Slenderness Effects for Concrete Columns in Sway Frame - Moment Magnification Method

Transcription

1 Slenderness Effets for Conrete Columns in Sway Frame - Moment Magnifiation Method

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3 Slender Conrete Column Design in Sway Frame Buildings Evaluate slenderness effet for olumns in a sway frame multistory reinfored onrete building by designing the first story exterior olumn. The lear height of the first story is 15 ft-6 in., and is 9 ft. for all of the other stories. Lateral load effets on the building are governed by wind fores. Compare the alulated results with the values presented in the Referene and with exat values from spcolumn engineering software program from StruturePoint. Figure 1 Slender Reinfored Conrete Column Cross-Setion Version: July

4 Contents 1. Fatored Axial Loads and Bending Moments Servie loads Load Combinations Fatored Loads Slenderness Effets and Sway or Nonsway Frame Designation Determine Slenderness Effets Moment Magnifiation at Ends of Compression Member Gravity Load Combination #2 (Gravity Loads Only) Lateral Load Combination #6 (Gravity Plus Wind Loads) Moment Magnifiation along Length of Compression Member Gravity Load Combination #2 (Gravity Loads Only) Load Combination 6 (Gravity Plus Wind Loads) Column Design , a, and strains in the reinforement Fores in the onrete and steel ϕp n and ϕm n Column Interation Diagram - spcolumn Software Summary and Comparison of Design Results Conlusions & Observations Version: July

5 Code Building Code Requirements for Strutural Conrete (ACI ) and Commentary (ACI 318R-14) Referene Reinfored Conrete Mehanis and Design, 7 th Edition, 2016, James Wight, Pearson, Example 12-3 Design Data f = 4,000 psi for olumns f y = 60,000 psi Slab thikness = 6 in. Exterior Columns = 18 in. x 18 in. Interior Columns = 18 in. x 18 in. Interior Beams = 18 in. x 30 in. x 30 ft Exterior Beams = 18 in. x 30 in. x 32 ft Floor superimposed dead load = 20 psf Floor live load = 80 psf Roof superimposed dead load = 25 psf Roof live load = 30 psf Wind loads omputed aording to ASCE 7-10 Total building loads in the first story from the referene: Table 1 Total building fatored loads ASCE 7-10 Referene No. Load Combination P u, kip D 10, D + 1.6L + 0.5L r 11, D + 0.5L L r 10, D + 1.6L r + 0.8W 9, D + 1.6L r - 0.8W 9, D + 0.5L + 0.5L r + 1.6W 10, D + 0.5L + 0.5L r - 1.6W 10, D + 1.6W 7, D - 1.6W 7,065 1

6 1. Fatored Axial Loads and Bending Moments 1.1. Servie loads Table 2 - Exterior olumn servie loads Load Case Axial Load, Bending Moment, ft-kip kip Top Bottom Dead, D Live, L Roof Live, L r Wind, W (N-S) Load Combinations Fatored Loads ASCE 7-10 (2.3.2) ASCE 7-10 Referene No. Load Combination Table 3 - Exterior olumn fatored loads Axial Load, kip Bending Moment, ft-kip Top Bottom M Top,ns ft-kip M Bottom,ns ft-kip D D + 1.6L + 0.5L r M Top,s ft-kip 3 1.2D + 0.5L L r M Bottom,s ft-kip 4 1.2D + 1.6L r + 0.8W D + 1.6L r - 0.8W D + 0.5L + 0.5L r + 1.6W D + 0.5L + 0.5L r - 1.6W D + 1.6W D - 1.6W

7 2. Slenderness Effets and Sway or Nonsway Frame Designation Columns and stories in strutures are onsidered as nonsway frames if the inrease in olumn end moments due to seond-order effets does not exeed 5% of the first-order end moments, or the stability index for the story (Q) does not exeed ACI ( ) P u is the total vertial load in the first story orresponding to the lateral loading ase for whih P u is greatest (without the wind loads, whih would ause ompression in some olumns and tension in others and thus would anel out). ACI ( and R ) V us is the fatored horizontal story shear in the first story orresponding to the wind loads, and Δ o is the first-order relative defletion between the top and bottom of the first story due to V us. ACI ( and R ) From Table 1, load ombination ( No. 2) provides the greatest value of P u. P 1.2 D 1.6 L 0.5 L 11, 400 kip ASCE 7-10 ( ) u r Sine there is no lateral load in this load ombination, the referene applied an arbitrary lateral load representing (V us) at the top of the first story and alulated the resulting story lateral defletion (Δ o). V 20 kip (given) us in. (given) o Pu o 11, Q V l us ACI (Eq ) Thus, the frame at the first story level is onsidered sway. 3

