SIDESWAY MAGNIFICATION FACTORS FOR STEEL MOMENT FRAMES WITH VARIOUS TYPES OF COLUMN BASES

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1 Advned Steel Constrution Vol., No., pp () 7 SIDESWAY MAGNIFICATION FACTORS FOR STEEL MOMENT FRAMES WIT VARIOUS TYPES OF COLUMN BASES J. ent sio Assoite Professor, Deprtment of Civil nd Environmentl Engineering, Southern Illinois University Crondle, Crondle, IL, USA E-mil: hsio@engr.siu.edu Reeived: 6 June ; Revised: July ; Aepted: 7 July ABSTRACT: The sideswy mgnifition ftor ws introdued to the design of olumns in moment frmes sujeted to the P-Δ effet. Three pprohes for the omputtion of the sideswy mgnifition ftor, nmely, the olumn-lterl-uling-strength pproh, the story-lterl-stiffness pproh, nd the modified-story-lterl-stiffness pproh, were suggested y the AISC Speifition for Struturl Steel Buildings, the ASCE/S 7- Minimum Design Lods for Buildings nd Other Strutures, nd the AISC Speifition for Struturl Steel Buildings, respetively. This pper evlutes the sideswy mgnifition ftors derived from the forementioned three pprohes for four different olumn se onditions, nmely, idel fixed-, idel pinned-, prtil fixed-, nd prtil pinned-se onditions. The results of the study onduted in this pper re tht () if only the flexurl defletions re onsidered in the frme nlysis, the sideswy mgnifition ftor derived from the modified story-lterl-stiffness pproh losely grees with tht derived from the olumn-lterl-uling-strength pproh nd () if the flexurl defletions, s well s the sher nd xil deformtions, re onsidered in the frme nlysis, the sideswy mgnifition ftor derived from the story-lterl-stiffness pproh losely grees with tht derived from the olumn-lterl-uling-strength pproh. eywords: Defletion, Effetive length, Lterl fores, Sideswy, Steel olumns, Steel frmes. INTRODUCTION Mny methods hve een introdued to the P-Δ nlysis of olumns in moment frmes. These methods inlude the mplifition ftor method, the diret method, the itertive method, the negtive property fititious memer methods, nd the seond-order omputer progrm method. The mplifition ftor method is rpid, ut very pproximte method while the diret method gives urte results for low- or medium-rise rigid frmes []. Both the mplifition ftor method nd the diret method re ommonly used y struturl engineers for the omputtion of the sideswy mgnifition ftor for the design of olumns in steel moment frmes sujeted to the P- Δ effet. The mplifition ftor method involves the omputtion of the lterl uling strength of the olumns in the story eing onsidered; this method ws suggested y the AISC Speifition for Struturl Steel Buildings []. The diret method involves the omputtion of the lterl stiffness of the story eing onsidered; this method ws suggested y the AISC Speifition for Struturl Steel Buildings, the ASCE/S 7- Minimum Design Lods for Buildings nd Other Strutures [], nd the AISC Speifition for Struturl Steel Buildings []. The following is summry of the pprohes for the omputtion of the sideswy mgnifition ftor used y the P-Δ effets suggested y the AISC, ASCE/S 7-, nd AISC, respetively. The AISC Speifition for Struturl Steel Buildings suggests the following two pprohes for the omputtion of the sideswy mplifier vlues (B) for the design of olumns in moment frmes sujeted to the P-Δ effet:

