III-47 MOMENT FRAME MODEL

Transcription

1 III-47 OENT FRAE ODEL Grids and 8 were modeled in onventional strutural analysis software as two-dimensional models. The seondorder option in the strutural analysis program was not used. Rather, for illustration purposes, seond-order effets are alulated separately, using the B -B approximation method given in Setion C.b. The olumn and beam layouts for the moment frames follow. Although the frames on Grids A and F are the same, slightly heavier seismi loads aumulate on grid F, after aounting for the atrium area on Grid A and aidental torsion. The models are half-building models. The frame was originally modeled with W4 8 interior olumns and W 44 non-omposite beams. This model had a drift that substantially exeeded the 0.05h sx story drift allowed per IBC Table 67.3., Seismi Use Group I. The olumn size was inremented up to a W4 99 and the W 44 beams were upsized to W4 55 (with minimum omposite studs) and the beams were modeled with a stiffness of I eq I s. Alternatively, the beams ould be modeled as Ieq 0.6ILB + 0.4I n (Formula 4), s. This equation is given in AISC Design Guide 8. These hanges resulted in a drift that satisfied the L / 400 limit. This layout is shown in the Grid A and F, Frame (½ building) layout that follows. All of the vertial loads on the frame were modeled as point loads on the frame. As noted in the desription of the dead load, W, in IBC Setion 67.5, 0.00 kip/ft is inluded in the dead load ombinations. The remainder of the half-building model gravity loads were aumulated in the leaning olumn, whih was onneted to the frame portion of the model with pinned ended links. See Geshwindner, AISC Engineering Journal, Fourth Quarter 994, A ratial Approah to the Leaning Column. The dead load and live load are shown in the load ases that follow. The wind and the seismi loads are modeled and distributed /4 to exterior olumns and /7 to the interior olumns. This approah minimizes the tendeny to aumulate too muh load in the lateral system nearest an externally applied load. There are four horizontal load ases. Two are the wind load and seismi load, per the previous disussion. In addition, notional loads of N i 0.00Y i were established. These load ases are shown in the load ases that follow. The same modeling proedures were used in the braed frame analysis. If olumn bases are not fixed in onstrution, they should not be fixed in the analysis.

2 The model layout, dead loads, live loads, wind loads, seismi loads, dead load notional loads, and live load notional loads for the moment frame are given as follows: III-48

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7 III-53 CALCULATION OF REQUIRED STRENGT - TREE ETODS Three methods of determining the required strength inluding seond-order effets are inluded below. A fourth method seond-order analysis by amplified first-order analysis is found in Setion C.b or the. The method requires the inlusion of notional loads in the analysis, but all required strengths an be determined from a first-order analysis. For guidane on applying these methods, see the disussion in anual art titled Required Strength, Effetive Length, and Seond-Order Effets. GENERAL INFORATION FOR ALL TREE ETODS Seismi load dominates over winds loads in the moment frame diretion of this example building. Although the frame analysis that follows was run for all LRFD and ASD load ombinations, the olumn unity design hek is highly dependent on the moment portion, and therefore, the ontrolling equations are those with the load ombinations shown below. A typial olumn near the middle of the frame is analyzed, but the first interior olumns and the end olumns were also heked. Beam analysis is overed after the three different methods are shown for the typial interior olumn. Note: The seond-order analysis and the unity heks are based on the moments and olumn loads for the partiular load ombination being heked, and not on the envelope of maximum values. ETOD. EFFECTIVE LENGT ETOD This method aounts for seond-order effets in frames by amplifying the axial fores and moments in members and onnetions from a first-order analysis. C.a A first-order frame analysis is run using the load ombinations for LRFD or ASD. A minimum lateral load (notional load) equal to 0.% of the gravity loads is inluded for any load ase for whih the lateral load is not already greater. The general load ombinations are in ASCE 7 and are summarized in art of the anual. A summary of the axial loads, moments and st floor drifts from the first-order omputer analysis is shown below: LRFD.D ±.0E + 0.5L + 0.S (Controls for olumns and beams) For Interior Column Design: u 335 kips u 57 kip-ft (from first-order analysis) u 9 kip-ft (from first-order analysis) First-order first floor drift 0.56 in. ASD D ± (W or 0.7 E) (Controls for olumns) D (W or 0.7E) L (L r or S or R) (Controls for beams) For Interior Column Design: a 47 kips a 0 kip-ft (from first-order analysis) a 6 kip-ft (from first-order analysis) First-order first floor drift in. The required seond-order flexural strength, r, and axial strength, r, are as follows: For typial interior olumns the gravity-load moments are approximately balaned, therefore, nt 0.0 kip-ft