8 3. Determine Slenderness Effets Iolumn ,124 in. ACI (Table (a)) E ACI ( b) ' 57,000 f 57,000 4,000 3,605 ksi For the olumn below level 2: E I l olumn 3,605 6, in.kip 18 For the olumn above level 2: E I l olumn 3,605 6, in.kip 11.5 For beams framing into the olumns: E b I l b beam 3,605 14, in.kip 3212 Where: E b ACI ( b) ' 57, 000 f 57, , 605 ksi I bh beam ,175 in. ACI (Table (a)) EI l olumns A 1.97 EI 133 l beams ACI (Figure R6.2.5) 1.0 (Column onsidered fixed at the base) ACI (Figure R6.2.5) B Using Figure R6.2.5 from ACI k = 1.44 as shown in the figure below for the exterior olumn. 4

9 Figure 2 Effetive Length Fator (k) for Exterior Column (Sway Frame) kl u r Consider Slenderness ACI (6.2.5a) Where: I g r radius of gyration = ( a) or (b) ACI ( ) A g 4 I g 18 /12 r in. 2 A 18 g 4. Moment Magnifiation at Ends of Compression Member A detailed alulation for load ombinations 2 and 6 is shown below to illustrate the slender olumn moment magnifiation proedure. Table 4 summarizes the magnified moment omputations for the exterior olumns Gravity Load Combination #2 (Gravity Loads Only) M M M ACI ( b) 2 2ns s 2s Where: MTop _ s MBottom _ s M2_ s 0ft.kip 5

10 M M 2 2ns M nd M _2 Top, ns 59.8 ft.kip Top M nd M _2 Bottom, ns 63 ft.kip Bottom M max M, M M 63 ft.kip M M 63 ft.kip nd nd nd nd st st 2 _ 2 Top _ 2 Bottom _ 2 Bottom _ 2 2 _1 Bottom _1 M min M, M M 59.8 ft.kip M M 59.8 ft.kip nd nd nd nd st st 1_ 2 Top _ 2 Bottom _ 2 Top _ 2 1_1 Top _1 P u = 413.3kip 4.2. Lateral Load Combination #6 (Gravity Plus Wind Loads) M M M ACI ( b) 2 2ns s 2s Where: 1 (a) 1 Q 1 s moment magnifier (b) P u P () Seond-order elasti analysis ACI ( ) There are three options for alulating δ s. ACI ( (b)) will be used sine it does not require a detailed strutural analysis model results to proeed and is also used by the solver engine in spcolumn. P u is the summation of all the fatored vertial loads in the first story, and P is the summation of the ritial bukling load for all sway-resisting olumns in the first story. 2 u 2 EI eff P ACI ( ) kl Where: EI eff 0.4EI g (a) 1 ds 0.2E I E I (b) 1 ds EI () 1 ds g s se ACI ( ) There are three options for alulating the effetive flexural stiffness of slender onrete olumns (EI) eff. The seond equation provides aurate representation of the reinforement in the setion and will be used in this example and 6

11 is also used by the solver in spcolumn. Further omparison of the available options is provided in Effetive Flexural Stiffness for Critial Bukling Load of Conrete Columns tehnial note Iolumn 8,748 in. ACI (Table (a)) E ACI ( a) ' 57, 000 f 57, , 605 ksi β ds is the ratio of maximum fatored sustained shear within a story to the maximum fatored shear in that story assoiated with the same load ombination. The maximum fatored sustained shear in this example is equal to zero leading to β ds = 0. ACI ( ) For exterior olumns with one beam framing into them in the diretion of analysis (8 olumns): With 8-#6 reinforement equally distributed on all sides I se = in. 4 (Ref. uses approximate value of 150 in. 4 in lieu of exat value alulated by spcolumn). EI eff 0.2E I E I 1 g s se ds ACI ( (b)) 0.23, 6058, , EI kip-in. eff k = 1.44 (alulated previously). 2 6 P ,313 kip For exterior olumns with two beams framing into them in the diretion of analysis (8 olumns): EI l olumns A 0.95 EI l beams ACI (Figure R6.2.5) 1.0 (Column essentially fixed at base) ACI (Figure R6.2.5) B Using Figure R6.2.5 from ACI k = 1.31 as shown in the figure below for the exterior olumns with two beams framing into them in the diretions of analysis. 7