2 7 Sideswy Mgnifition Ftors for Steel Moment Frmes with Vrious Types of Column Bses B Pnt L () B () Pnt L RM where =. for the Lod nd Resistne Ftor Design (LRFD); Pnt = the totl vertil lod supported y the story; E = the modulus of elstiity of steel, 9, si ( MP); I = the moment of inerti in the plne of ending; = the effetive length ftor in the plne of ending, sed on sideswy uling; L = the story height; RM =.8 for moment-frme systems; = the story sher produed y the lterl fores used to ompute Δ; nd Δ = the first-order interstory drift resulting from lterl fores. The first pproh (Eq. ) uses the sme method s the forementioned mplifition ftor method, whih involves the omputtion of the Euler lod of the olumn [] nd the elsti story sideswy uling resistne. The effetive length ftor () presented in Eq. n e determined y using the Alignment Chrt-Sideswy Uninhiited (Moment Frme) presented in the AISC Speifition for Struturl Steel Buildings. The seond pproh (Eq. ) uses the sme method s the forementioned diret method, whih involves the elsti nlysis of the first-order interstory drifts due to lterl fores [,6]. The ASCE/S 7- Minimum Design Lods for Buildings nd Other Strutures suggests the following pproh for the omputtion of the sideswy mgnifition ftor for the design of olumns in moment frmes sujeted to the P-Δ effet: mgnifition ftor δ xe p x Vxh δ sx x e () where Px = the totl vertil design lod t nd ove level x; Vx = the sher fore ting etween levels x nd x-; hsx = the story height elow level x; δ xe = the displement t level x y first-order elsti nlysis; nd = the displement t level x- y first-order elsti nlysis. δ( x ) e This pproh (Eq. ) uses the sme method s the forementioned diret method, whih involves the omputtion of the first-order interstory drifts due to lterl fores. Note tht the only differene etween Eqs. nd is tht Eq. onsiders the lterl stiffness modifition oeffiient, RM, while Eq. does not. The AISC Speifition for Struturl Steel Buildings suggests the following pproh (Eq. ) for the omputtion of the sideswy mplifier vlue (B) for the design of olumns in moment frmes sujeted to the P-Δ effet:

3 J. ent sio 76 L RM B Pstory () where =. for the Lod nd Resistne Ftor Design (LRFD); Pstory = the totl vertil lod supported y the story; RM =. (Pmf /Pstory); Pmf = the totl vertil lod in olumns in the story tht re prt of moment frmes; L = the story height; = the story sher produed y the lterl fores used to ompute Δ; nd Δ = the first-order interstory drift resulting from lterl fores. Note tht for uilding in whih lterl stiffness is provided entirely y moment frmes, Pmf = Pstory, whih in turn results in RM =.8. Therefore, Eqs. nd re identil.. EFFECTIVE LENGT FACTORS FOR COLUMNS IN SWAY FRAMES The effetive length ftor () for olumns in swy frme n e determined y using the lignment hrt for uninhiited sideswy (presented in Chpter C Stility Anlysis nd Design in the AISC Speifition for Struturl Steel Buildings) or y using the following eqution []: B GA G / 6 G G A B 6 / tn / where G = the restrint ftor t the olumn end (the susripts A nd B refer to the joints t the top nd ottom of the olumn eing onsidered) = (I/L) / (I/L) The pproximte vlue n e otined y using the following eqution [7]:. 6G AGB. GA GB 7. () G G 7. A B For olumn with omplete fixity t its se, GB =, Eq. eomes. G A 7. (6) G 7. A For olumn with true frition-free pinned se, GB =, Eq. eomes. 6G. (7) A The AISC Speifition, however, suggests tht if the olumn end is rigidly tthed to properly designed footing, GB my e ten s.. Smller vlues my e used if justified y nlysis. The reder is referred to the AISC Steel Design Guide Bse Plte nd Anhor Rod Design [8] for the design of olumn se pltes with moments. The AISC Speifition lso suggests tht if the olumn se is not rigidly onneted to footing or foundtion, GB my e ten s for prtil designs. GB is theoretilly infinity if the olumn se is designed s true frition-free pin.

4 77 Sideswy Mgnifition Ftors for Steel Moment Frmes with Vrious Types of Column Bses. STORY DRIFTS OF MOMENT FRAMES UNDER LATERAL LOADS Referring to the frme shown in Figure, the olumn sizes re identil nd oth olumns hve fixed se onnetion. The horizontl lod,, is ting t joint. Using the stiffness method, the following eqution n e derived [9]: L R θ θ 6 (8) where i-j = Ei-jIi-j/Li-j; θi = rottion t joint i; R = Δ/L. Sine the sizes nd lengths of the two olumns re identil, - = - = /L. Setting λ = (I/L) / (I/L), Eq. 8 n e rewritten s L R L θ θ λ λ λ λ (9) From Eq. 9, 6λ 8 L R Sine R = Δ/L, 6λ 8 L () Figure. Typil Moment Frme with Idel Fixed-Bse Connetions Notes: I = moment of inerti of olumn I = moment of inerti of em L = olumn length L = em length E = modulus of elstiity, onstnt for ll memers Δ = story drift resulting from lterl fore Δ L L