8 III-54 r B nt + B lt LRFD r B nt + B lt ASD Eqn C-a Determine B Determine B r required seond-order axial strength using LRFD or ASD load ombinations, kips. Note that for members in axial ompression r may be taken as r nt + lt. For a long frame, suh as this one, the hange in load to the interior olumns, assoiated with lateral load is negligible. Therefore, r 335 kips (from previous alulations) and I,0 in. 4 (W4 99) r required seond-order axial strength using LRFD or ASD load ombinations, kips. Note that for members in axial ompression r may be taken as r nt + lt. For a long frame, suh as this one, the hange in load to the interior olumns, assoiated with lateral load is negligible. Therefore, r 47 kips (from previous alulations) and I,0 in. 4 (W4 99) Setion C.b el π EI ( KL) el π EI ( KL) Eqn C-5 π 4 ( 9, 000 ksi)(,0 in ) ( ) (.0) ( 3.5 ft)( in/ft) π 4 ( 9,000 ksi)(,0 in ) ( ) (.0) ( 3.5 ft)( in/ft),00 kips C m ( / ) (57 kip-ft / 9 kip-ft) 0.36 α.0, Cm B α r e,00 kips C m ( / ) (0 kip-ft / 6 kip-ft) 0.36 α.6, Cm B α r e Eqn C-4 Eqn C (.0)( 335 kips),00 kips 0.36 (.6)( 47 kips),00 kips ; Use ; Use.0

9 III-55 Calulate B B α where: α.0, nt e nt 5, 50 kips (from omputer output) and L e may be taken as R where R is taken as 0.85 for moment frames.d+.0e+ 0.5L+ 0.S 95 kips (orizontal) (from previous seismi fore distribution alulations) Calulate B B α where: α.6, nt e nt 3,750 kips (from omputer output) and L e may be taken as R where R is taken as 0.85 for moment frames D+ 0.7E 37 kips (orizontal) (from previous seismi fore distribution alulations) Setion C.b 0.56 in. (from omputer output) in. (from omputer output) 0.85 e ( 95 kips)( 3.5 ft)( in/ft) 0.56 in e ( 37 kips) ( 3.5 ft)( in/ft) in. Eqn C-6b 47,800 kips B α nt e (.0)( 5, 50 kips) 47,800 kips 47,900 kips B α nt e (.6)( 3, 750 kips) 47,900 kips Eqn C-3. Calulate amplified moment (.0)(0.0 kip-ft) + (.)(9 kip-ft) 56 kip-ft.4 Calulate amplified moment (.0)(0.0 kip-ft) + (.4)(6 kip-ft) 84 kip-ft Calulate amplified axial load r 335 kips (from omputer analysis) Calulate amplified axial load r 47 kips (from omputer analysis)

10 III-56 r nt + B lt 335 kips + (.)(0.0 kips) 335 kips Determine the ontrolling effetive length For out-of-plane bukling in the braed frame K y.0 For in-plane bukling in the moment frame, use the nomograph K x.43 To aount for leaning olumns in the ontrolling load ase For leaning olumns, Q 300 kips 490 kips K K + o KLx.3(3.5) 7.3 r / r.66 x y Q r nt + B lt 47 kips + (.4)(0.0 kips) 47 kips Determine the ontrolling effetive length For out-of-plane bukling in the braed frame K y.0 For in-plane bukling in the moment frame, use the nomograph K x.43 To aount for leaning olumns in the ontrolling load ase For leaning olumns, Q 40 kips 740 kips K K + o KLx.4(3.5) 7.4 r / r.66 x y Q Commentary Setion C.b

11 III-57,04 kips ( W4 KL 7.3 ft) r 335 kips ,04 kips x 646 kip-ft ( W4 99) 69 kips ( W4 KL 7.4 ft) r 47 kips kips x 430 kip-ft ( W4 99) Set. anual Table 4- anual Table 3- r 8 rx ry x y r 8 rx ry x y Eqn -a 856 kip-ft kip-ft 884 kip-ft kip-ft o.k o.k.

