Slenderness Effets for Conrete Columns in Sway Frame - Moment Magnifiation Method
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 13 ft-4 in., and is 10 ft-4in. 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 Reinfored Conrete Column Cross-Setion Version: July-19-2017
Contents 1. Fatored Axial Loads and Bending Moments... 2 1.1. Servie loads... 2 1.2. Load Combinations Fatored Loads... 2 2. Slenderness Effets and Sway or Nonsway Frame Designation... 3 3. Determine Slenderness Effets... 4 4. Moment Magnifiation at Ends of Compression Member... 5 5. Moment Magnifiation along Length of Compression Member... 11 6. Column Design... 16 6.1., a, and strains in the reinforement... 16 6.2. Fores in the onrete and steel... 17 6.3. ϕp n and ϕm n... 17 7. Column Interation Diagram - spcolumn Software... 19 8. Summary and Comparison of Design Results... 28 9. Conlusions & Observations... 30 Version: July-19-2017
Code Building Code Requirements for Strutural Conrete (ACI 318-14) and Commentary (ACI 318R-14) Referene Notes on ACI 318-11 Building Code Requirements for Strutural Conrete, Twelfth Edition, 2013 Portland Cement Assoiation, Example 11-2 Design Data f = 6,000 psi for olumns in the bottom two stories = 4,000 psi elsewhere f y = 60,000 psi Slab thikness = 7 in. Exterior Columns = 22 in. x 22 in. Interior Columns = 24 in. x 24 in. Beams = 24 in. x 20 in. x 24 ft Superimposed dead load = 30 psf Roof live load = 30 psf Floor live load = 50 psf Wind loads omputed aording to ASCE 7-10 Total building loads in the first story from strutural analysis: D = 17,895 kip L = 1,991 kip L r = 270 kip W = 0 kip, wind loads in the story ause ompression in some olumns and tension in others and thus would anel out. 1
1. Fatored Axial Loads and Bending Moments 1.1. Servie loads Table 1 - Exterior olumn servie loads Load Case Axial Load, Bending Moment, ft-kip kip Top Bottom Dead, D 622.4 34.8 17.6 Live, L 73.9 15.4 7.7 Roof Live, L r 8.6 0.0 0.0 Wind, W (N-S) -48.3 17.1 138.0 Wind, W (S-N) 48.3-17.1-138.0 1.2. Load Combinations Fatored Loads ASCE 7-10 (2.3.2) ASCE 7-10 Referene No. Load Combination Table 2 - Exterior olumn fatored loads Axial Load, kip Bending Moment, ft-kip Top Bottom M Top,ns ft-kip M Bottom,ns ft-kip 2.3.2-1 1 1.4D 871.4 48.7 24.6 48.7 24.6 --- --- 2.3.2-2 2 1.2D + 1.6L + 0.5L r 869.4 66.4 33.4 66.4 33.4 --- --- 2.3.2-3 2.3.2-4 2.3.2-6 M Top,s ft-kip 3 1.2D + 0.5L + 1.6 L r 797.6 49.5 25.0 49.5 25.0 --- --- M Bottom,s ft-kip 4 1.2D + 1.6L r + 0.8W 722.0 55.4 131.5 41.8 21.1 13.7 110.4 5 1.2D + 1.6L r - 0.8W 799.3 28.1-89.3 41.8 21.1-13.7-110.4 6 1.2D + 0.5L + 0.5L r + 1.6W 710.9 76.8 245.8 49.5 25.0 27.4 220.8 7 1.2D + 0.5L + 0.5L r - 1.6W 865.4 22.1-195.8 49.5 25.0-27.4-220.8 8 0.9D + 1.6W 482.9 58.7 236.6 31.3 15.8 27.4 220.8 9 0.9D - 1.6W 637.4 4.0-205.0 31.3 15.8-27.4-220.8 2