5 J. ent sio 78 Eq. is to e used for the omputtion of the story drift for the fixed-se frme loded under lterl fore, s shown in Figure. Note tht this eqution onsiders the flexurl defletions of the memers only. Axil nd sher deformtions of the memers hve een negleted. Similr to the derivtion of Eq. 8 from the moment frme with fixed-se onnetions shown in Figure, Eq. n e derived from the moment frme with pinned-se onnetions shown in Figure : θ θ R L () where i-j = Ei-jIi-j/Li-j; θi = rottion t joint i; R = Δ/L. Δ L L Notes: I = moment of inerti of olumn I = moment of inerti of em L = olumn length L = em length E = modulus of elstiity, onstnt for ll memers Δ = story drift resulting from lterl fore Figure. Typil Moment Frme with Idel Pinned-Bse Connetions Sine the sizes nd lengths of the two olumns re identil, - = - = /L. Setting λ = (I/L) / (I/L), Eq. n e rewritten s L λ. λ. λ λ... θ. θ R L () From Eq., L R 6 λ Sine R = Δ/L,

6 79 Sideswy Mgnifition Ftors for Steel Moment Frmes with Vrious Types of Column Bses L 6 λ () Eq. is to e used for the omputtion of the story drift for the pinned-se frme loded under lterl fore, s shown in Figure. Also note tht this eqution onsiders the flexurl defletions of the memers only. Axil nd sher deformtions of the memers hve een negleted.. EFFECTIVE LENGT AND DIRECT ANALYSIS METODS The AISC Speifition ddresses two mjor methods, the Effetive Length nd Diret Anlysis Methods, for the lultion of the required strengths for the stility design of memers nd onnetions. The Effetive Length Method is vlid so long s the rtio of seond-order defletion to first-order defletion in ll stories is equl to or less thn. (tht is: B = seond-order/ first-order.). The Diret Anlysis Method hs the dvntge of not hving to lulte the effetive length ftor,. This mens tht in designing ompression memers, the effetive length ftor is ten s.. To omplish this the Speifition requires tht redued xil nd flexurl stiffness shll e used for ll elements ontriuting to the lterl lod resistne of the struture to ount for the influene of inelstiity nd residul stresses on seond-order effets. Note tht the omputtion of L the first-order interstory drift,, in RM, nd, ( δxe δ( x ) e ), in V x h sx, respetively, δ xe δx e in Setion uses the Effetive Length Method. Therefore, the redued xil nd flexurl stiffness in the memers of the struture hve een ignored. owever, the nlysis using oth methods shll onsider flexurl, sher nd xil memer deformtions nd ll other omponent nd onnetion deformtions tht ontriute to displements of the struture.. COMPARISON OF EQUATIONS,, AND This setion ddresses the omprison of the L, R M L, nd V x h sx xe x e vlues in Eqs. δ δ,, nd, respetively, for vrious olumn se onditions, whih inlude the idel fixed-se ( omplete fixity, GB = ) ondition, the idel pinned-se ( true frition-free pin, GB = ) ondition, the prtil fixed-se (GB =.) ondition, nd the prtil pinned-se (GB = ) ondition. In this pper, (Pe), (Pe), nd (Pe) represent the Eqs.,, nd, respetively.. Idel Fixed-Bse Condition (G B = ) L, R M L, nd δ V x h sx xe x e δ vlues in Consider the frme shown in Figure () for the ondition where GA = (I/L) / (I/L) =.. From Eq. 6, one hs =.. Therefore, the (Pe) vlue in Eq. n e omputed s