12 Figure 3 Effetive Length Fator (k) for Exterior Columns with Two Beams Framing into them in the Diretion of Analysis 2 6 P ,586 kip For interior olumns (8 olumns): EI l olumns A 0.95 EI l beams ACI (Figure R6.2.5) 1.0 (Column essentially fixed at base) ACI (Figure R6.2.5) B Using Figure R6.2.5 from ACI k = 1.31 as shown in the figure below for the interior olumns. 8

13 Figure 4 Effetive Length Fator (k) Calulations for Interior Columns With 8-#8 reinforement equally distributed on all sides I se = in. 4 EI eff 0.2E I E I 1 g s se ds ACI ( (b)) 0.23,605 6,124 29, kip-in EI eff 2 6 P ,977 kip P n P n P n P ,313 81,586 81,977 39, 005 kip P 10,100 kip (Table 1) P u 1 s Pu P ACI ( (b)) 1 s = , ,005 smtop, s ft.kip 9

14 M M M ACI ( ) _2,, ft.kip Top nd Top ns s Top s sm, ft.kip Bottom s M M M ACI ( ) _2,, ft.kip Bottom nd Bottom ns s Bottom s M max M, M M ft.kip M M ft.kip nd nd nd nd st st 2 _ 2 Top _ 2 Bottom _ 2 Top _ 2 2 _1 Top _1 M min M, M M ft.kip M M ft.kip nd nd nd nd st st 1_ 2 Top _ 2 Bottom _ 2 Bottom _ 2 1_1 Bottom _1 P u = kip A summary of the moment magnifiation fators and magnified moments for the exterior olumn for all load ombinations using both equation options ACI ( (a)) and ( (b)) to alulate δ s is provided in the table below for illustration and omparison purposes. Note: The designation of M 1 and M 2 is made based on the seond-order (magnified) moments and not based on the first-order (unmagnified) moments. Table 4 - Fatored Axial loads and Magnified Moments at the Ends of Exterior Column No. Load Combination Axial Load, Using ACI (a) Using ACI (b) kip δ s M 1, ft-kip M 2, ft-kip δ s M 1, ft-kip M 2, ft-kip 1 1.4D * * * D + 1.6L + 0.5L r D + 0.5L L r * * * D + 1.6L r + 0.8W 363 * * * D + 1.6L r - 0.8W * * * D + 0.5L + 0.5L r + 1.6W D + 0.5L + 0.5L r - 1.6W * * * D + 1.6W * * * D - 1.6W * * * * Not overed by the referene 10

15 5. Moment Magnifiation along Length of Compression Member In sway frames, seond-order effets shall be onsidered along the length of olumns. It shall be permitted to aount for these effets using ACI ( ) (Nonsway frame proedure), where C m is alulated using M 1 and M 2 from ACI ( ) as follows: ACI ( ) M M 2 2 ACI ( ) Where: M 2 = the seond-order fatored moment. Cm magnifiation fator 1.0 Pu P ACI ( ) 2 u 2 EI eff P ACI ( ) kl Where: EI eff 0.4EI g (a) 1 dns 0.2E I E I (b) 1 dns EI () 1 dns g s se ACI ( ) There are three options for alulating the effetive flexural stiffness of slender onrete olumns (EI) eff. The seond equation provides aurate representation of the reinforement in the setion and will be used in this example and is also used by the solver in spcolumn. Further omparison of the available options is provided in Effetive Flexural Stiffness for Critial Bukling Load of Conrete Columns tehnial note Gravity Load Combination #2 (Gravity Loads Only) Iolumn 6,124 in. ACI (Table (a)) E ACI ( a) ' 57, 000 f 57, , 605 ksi β dns is the ratio of maximum fatored sustained axial load to maximum fatored axial load assoiated with the same load ombination. ACI ( ) Pu, sustained kip 11