2. Slenderness Effets and Sway or Nonsway Frame Designation Columns and stories in strutures are onsidered as non-sway 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 0.05. ACI 318-14 (6.6.4.3) 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 318-14 (6.6.4.4.1 and R6.6.4.3) 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 u. ACI 318-14 (6.6.4.4.1 and R6.6.4.3) From Table 2, load ombinations (2.3.2-4 No. 5 and 6) provide the greatest value of P u. P 1.2 D 0.5 L 0.5 L 1.217,895 0.51,991 0.5 270 22, 605 kip ASCE 7-10 (2.3.2-4) V us u r 1.6V 1.6 302.6 484.2 kip ASCE 7-10 (2.3.2-6) s 1.6 1.6 0.28 0 0.45 in. o Pu o 22, 605 0.45 Q 0.12 0.05 V l 484.2 1512 20 / 2 us ACI 318-14 (Eq. 6.6.4.4.1) Thus, the frame at the first story level is onsidered sway. 3
3. Determine Slenderness Effets 22 12 12 4 4 4 Iolumn 0.7 0.7 13, 665 in. ACI 318-14 (Table 6.6.3.1.1(a)) E ACI 318-14 (19.2.2.1.b) ' 57, 000 f 57, 000 6000 4, 415 ksi For the olumn below level 2: E I l olumn 4, 41513,665 1512 20 / 2 3 355 10 in.kip For the olumn above level 2: E I l olumn 4, 41513,665 1212 3 419 10 in.kip For beams framing into the olumns: E b I l b beam 3,605 5,600 2412 3 70 10 in.kip Where: E b ACI 318-14 (19.2.2.1.b) ' 57, 000 f 57, 000 4000 3, 605 ksi I bh 2420 12 12 3 3 4 beam 0.35 0.35 5, 600 in. ACI 318-14 (Table 6.6.3.1.1(a)) EI l olumns 355 419 A 11 EI 70 l beams ACI 318-14 (Figure R6.2.5) 1.0 (Column essentially fixed at base) ACI 318-14 (Figure R6.2.5) B Using Figure R6.2.5 from ACI 318-14 k = 1.9 as shown in the figure below for the exterior olumns with one beam framing into them in the diretions of analysis. 4
Figure 2 Effetive Length Fator (k) Calulations for Exterior Columns with One Beam Framing into them in the Diretion of Analysis (Sway Frame) kl u r 1.913.333 47.87 22 Consider Slenderness ACI 318-14 (6.2.5a) 6.6 Where: I g r radius of gyration = ( a) or (b) 0.3 1 ACI 318-14 (6.2.5.1) A g 2 2 I g 1 22 r 6.35 in. A 12 12 g 4. Moment Magnifiation at Ends of Compression Member A detailed alulation for load ombination 4 (gravity plus wind) is shown below to illustrate the proedure. Table 3 summarizes the magnified moment omputations for the exterior olumns. M M M ACI 318-14 (6.6.4.6.1b) 2 2ns s 2s Where: 5
1 (a) 1 Q 1 s moment magnifier (b) P u 1 0.75 P () Seond-order elasti analysis ACI 318-14 (6.6.4.6.2) ACI 318-14 (6.6.4.6.2(b)) will be used for omparison purposes with results obtained from spcolumn model. However, (a) and () an also be used to alulate the moment magnifier. 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 EI 2 eff ACI 318-14 (6.6.4.4.2) P Where: kl u EI eff 0.4EI g (a) 1 ds 0.2E I E I (b) 1 ds EI () 1 ds g s se ACI 318-14 (6.6.4.4.4) 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. 22 12 12 4 4 4 Iolumn 19,521in. ACI 318-14 (Table 6.6.3.1.1(a)) E ACI 318-14 (19.2.2.1.a) ' 57, 000 f 57, 000 6000 4, 415 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 318-14 (6.6.3.1.1) For exterior olumns with one beam framing into them in the diretion of analysis (12 olumns): With 8-#8 reinforement equally distributed on all sides and 22 in. x 22 in. olumn setion I se = 352.6 in. 4. EI eff 0.2E I E I 1 g s se ds ACI 318-14 (6.6.4.4.4(b)) 6