7 J. ent sio 8 (Pe). 9. L L 8in. (9, si )(999in. ). 9 =, ips ( N) Note tht the moment of inerti out the x-xis for W 9, Ix = 999 Sine GA =., λ = (I/L) / (I/L) = /GA =.7. From Eq., L L =.6 8. From whih the (Pe) vlue in Eq. n e omputed s in. ( 8 m ) []. P R L e M = L. 8 L. 6. L =,8 ips ( 9 N) (. N) (. N) (. N) (. N) (. N) (. N) ' (.7 m) ' (.7 m) ' (.7 m) ' (6. m) ' (.7 m) '(. m) () G A =. Notes: () G A =. () G A =.6667 () All memers re W 9. () All memers re ent out their x-xes due to the Figure. Moment Frmes with Idel Fixed-Bse Connetions (GB = ) Tle summrizes nd ompres the omputed (Pe) nd (Pe) vlues in Eqs. nd for GA =.,, nd.6667 for the lterlly loded frmes shown in Figure for idel fixed-se onnetions (GB = ). The results shown in Tle indite tht the omputed (Pe) vlues for Eq. losely gree with the omputed (Pe) vlues for Eq.. The differenes etween the (Pe) nd (Pe) vlues vry from % for GA =.6667 to % for GA =.. Note tht the Δ vlues shown in Tle re derived from Eq., whih onsiders the flexurl defletions of the memers only. Axil nd sher deformtions of the memers hve een negleted. Tle lso summrizes nd ompres the omputed (Pe) nd (Pe) vlues in Eqs. nd for GA =.,, nd.6667 for the lterlly loded frmes shown in Figure for idel fixed-se onnetions (GB = ). owever, the Δ vlues shown in Tle re otined y using the omputer softwre SAP [], whih onsiders the flexurl defletions, s well s the xil nd sher deformtions, of the memers. The results shown in Tle indite tht the omputed (Pe) vlues for Eq. re ll quite lrger thn the omputed (Pe) vlues for Eq.. The differenes etween the (Pe) nd (Pe) vlues vry from 6% for GA =. to % for GA =.6667.

8 8 Sideswy Mgnifition Ftors for Steel Moment Frmes with Vrious Types of Column Bses Tle. Comprison of (Pe) nd (Pe) for the Lterlly Loded Frmes Shown in Figure (Idel fixed-se onnetions, GB = ) Figure G A (P e) Δ (P e) (P e) vs. (P e) ()... 9 L (P e) > (P e) y %. L 8. L =, ips ( N) =. in. (. m) =,8 ips ( 9 N) () L =, ips (8 N) L 8 7 =. in. (. m). 8 L =,77 ips (6 8 N) (P e) > (P e) y % () L =,8 ips (6 7 N) L 8 =.9 in. (.77 m). 69 L =, ips (6 N) (P e) > (P e) y % (Pe)= s shown in Eq.. L Only the flexurl defletions of the memers re onsidered for the omputtion of Δ. (Pe)= R M L s shown in Eq.. Tle. Comprison of (Pe) nd (Pe) for the Lterlly Loded Frmes Shown in Figure (Idel fixed-se onnetions, GB = ) Figure G A (P e) Δ (P e) (P e) vs. (P e) ().., ips.6 in.,8 ips (P e) > (P e) y 6 % ( N) (.7 m) (6 6 N) ()..6, ips.8 in.,9 ips (P e) > (P e) y 8 % (8 N) () ,8 ips (6 7 N) (. m). in. (. m) (9 N),9 ips ( N) (P e) > (P e) y % (Pe)= s shown in Eq.. L The flexurl defletions, s well s the xil nd sher deformtions, re onsidered for the omputtion of Δ. L (Pe)= R s shown in Eq.. M Tle summrizes nd ompres the omputed (Pe) nd (Pe) vlues in Eqs. nd for GA =.,, nd.6667 for the lterlly loded frmes shown in Figure for idel fixed-se onnetions (GB = ). The ( δ xe δ( x ) e ) vlues shown in Tle re otined y using the omputer softwre SAP, whih onsiders the flexurl defletions, s well s the xil nd sher deformtions, of the memers. The results shown in Tle indite tht the omputed (Pe) vlues for Eq. losely gree with the omputed (Pe) vlues for Eq.. The differenes etween the (Pe) nd (Pe) vlues re within %.