12 III-58 ETOD. SILIFIED DETERINATION OF REQUIRED STRENGT A method of seond-order analysis based upon drift limits and other assumptions is desribed in Chapter of the anual. A first-order frame analysis is run using the load ombinations for LRFD or ASD. A minimum lateral load (notional load) equal to 0.% of the gravity loads is inluded for any load ase for whih the lateral load is not already greater. LRFD.D ±.0E + 0.5L + 0.S (Controls olumns and beams) For a first-order analysis For Interior Column Design: u 335 kips u 57 kip-ft (from first-order analysis) u 9 kip-ft (from first-order analysis) First-floor first-order drift 0.56 in. ASD D ± (W or 0.7 E) (Controls olumns) D (W or 0.7E) L (L r or S or R) (Controls beams) For a first-order analysis For Interior Column Design: a 47 kips a 0 kip-ft (from first-order analysis) a 6 kip-ft (from first-order analysis) First-floor first-order drift in. Then the following steps are exeuted. Step : LRFD Step : ASD Lateral load 95 kips Defletion due to first-order elasti analysis 0.56 in. between first and seond floor Floor height 3.5 ft Drift ratio (3.5 ft)( in/ft) / 0.56 in 88 Step : Design story drift limit 0.05 h xs h/400 Adjusted Lateral load (88/ 400)(95 kips) 4 kips Lateral load 40 kips Defletion due to first-order elasti analysis in. between first and seond floor Floor height 3.5 ft Drift ratio (3.5 ft)( in/ft) / in 4 Step : Design story drift limit 0.05 h xs h/400 Adjusted Lateral load (4 / 400)(37 kips) 4 kips IBC Table 67.3.

13 III-59 Step 3: total story load Load ratio (.0) lateral load 5, 50 kips (.0) 4 kips Step 3: (for an ASD design the ratio must be fatored by.6) total story load Load ratio (.6) lateral load 3,750 kips (.6) 4 kips 37. Interpolating from the table: B. Whih mathes the value obtained in the first method to the signifiant figures of the table 4.6 Interpolating from the table: B. Whih mathes the value obtained in the first method to the signifiant figures of the table Note: Beause the table is intentionally based on two signifiant figures, this value is taken as. rather than an interpolated value >.. This onvenient seletion is within the auray of the method. Sine the seletion is in the shaded area of the hart, K.0. anual age - Step 4. ultiply all the fores and moment from the first-order analysis by the value obtained from the table. r B ( nt + lt ) LRFD r B ( nt + lt ) ASD.(0 kip-ft + 9 kip-ft) 5 kip-ft r.( nt + lt ).(335 kips +0.0 kips) 368 kips r 368 kips For ,40 kips where x,40 kips ( W4 KL 3.5 ft) 646 kip-ft ( W4 99) r 8 rx ry x y.(0 kip-ft + 6 kip-ft) 77 kip-ft r.( nt + lt ).(47 kips +0.0 kips) 7 kips r 7 kips For kips x 759 kips ( W4 KL 3.5 ft) 430 kip-ft ( W4 99) r 8 rx ry x y Set. anual Table 4- anual Table 3- Eqn -a 85 kip-ft kip-ft 877 kip-ft kip-ft o.k o.k.

14 III-60 ETOD 3. DIRECT ANALYSIS ETOD Seond-order analysis by the diret analysis method is found in Appendix 7 of the. This method requires that both the flexural stiffness and axial stiffness be redued and that 0.% notional lateral loads be applied in the analysis. The ombination of these two modifiations aount for the seond-order effets and the results for design an be taken diretly from the analysis. A summary of the axial loads, moments and st floor drifts from first-order analysis is shown below: LRFD.D ±.0E + 0.5L + 0.S (Controls olumns and beams) For a st order analysis with notional loads and redued stiffness: For Interior Column Design: u 335 kips u 57 kip-ft (from first-order analysis) u 9 kip-ft (from first-order analysis) First-floor drift due to redued stiffnesses in. ASD D ± (W or 0.7 E) (Controls olumns) D (W or 0.7E) L (L r or S or R) (Controls beams) For a st order analysis with notional loads and with redued stiffness: For Interior Column Design: a 47 kips a 0 kip-ft a 6 kip-ft First-floor drift due to redued stiffnesses in. Note: For ASD, this method requires multiplying the ASD load ombinations by a fator of.6 in analyzing the drift of the struture, and then dividing the results by.6 to obtain the required strengths. The ASD fores shown above inlude the multiplier of.6. LRFD For this method, K.0.,40 kips ( W4 KL 3.5 ft) r 335 kips ,40 kips x 646 kip-ft ( W4 99) ASD For this method, K kips ( W4 KL 3.5 ft) r 47 kips kips x 430 kip-ft ( W4 99) anual Table 4- Set. anual Table 3- r B nt + B lt r B nt + B lt Determine B r required seond-order axial strength using LRFD or ASD load ombinations, kips. Note that for members in axial ompression r may be taken as r nt + lt. For a long frame, suh as this one, the hange in load to the interior olumns, assoiated with lateral load is negligible. Therefore, r 335 kips (from previous alulations) and Determine B r required seond-order axial strength using LRFD or ASD load ombinations, kips. Note that for members in axial ompression r may be taken as r nt + lt. For a long frame, suh as this one, the hange in load to the interior olumns, assoiated with lateral load is negligible. Therefore, r 47 kips (from previous alulations) and