16 P kip u dns Pu, sustained dns 0.82 P u EI l olumns A 1.97 (Calulated previously) EI 133 l B 1.0 beams (Column essentially fixed at base) ACI (Figure R6.2.5) ACI (Figure R6.2.5) Using Figure R6.2.5(a) from ACI k = 0.81 as shown in the figure below for the exterior olumn. Figure 5 Effetive Length Fator (k) Calulations for Exterior Column (Nonsway) With 8-#6 reinforement equally distributed on all sides I se = in. 4 (Ref. uses approximate value of 150 in. 4 in lieu of exat value alulated by spcolumn). EI eff 0.2E I E I 1 g s se dns ACI ( (b)) 0.23, 605 6,124 29, EI kip-in. eff 12

17 P ,277 kip P kip ASCE 7-10 ( ) u C M 1 m ACI ( a) M 2 M2 M nd 2_ ft.kip (as onluded from setion 4) ACI ( ) M1 M nd 1_ ft.kip (as onluded from setion 4) ACI ( ) Sine the olumn is bent in double urvature, M 1/M 2 is positive. ACI ( ) C m Cm Pu P 1.0 ACI ( ) = ,277 Mmin P h ACI ( ) u Where P u = kip, and h = the setion dimension in the diretion being onsidered = 18 in. M min ft.kip 12 M 59.8 ft.kip M 39.3 ft.kip M 59.8 ft.kip ACI ( ) 1 min 1 M M ACI ( ) 1 1 M ft.kip M ft.kip M 39.3 ft.kip M ft.kip ACI ( ) 2 2,min 2 M M ACI ( ) 2 2 M ft.kip M 1 and M 2 will be onsidered separately to ensure proper omparison of resulting magnified moments against negative and positive moment apaities of unsymmetrial setions as an be seen in the following figure. 13

18 Figure 6 Column Interation Diagram for Unsymmetrial Setion 5.2. Load Combination #6 (Gravity Plus Wind Loads) Iolumn 6,124 in. ACI (Table (a)) E ACI ( a) ' 57, 000 f 57, , 605 ksi β dns is the ratio of maximum fatored sustained axial load to maximum fatored axial load assoiated with the same load ombination. ACI ( ) Pu, sustained kip P kip u dns Pu, sustained dns 0.89 P u EI l olumns A 1.97 (Calulated previously) EI 133 l beams ACI (Figure R6.2.5) 1.0 (Column essentially fixed at base) ACI (Figure R6.2.5) B Using Figure R6.2.5(a) from ACI k = 0.81 With 8-#6 reinforement equally distributed on all sides I se = in. 4 (Ref. uses approximate value of 150 in. 4 in lieu of exat value alulated by spcolumn). 14

19 EI eff 0.2E I E I 1 g s se dns ACI ( (b)) 0.23, 605 6,124 29, EI kip-in. eff P ,192 kip P kip ASCE 7-10 ( ) u C M 1 m ACI ( a) M 2 M2 M nd 2_ ft.kip (as onluded from setion 4) ACI ( ) M1 M nd 1_ ft.kip (as onluded from setion 4) ACI ( ) Sine the olumn is bent in double urvature, M 1/M 2 is positive. ACI ( ) C m Cm Pu P 1.0 ACI ( ) = ,192 Mmin P h ACI ( ) u Where P u = kip, and h = the setion dimension in the diretion being onsidered = 18 in. M min ft.kip 12 M ft.kip M 36.1 ft.kip M ft.kip ACI ( ) 1 min 1 M M ACI ( ) 1 1 M ft.kip M ft.kip M 36.1 ft.kip M ft.kip ACI ( ) 2 2,min 2 M M ACI ( )

20 M ft.kip M 1 and M 2 are onsidered separately to ensure proper omparison of resulting magnified moments against negative and positive moment apaities of unsymmetrial setions. A summary of the moment magnifiation fators and magnified moments for the exterior olumn for all load ombinations using both equation options ACI ( (a)) and ( (b)) to alulate δ s is provided in the table below for illustration and omparison purposes. Table 5 - Fatored Axial loads and Magnified Moments along Exterior Column Length No. Load Combination Axial Load, kip Using ACI (a) Using ACI (b) δ M 1,ft-kip M 2, ft-kip δ M 1, ft-kip M 2, ft-kip 1 1.4D * * * D + 1.6L + 0.5L r D + 0.5L L r * * * D + 1.6L r + 0.8W 363 * * * D + 1.6L r - 0.8W * * * D + 0.5L + 0.5L r + 1.6W D + 0.5L + 0.5L r - 1.6W * * * D + 1.6W * * * D - 1.6W * * * * Not overed by the referene For olumn design ACI 318 requires the seond-order moment to first-order moment ratios should not exeed If this value is exeeded, the olumn design needs to be revised. ACI (6.2.6) 16