9 J. ent sio 8 Tle. Comprison of (Pe) nd (Pe) for the Lterlly Loded Frmes Shown in Figure (Idel fixed-se onnetions, GB = ) Figure G A (P e) ( δ xe δ( x ) e ) (P e) (P e) vs. (P e) ().., ips ( N) ()..6, ips (8 N) () ,8 ips (6 7 N).6 in. (.7 m).8 in. (. m). in. (. m), ips ( 8 N), ips (8 N),6 ips (6 67 N) (P e) < (P e) y % (P e) (P e) (P e) > (P e) y % (Pe)= s shown in Eq.. L The flexurl defletions, s well s the xil nd sher deformtions, re onsidered for the omputtion of ( δ xe δ( x ) e ). (Pe)= δ V x h sx xe δx e s shown in Eq... Idel Pinned-Bse Condition (G B = ) Tle summrizes nd ompres the omputed (Pe) nd (Pe) vlues in Eqs. nd for GA =.,, nd.6667 for the lterlly loded frmes shown in Figure for idel pinned-se onnetions (GB = ). The results shown in Tle indite tht the omputed (Pe) vlues for Eq. losely gree with the omputed (Pe) vlues for Eq.. The differenes etween the vlues vry from % for GA =.6667 to % for GA =.. Note tht the omputed Δ vlues shown in Tle re derived from Eq., whih onsiders the flexurl defletions of the memers only. Axil nd sher deformtions of the memers hve een negleted. Tle. Comprison of (Pe) nd (Pe) for the Lterlly Loded Frmes Shown in Figure (Idel pinned-se onnetions, GB = ) Figure G A (P e) Δ (P e) (P e) vs. (P e) () L =,88 ips ( 8 N) L 6. =.9 in. (. m). 6 L =,7 ips ( 9 N) (P e) > (P e) y % () L =, ips ( N) L 6 =. in. (.78 m). L =, ips ( N) (P e) > (P e) y % () L =,8 ips ( 8 N) L 6 =.7 in. (. m). 8 L =, ips ( N) (P e) > (P e) y % (Pe)= s shown in Eq.. L Only the flexurl defletions of the memers re onsidered for the omputtion of Δ. (Pe)= L RM s shown in Eq..

10 8 Sideswy Mgnifition Ftors for Steel Moment Frmes with Vrious Types of Column Bses (. N) (. N) (. N) (. N) (. N) (. N) ' (.7 m) ' (.7 m) ' (.7 m) ' (6. m) ' (.7 m) '(. m) () G A =. () G A =. () G A =.6667 Notes: () All memers re W 9. () All memers re ent out their x-xes due to the pplied Figure. Moment Frmes with Idel Pinned-Bse Connetions (GB = ) Tle lso summrizes nd ompres the omputed (Pe) nd (Pe) vlues in Eqs. nd for GA =.,, nd.6667 for the lterlly loded frmes shown in Figure for idel pinned-se onnetions (GB = ). owever, the Δ vlues shown in Tle re otined y using the omputer softwre SAP, whih onsiders the flexurl defletions, s well s the xil nd sher deformtions, of the memers. The results shown in Tle indite tht the omputed (Pe) vlues for Eq. re ll quite lrger thn the omputed (Pe) vlues for Eq.. The differenes etween the vlues vry from % for GA =. to 6% for GA = Tle. Comprison of (Pe) nd (Pe) for the Lterlly Loded Frmes Shown in Figure (Idel pinned-se onnetions, GB = ) Figure G A (P e) Δ (P e) (P e) vs. (P e) ()..77,88 ips.9 in.,7 ips (P e) > (P e) y % ( 8 N) (. m) ( N) ()..66, ips.7 in.,8 ips (P e) > (P e) y % ( N) ().6667.,8 ips ( 8 N) (.89 m). in. (.9 m) ( N), ips ( N) (P e) > (P e) y 6 % (Pe)= s shown in Eq.. L The flexurl defletions, s well s the xil nd sher deformtions, re onsidered for the omputtion of Δ. L (Pe)= R s shown in Eq.. M Tle 6 summrizes nd ompres the omputed (Pe) nd (Pe) vlues in Eqs. nd for GA =.,, nd.6667 for the lterlly loded frmes shown in Figure for idel pinned-se onnetions (GB = ). The ( δxe δ( x ) e ) vlues shown in Tle 6 re otined y using the omputer softwre SAP, whih onsiders the flexurl defletions, s well s the xil nd sher deformtions, of the memers. The results shown in Tle 6 indite tht the omputed (Pe) vlues for Eq. losely gree with the omputed (Pe) vlues for Eq.. The differenes etween the vlues vry from % for GA =.6667 to % for GA =..