15 III-6 I,0 in. 4 (W4 99) I,0 in. 4 (W4 99) el π EI ( KL) el π EI ( KL) π 4 ( 9, 000 ksi)(,0 in ) ( ) (.0) ( 3.5 ft)( in/ft) π 4 ( 9,000 ksi)(,0 in ) ( ) (.0) ( 3.5 ft)( in/ft),06 kips,00 kips C m ( / ) (57 kip-ft / 9 kip-ft) 0.36 α.0, Cm B α r e C m ( / ) (0 kip-ft / 6 kip-ft) 0.37 α.6, Cm B α r e 0.36 (.0)( 335 kips),00 kips 0.37 (.6)( 47 kips),00 kips ; Use ; Use.0 Calulate B B α nt e where: α.0, nt 5, 50 kips (from omputer output) L e may be taken as R where R is taken as 0.85 for moment frames.d+.0e+ 0.5L+ 0.S 95 kips (orizontal) (from previous seismi fore distribution alulations) Calulate B B α nt e where: α.6, nt 3,750 kips (from omputer output) L e may be taken as R where R is taken as 0.85 for moment frames D+ 0.7E 37 kips (orizontal) (from previous seismi fore distribution alulations)

16 III in in. (from omputer output) 0.85 e ( 95 kips)( 3.5 ft)( in/ft) in e ( 37 kips) ( 3.5 ft)( in/ft) in. 38,00 kips B α nt e (.0)( 5, 50 kips) 38,00 kips 38,300 kips B α nt e (.6)( 3, 750 kips) 38,300 kips.6 Calulate amplified moment (.0)(0.0 kip-ft) + (.6)(9 kip-ft) 66 kip-ft.9 Calulate amplified moment (.0)(0.0 kip-ft) + (.9)(6 kip-ft) 9 kip-ft Calulate amplified axial load r 335 kips (from omputer analysis) Calulate amplified axial load r 47 kips (from omputer analysis) r nt + B lt 335 kips + (.6)(0.0 kips) 335 kips r 8 rx ry x y r nt + B lt 47 kips + (.9)(0.0 kips) 47 kips r 8 rx ry x y Eqn -b 866 kip-ft kip-ft 89 kip-ft kip-ft

17 III-64 BRACED FRAE ANALYSIS The braed frames at Grids and 8 were analyzed for their lateral loads. The same stability design requirements from Chapter C were applied to this system. Seond-order analysis by amplified first-order analysis The following is a method to aount for seond-order effets in frames by amplifying the axial fores and moments in members and onnetions from a first-order analysis. C.b First a first-order frame analysis is run using the load ombinations for LRFD and ASD. From this analysis the ritial axial loads, moments, and defletions are obtained. The required seond-order flexural strength, r, and axial strength, r, are as follows: el π EI ( KL) LRFD el π EI ( KL) ASD Eqn C-5 π 4 ( 9,000 ksi)( 45 in ) ( ) (.0) ( 3.5 ft)( in/ft) π 4 ( 9,000 ksi)( 45 in ) ( ) (.0) ( 3.5 ft)( in/ft) 4,635 kips 4,635 kips I W 4 45 in ( 58) I W 4 45 in ( 58) B α nt e B α nt e Eqn C-3 (.0)( 5, 45 kips) 50,7 kips (.6)( 3,894 kips) 50, 7 kips nt 5, 45 kips (from omputer output) nt 3,894 kips (from omputer output) e R.0 L ( 95 kips)( 3.5 ft)( in/ft) 0..0 in. e R.0 L ( 37 kips)( 3.5 ft)( in/ft) 0.47 in. Eqn C-6b 50,7 kips 50,7 kips

18 III-65.D+.0E+ L+ 0.S 95 kips (from previous alulations) D+ 0.7E 37 kips (from previous alulations) 0.0 in. (from omputer output) 0.47 in. (from omputer output) r nt + B lt 4 kips + (.04)(0 kips) 470 kips 553 kips ( W 58) r 470 kips For kips r nt + B lt 73 kips + (.04)(79 kips) 360 kips 368 kips ( W 58) r 360 kips For kips Eqn C-b anual Table 4- Note: Notie that the lower displaements of the braed frame produe muh lower values for B. Similar values ould be expeted for the other two methods of analysis.

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