21 No. Table 6 - Seond-Order Moment to First-Order Moment Ratios Load Combination Using ACI (a) Using ACI (b) M 1/M 1(1st) M 2/M 2(1st) M 1/M 1(1st) M 2/M 2(1st) 1 1.4D ** ** 1.00 * 1.00 * 2 1.2D + 1.6L + 0.5L r 1.00 * 1.00 * 1.00 * 1.00 * 3 1.2D + 0.5L L r ** ** 1.00 * 1.00 * 4 1.2D + 1.6L r + 0.8W ** ** D + 1.6L r - 0.8W ** ** 1.00 * 1.00 * 6 1.2D + 0.5L + 0.5L r + 1.6W D + 0.5L + 0.5L r - 1.6W ** ** 1.40 < < D + 1.6W ** ** D - 1.6W ** ** 1.40 < < 1.54 * Cutoff value of M min is applied to M 1(1st) and M 2(1st) in order to avoid unduly large ratios in ases where M 1(1st) and M 2(1st) moments are smaller than M min. ** Not overed by the referene 17

22 6. Column Design Based on the fatored axial loads and magnified moments onsidering slenderness effets, the apaity of the assumed olumn setion (18 in. x 18 in. with 8-#6 bars distributed all sides equal) will be heked and onfirmed to finalize the design. A olumn interation diagram will be generated using strain ompatibility analysis, the detailed proedure to develop olumn interation diagram an be found in Interation Diagram Tied Reinfored Conrete Column example. The axial ompression apaity ϕp n for all load ombinations will be set equals to P u, then the moment apaity ϕm n assoiated to ϕp n will be ompared with the magnified applied moment M u. The design hek for load ombination #6 is shown below for illustration. The rest of the heks for the other load ombinations are shown in the following Table. Figure 7 Strains, Fores, and Moment Arms (Load Combination 6) The following proedure is used to determine the nominal moment apaity by setting the design axial load apaity, ϕp n, equal to the applied axial load, P u and iterating on the loation of the neutral axis. 6.1., a, and strains in the reinforement Try in. Where is the distane from the fiber of maximum ompressive strain to the neutral axis. ACI ( ) a in. ACI ( ) Where: ' f ACI (Table ) ACI ( ) u 18

23 f y 60 y E 29, 000 s s ( d1 ) ( ) (Tension) < y tension reinforement has not yielded 0.65 ACI (Table ) ' s 1 ( d2) ( ) (Compression) > y ' h s2 ( ) (Tension) < y Fores in the onrete and steel C f ab ACI ( ) ' , kip f E , 000, , 620 psi s s s T f A 39, kip s y s1 Sine > ompression reinforement has yielded ' s1 y f 1 f 60,000 psi ' s y Sine < ompression reinforement has not yielded ' s2 y ' ' fs2 s2 Es , 000, , 479 psi The area of the reinforement in this layer has been inluded in the area (ab) used to ompute C. As a result, it is neessary to subtrat 0.85f from f s before omputing C s: ' ' ' f f A C , , kip s1 s1 s1 ' ' ' f f A C , , kip s2 s2 s ϕp n and ϕm n Pn C Cs 1Cs 2 Ts kip P kip = P n The assumed value of = in. is orret. u 19

24 h a h h h h M n C Cs 1 d2 Cs2 Ts d M n kip.ft M kip.ft M M kip.ft n u 2 Table 7 Exterior Column Axial and Moment Capaities No. P u, kip M u = M 2(2nd), ft-kip, in. ε t = ε s φ φp n, kip φm n, kip.ft Sine ϕm n > M u for all ϕp n = P u, use 18 x 18 in. olumn with 8-#6 bars. 20