11 J. ent sio 8 Tle 6. Comprison of (Pe) nd (Pe) for the Lterlly Loded Frmes Shown in Figure (Idel pinned-se onnetions, GB = ) Figure G A (P e) ( δ xe δ( x ) e ) (P e) (P e) vs. (P e) ()..77,88 ips ( 8 N) ()..66, ips ( N) ().6667.,8 ips ( 8 N).9 in. (. m).7 in. (.89 m). in. (.9 m), ips ( 7 N),9 ips ( 6 N), ips ( 7 N) (P e) < (P e) y % (P e) < (P e) y % (P e) < (P e) y % (Pe)= s shown in Eq.. L The flexurl defletions, s well s the xil nd sher deformtions, re onsidered for the omputtion of ( δ xe δ( x ) e ). (Pe)= δ V x h sx xe δx e s shown in Eq... Prtil Fixed-Bse Condition (G B = ) In order to hieve the prtil fixed-se ondition with GB =, the omputer model of the frme shown in Figure () is modified to eome the frme shown in Figure y dding se em W 9 to onnet the ses of the W 9 olumns, s well s y hnging the fixed ses to pinned ses. Note tht oth ends of the dded se em re rigidly onneted to the ottom ends of the olumns, while the newly dded pinned ses re loted right elow the new rigid onnetions t the ottom ends of the olumns. Referring Figure, GB n e omputed s I / L 999 in. /8 in. G B I / L 999 in. /8 in. 8 m / 7 m 8 m / 7 m = Note tht I nd L shown in the ove eqution re the moment of inerti (out the x-xis) nd the length of the se em, while I nd L re the moment of inerti (out the x-xis) nd the length of the olumn. (. N) (. N) W 9 Bse em W 9 ' (.7 m) ' (.7 m) Note: All memers re ent out their x-xes due to the pplied fores. Figure. Moment Frme with Modified Bse Stiffness for the Prtil Fixed-Bse Condition with GB =

12 8 Sideswy Mgnifition Ftors for Steel Moment Frmes with Vrious Types of Column Bses Tle 7 summrizes nd ompres the omputed (Pe) nd (Pe) vlues in Eqs. nd for the lterlly loded frme with GA = nd GB = s shown in Figure. Note tht the Δ vlue shown in Tle 7 is otined y using the omputer softwre SAP, whih onsiders the flexurl defletions, s well s the xil nd sher deformtions, of the memers. The results shown in Tle 7 indite tht the omputed (Pe) vlue for Eq. is out % lrger thn the omputed (Pe) vlue for Eq.. Tle 7. Comprison of (Pe) nd (Pe) for the Lterlly Loded Frmes Shown in Figure (Prtil fixed-se onnetions) G A G B (P e) Δ (P e) (P e) vs. (P e)... 9,8 ips ( 9 N).96 in. (.98 m) 7,8 ips ( 7 N) (P e) > (P e) y % (Pe)= s shown in Eq.. L The flexurl defletions, s well s the xil nd sher deformtions, re onsidered for the omputtion of Δ. L (Pe)= R s shown in Eq.. M Tle 8 summrizes nd ompres the omputed (Pe) nd (Pe) vlues in Eqs. nd for the lterlly loded frme with GA = nd GB = s shown in Figure. Note tht the ( δxe δ( x ) e ) vlue shown in Tle 8 is otined y using the omputer softwre SAP, whih onsiders the flexurl defletions, s well s the xil nd sher deformtions, of the memers. The results shown in Tle 8 indite tht the omputed (Pe) vlue for Eq. is out 7% lrger thn the omputed (Pe) vlue for Eq.. Tle 8. Comprison of (Pe) nd (Pe) for the Lterlly Loded Frmes Shown in Figure (Prtil fixed-se onnetions) G A G B (P e) ( δ xe δ( x ) e ) (P e) (P e) vs. (P e)... 9,8 ips ( 9 N).96 in. (.98 m) 9,8 ips ( 8 N) (P e) > (P e) y 7 % (Pe)= s shown in Eq.. L The flexurl defletions, s well s the xil nd sher deformtions, re onsidered for the omputtion of ( xe δ ). (Pe)= δ V x h sx xe δx e δ ( x ) e s shown in Eq... Prtil Pinned-Bse Condition (G B = ) In order to hieve the prtil fixed-se ondition with GB =, the omputer model of the frme shown in Figure () is modified to eome the frme shown in Figure 6 y dding se em W8 8 to onnet the ses of the W 9 olumns. Note tht oth ends of the dded se em re rigidly onneted to the ottom ends of the olumns, while the pinned ses re loted right elow the new rigid onnetions t the ottom ends of the olumns. Referring Figure 6, the GB n e omputed s