0.2 4, 41519, 521 29, 000352.6 1 0 6 2 EI 27.510 kip-in. eff k = 1.9 (alulated previously). 27.510 2 6 P 1 2 1.913.333 2,933 kip For exterior olumns with two beams framing into them in the diretion of analysis (4 olumns): EI l olumns 355 419 A 5.5 EI 70 70 l B 1.0 beams (Column essentially fixed at base) ACI 318-14 (Figure R6.2.5) ACI 318-14 (Figure R6.2.5) Using Figure R6.2.5 from ACI 318-14 k = 1.71 as shown in the figure below for the exterior olumns with two beams framing into them in the diretions of analysis. Figure 3 Effetive Length Fator (k) Calulations for Exterior Columns with Two Beams Framing into them in the Diretion of Analysis 2 6 P 2 2 27.510 1.7113.333 12 3,621 kip For interior olumns (8 olumns): 7
24 12 12 4 4 4 Iolumn 0.7 0.7 19, 354 in. ACI 318-14 (Table 6.6.3.1.1(a)) E ACI 318-14 (19.2.2.1.a) ' 57, 000 f 57, 000 6000 4, 415 ksi For the olumn below level 2: E I l olumn 4, 41519, 354 15 20 / 2 3 503 10 in.kip For the olumn above level 2: E I l olumn 4, 41519, 354 12 3 593 10 in.kip For beams framing into the olumns: E b I l b beam 3,605 5,600 24 3 70 10 in.kip Where: E b ACI 318-14 (19.2.2.1.a) ' 57, 000 f 57, 000 4000 3, 605 ksi I bh 2420 12 12 4 4 4 beam 0.35 0.35 5, 600 in. ACI 318-14 (Table 6.6.3.1.1(a)) EI l olumns 503 593 A 7.8 EI 70 70 l beams ACI 318-14 (Figure R6.2.5) 1.0 (Column essentially fixed at base) ACI 318-14 (Figure R6.2.5) B Using Figure R6.2.5 from ACI 318-14 k = 1.81 as shown in the figure below for the interior olumns. 8
Figure 4 Effetive Length Fator (k) Calulations for Interior Columns With 8-#8 reinforement equally distributed on all sides and 24 in. x 24 in. olumn setion I se = 439.1 in. 4. EI eff 0.2E I E I 1 g s se ds ACI 318-14 (6.6.4.4.4(b)) 0.2 4,415 27,648 29,000 439.1 37.1 10 kip-in. 1 0 6 2 EI eff 2 6 P 3 2 37.110 1.8113.333 12 4,372 kip P n P n P n P 1 1 2 2 3 3 12 2,933 43, 621 8 4,372 84, 652 kip P For load ombination 4: P 1.2 D 1.6 L 1.217,895 1.6 270 21,906 kip ASCE 7-10 (2.3.2-3) u r 1 s Pu 1 0.75 P ACI 318-14 (6.6.4.6.2(b)) 9
1 s =1.53 21, 906 1 0.75 84,652 sm, 1.5313.7 20.9 ft.kip Top s M M M ACI 318-14 (6.6.4.6.1) _2,, 41.8 20.9 62.7 ft.kip Top nd Top ns s Top s sm, 1.53110.4 168.6 ft.kip Bottom s M M M ACI 318-14 (6.6.4.6.1), _2, 21.1 168.6 189.7 ft.kip Bottom nd Bottom ns s Bottom s M max M, M M 189.7 ft.kip M M 131.5 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 62.7 ft.kip M M 55.4 ft.kip nd nd nd nd st st 1_ 2 Top _ 2 Bottom _ 2 Top _ 2 1_1 Top _1 P u = 722.0 kip A summary of the moment magnifiation fators and magnified moments for the exterior olumn for all load ombinations using both equation options ACI 318-14 (6.6.4.4.4(a)) and (6.6.4.4.4(b)) to alulate (EI) eff 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 3 - Fatored Axial loads and Magnified Moments for Exterior Column No. Load Combination Axial Load, Using ACI 6.6.4.4.4(a) Using ACI 6.6.4.4.4(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 871.4 --- 24.6 48.7 --- 24.6 48.7 2 1.2D + 1.6L + 0.5L r 869.4 --- 33.4 66.4 --- 33.4 66.4 3 1.2D + 0.5L + 1.6 L r 797.6 --- 25.0 49.5 --- 25.0 49.5 4 1.2D + 1.6L r + 0.8W 722.0 1.37 60.6 172.3 1.53 62.7 189.7 5 1.2D + 1.6L r - 0.8W 799.3 1.37 23.0-130.1 1.53 20.9-147.5 6 1.2D + 0.5L + 0.5L r + 1.6W 710.9 1.39 87.5 330.9 1.55 92.0 367.9 7 1.2D + 0.5L + 0.5L r - 1.6W 865.4 1.39 11.5-280.9 1.55 7.0-317.9 8 0.9D + 1.6W 482.9 1.25 65.5 291.2 1.34 68.0 311.6 9 0.9D - 1.6W 637.4 1.25-2.9-259.6 1.34-5.4-280.0 10