25 7. Column Interation Diagram - spcolumn Software spcolumn program performs the analysis of the reinfored onrete setion onforming to the provisions of the Strength Design Method and Unified Design Provisions with all onditions of strength satisfying the appliable onditions of equilibrium and strain ompatibility and inludes slenderness effets using moment magnifiation method for sway and nonsway frames. For this olumn setion, we ran in investigation mode with ontrol points using the In lieu of using program shortuts, spsetion (Figure 8) was used to plae the reinforement and define the over to illustrate handling of irregular shapes and unusual bar arrangement. Figure 8 spcolumn Model Editor (spsetion) 21

26 Figure 9 spcolumn Model Input Wizard Windows 22

27 Figure 10 Column Setion Interation Diagram about X-Axis Design Chek for Load Combination 6 (spcolumn) 23

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34 8. Summary and Comparison of Design Results Analysis and design results from the hand alulations above are ompared for the one load ombination used in the referene (Example 12-3) and exat values obtained from spcolumn model. Table 8 Parameters for Moment Magnifiation along the Column Length Q k β dns C m I se P, kip δ M 2(min), ft-kip M 2(2nd), ft-kip Hand * ,277 1 > Referene ,660 1 > spcolumn ,259 1 > * From nomographs (ACI 318 harts) Conservatively estimated not using exat formulae without impat on the final results in this speial ase Exat formulated answer In this table, a detailed omparison for all onsidered load ombinations are presented for omparison. Table 9 - Fatored Axial loads and Magnified Moments at Column Ends No. P u, kip δ s M 1(2nd), ft-kip M 2(2nd), ft-kip Hand spcolumn Hand spcolumn Hand spcolumn Hand spcolumn N/A N/A N/A N/A N/A N/A

35 Table 10 - Fatored Axial loads and Magnified Moments along Column Length No. δ M 1, ft-kip M 2, ft-kip M 1/M 1(1st) M 2/M 2(1st) Hand spcolumn Hand spcolumn Hand spcolumn Hand spcolumn Hand spcolumn Table 11 - Design Parameters Comparison No., in. ε t = ε s φ φp n, kip φm n, kip.ft Hand spcolumn Hand spcolumn Hand spcolumn Hand spcolumn Hand spcolumn All the results of the hand alulations illustrated above are in preise agreement with the automated exat results obtained from the spcolumn program. 31

36 9. Conlusions & Observations The analysis of the reinfored onrete setion performed by spcolumn onforms to the provisions of the Strength Design Method and Unified Design Provisions with all onditions of strength satisfying the appliable onditions of equilibrium and strain ompatibility and inludes slenderness effets using moment magnifiation method for sway and nonsway frames. ACI 318 provides multiple options for alulating values of k, (EI) eff, δ s, and δ leading to variability in the determination of the adequay of a olumn setion. Engineers must exerise judgment in seleting suitable options to math their design ondition as is the ase in the referene where the author onservatively made assumptions to simplify and speed the alulation effort. The spcolumn program utilizes the exat methods whenever possible and allows user to override the alulated values with diret input based on their engineering judgment wherever it is permissible. In load ombinations 7 and 9, M u inluding seond-order effets exeeds 1.4 M u due to first-order effets (see Table 6). This indiates that in this building, the weight of the struture is high in proportion to its lateral stiffness leading to exessive PΔ effet (seondary moments are more than 25 perent of the primary moments). The PΔ effets will eventually introdue singularities into the solution to the equations of equilibrium, indiating physial strutural instability. It was onluded in the literature that the probability of stability failure inreases rapidly when the stability index Q exeeds 0.2, whih is equivalent to a seondary-to-primary moment ratio of The maximum value of the stability oeffiient θ (aording to ASCE/SEI 7) whih is lose to stability oeffiient Q (aording to ACI 318) is The value 0.25 is equivalent to a seondary-to-primary moment ratio of Hene, the upper limit of 1.4 on the seondary-to-primary moment ratio was seleted by the ACI 318. As an be seen in Table 6 of this example, exploring the impat of other ode permissible equation options provides the engineer added flexibility in deision making regarding design. For load ombinations 7 & 9 resolving the stability onern may be viable through a frame analysis providing values for V us and Δ o to alulate magnifiation fator δ s and may allow the proposed design to be aeptable by inspetion of load ombination #6. Creating a omplete model with detailed lateral loads and load ombinations to aount for seond order effets may not be warranted for all ases of slender olumn design nor is it disadvantageous to have a higher margin of safety when it omes to olumn slenderness and frame stability onsiderations. 32