13 J. ent sio 86 I / L 999 in. /8 in. GB I / L 98 in. /8 in. 8 m / 7m 8 m / 7m Note tht I nd L shown in the ove eqution re the moment of inerti (out the x-xis) nd the length of the se em, while I nd L re the moment of inerti (out the x-xis) nd the length of the olumn. (. N) (. N) W 9 Bse em W8 8 ' (.7 m) ' (.7 m) Note: All memers re ent out their x-xes due to the pplied fores. Tle 9 summrizes nd ompres the omputed (Pe) nd (Pe) vlues in Eqs. nd for the lterlly loded frme with GA = nd GB = s shown in Figure 6. Note tht the Δ vlue shown in Tle 9 is otined y using the omputer softwre SAP, whih onsiders the flexurl defletions, s well s the xil nd sher deformtions, of the memers. The results shown in Tle 9 indite tht the omputed (Pe) vlue for Eq. is out % lrger thn the omputed (Pe) vlue for Eq.. Tle 9. Comprison of (Pe) nd (Pe) for the Lterlly Loded Frmes Shown in Figure 6 (Prtil pinned-se onnetions) G A G B (P e) Δ (P e) (P e) vs. (P e)..9,8 ips ( N).8 in. (.97 m),98 ips (7 7 N) (P e) > (P e) y % (Pe)= s shown in Eq.. L The flexurl defletions, s well s the xil nd sher deformtions, re onsidered for the omputtion of Δ. L (Pe)= R s shown in Eq.. M Figure 6. Moment Frme with Modified Bse Stiffness for the Prtil Pinned-Bse Condition with GB = Tle summrizes nd ompres the omputed (Pe) nd (Pe) vlues in Eqs. nd for the lterlly loded frme with GA = nd GB = s shown in Figure 6. Note tht the ( δxe δ( x ) e ) vlue shown in Tle is otined y using the omputer softwre SAP, whih onsiders the flexurl defletions, s well s the xil nd sher deformtions, of the memers. The results shown in Tle indite tht the omputed (Pe) vlue for Eq. is out % lrger thn the omputed (Pe) vlue for Eq..

