly Excite Excited d Nonlinear Buildings, http:== www.nd.edu= ∼ quake= , Notre Dame University, Indiana, 2000. Copyright ? 2001 John Wiley & Sons, Ltd.


A Modal Pushover Analysis Procedure for Estimating Seismic Demands for Buildings - Chopra

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