14 87 Sideswy Mgnifition Ftors for Steel Moment Frmes with Vrious Types of Column Bses Tle. Comprison of (Pe) nd (Pe) for the Lterlly Loded Frmes Shown in Figure 6 (Prtil pinned-se onnetions) G A G B (P e) ( δ xe δ( x ) e ) (P e) (P e) vs. (P e)..9,8 ips ( N).8 in. (.97 m),69 ips ( 86 N) (P e) > (P e) y % (Pe)= s shown in Eq.. L The flexurl defletions, s well s the xil nd sher deformtions, re onsidered for the omputtion of ( xe δ ). (Pe)= δ V x h sx xe δx e δ ( x ) e s shown in Eq.. 6. DISCUSSION AND CONCLUSION The pprohes tht hve een ommonly used y struturl engineers for the omputtion of the sideswy mgnifition ftor for the design of olumns in steel moment frmes sujeted to the P-Δ effet re the olumn-lterl-uling-strength pproh (suggested y the AISC Speifition for Struturl Steel Buildings), the story-lterl-stiffness pproh (suggested y the ASCE/S 7- Minimum Design Lods for Buildings nd Other Strutures), nd the modified-story-lterl-stiffness pproh (suggested y the nd AISC Speifition for Struturl Steel Buildings). The olumn-lterl-uling-strength pproh involves the omputtion of the elsti story sideswy uling resistne, the story-lterl-stiffness pproh involves the omputtion of the elsti nlysis of the first-order interstory drift due to lterl fores, nd the modified-story-lterl-stiffness pproh involves the omputtion of the first-order interstory drift using the lterl stiffness modifition oeffiient, RM, for moment-frme systems. The story-lterl-stiffness pproh hs een reognized s n urte pproh for low- or medium-rise rigid frmes, while the olumn-lterl-uling-strength pproh hs een lssified s very pproximte pproh. Nevertheless, the results of the study onduted in this pper re tht () the sideswy mgnifition ftor derived from the olumn-lterl-uling-strength pproh losely grees with tht derived from the modified story-lterl-stiffness pproh if only the flexurl defletions re onsidered in the frme nlysis for the omputtion of the interstory drifts of the frme nd () the sideswy mgnifition ftor derived from the olumn-lterl-uling-strength pproh losely grees with tht derived from the story-lterl-stiffness pproh if the flexurl defletions, s well s the sher nd xil deformtions, re onsidered in the frme nlysis for the omputtion of the interstory drifts of the frme. Sine the olumn-lterl-uling-strength pproh is hnd-lulted pproh, it n e used rpidly nd prtilly to onfirm the ury nd vlidity of the results derived from the story-lterl-stiffness pproh, whih is usully rried out with omputer softwre for the omputtion of sideswy mgnifition ftors for steel moment frmes sujeted to the P-Δ effet. NOTATION The following symols re used in this pper: B = sideswy mplifier E = the modulus of elstiity of steel, 9, si ( MP)

15 J. ent sio 88 G = the restrint ftor t the olumn end = the story sher produed y the lterl fores used to ompute Δ = the story sher produed y the lterl fores used to ompute Δ hsx = the story height elow level x I = the moment of inerti in the plne of ending I = moment of inerti of em I = moment of inerti of olumn = the effetive length ftor in the plne of ending, sed on sideswy uling L = the story height L = the em length L = the olumn length Pmf = the totl vertil lod in olumns in the story tht re prt of moment frmes Pnt = the totl vertil lod supported y the story Pstory = the totl vertil lod supported y the story Px = the totl vertil design lod t nd ove level x RM =.8 for moment-frme systems Vx = the sher fore ting etween levels x nd x- =. for LRFD Δ = the first-order interstory drift resulting from lterl fores δ = the displement t level x y first-order elsti nlysis xe δ( x )e = the displement t level x- y first-order elsti nlysis θi = rottion t joint i REFERENCES [] Giotti, R. nd Smith, B.S., P-delt Anlysis for Building Strutures, Journl of Struturl Engineering, ASCE, 989, Vol., No., pp [] AISC, Speifition for Struturl Steel uildings, Amerin Institute of Steel Constrution, In., Chigo, IL.,. [] ASCE, ASCE/S 7- Minimum Design Lods for Buildings nd Other Strutures, Amerin Soiety of Civil Engineers, Reston, Virgini,. [] AISC, Speifition for Struturl Steel Buildings, Amerin Institute of Steel Constrution, In., Chigo, IL.,. [] Timosheno, S.P. nd Gere, J.M., Theory of Elsti Stility, nd Ed., MGrw-ill Boo Co., New Yor, N.Y., 96. [6] Slmon, C.G., Johnson, J.E. nd Mlhs, F.A., Steel Strutures, Design nd Behvior, th Ed., Person Prentie ll, Upper Sddle River, N.J., 9. [7] Dumonteil, P., Simple Equtions for Effetive Length Ftors, Engineering Journl, AISC, rd qurter, 99, pp. -. [8] AISC, Steel Design Guide Bse Plte nd Anhor Rod Design, nd Ed., Amerin Institute of Steel Constrution, Chigo, IL.,. [9] sio, J.., Computtion of Fundmentl Periods for Moment Frmes Using nd-lulted Approh, Eletroni Journl of Struturl Engineering, 9, Vol. 9, pp [] AISC, Steel Constrution Mnul, th Ed., Amerin Institute of Steel Constrution, In., Chigo, IL.. [] SAP Edutionl, SAP Mnuls, Computers nd Strutures, In., Bereley, CA., 997.