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 318-14 (6.6.4.5) (Nonsway frame proedure), where C m is alulated using M 1 and M 2 from ACI 318-14 (6.6.4.6.1) as follows: ACI 318-14 (6.6.4.6.4) M M 2 2 ACI 318-14 (6.6.4.5.1) Where: M 2 = the seond-order fatored moment. Cm magnifiation fator 1.0 Pu 1 0.75 P ACI 318-14 (6.6.4.5.2) 2 EI 2 eff ACI 318-14 (6.6.4.4.2) P Where: kl u EI eff 0.4EI g (a) 1 dns 0.2E I E I (b) 1 dns EI () 1 dns g s se ACI 318-14 (6.6.4.4.4) 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. 22 12 12 4 4 4 Iolumn 19,521in. ACI 318-14 (Table 6.6.3.1.1(a)) E ACI 318-14 (19.2.2.1.a) ' 57, 000 f 57, 000 6000 4, 415 ksi β dns is the ratio of maximum fatored sustained axial load to maximum fatored axial load assoiated with the same load ombination. ACI 318-14 (6.6.4.4.4) For load ombination 4: Pu, sustained 1.2 622.4 746.9 kip 11
P 1.2 622.4 1.68.6 0.848.3 722 kip u dns Pu, sustained 746.9 1.03 1.00 dns 1.0 P 722 u EI l olumns 355 419 A 11 (Calulated previously) EI 70 l B 1.0 beams (Column essentially fixed at base) ACI 318-14 (Figure R6.2.5) ACI 318-14 (Figure R6.2.5) Using Figure R6.2.5(a) from ACI 318-14 k = 0.86 as shown in the figure below for the exterior olumn. Figure 5 Effetive Length Fator (k) Calulations for Exterior Column (Nonsway) With 8-#8 reinforement equally distributed on all sides and 22 in. x 22 in. olumn setion I se = 352.6 in. 4. EI eff 0.2E I E I 1 g s se dns ACI 318-14 (6.6.4.4.4(b)) 0.2 4,415 19,521 29,000 352.6 13.7 10 kip-in. 11 6 2 EI eff 12
P 13.7 10 2 6 0.8613.333 12 2 7,158 kip For load ombination 4: P 1.2 622.4 1.68.6 0.848.3 722 kip ASCE 7-10 (2.3.2-3) u C M 1 m 0.6 0.4 ACI 318-14 (6.6.4.5.3a) M 2 M2 M nd 2_ 2 189.7 ft.kip (as onluded from setion 4) ACI 318-14 (6.6.4.6.4) M1 M nd 1_ 2 62.7 ft.kip (as onluded from setion 4) ACI 318-14 (6.6.4.6.4) Sine the olumn is bent in double urvature, M 1/M 2 is positive. ACI 318-14 (6.6.4.5.3) C m 62.7 0.6 0.4 0.468 189.7 Cm Pu 1 0.75 P 1.0 ACI 318-14 (6.6.4.5.2) 0.468 =0.54 1.00 1.00 722 1 0.75 7,158 Mmin P 0.6 0.03h ACI 318-14 (6.6.4.5.4) u Where P u = 722 kip, and h = the setion dimension in the diretion being onsidered = 22 in. M min 0.6 0.0322 722 75.81 ft.kip 12 M 62.70 ft.kip M 75.81 ft.kip M 75.81 ft.kip ACI 318-14 (6.6.4.5.4) 1 min 1 M M ACI 318-14 (6.6.4.5.1) 1 1 M1 1.00 75.81 75.81 ft.kip M 189.7 ft.kip M 75.81ft.kip M 189.7 ft.kip ACI 318-14 (6.6.4.5.4) 2 2,min 2 M M ACI 318-14 (6.6.4.5.1) 2 2 M2 1.00 189.7 189.7 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
Figure 6 Column Interation Diagram for Unsymmetrial Setion A summary of the moment magnifiation fators and magnified moments for the exterior olumn for all load ombinations using both equation options ACI 318-14 (6.6.4.4.4(a)) and (6.6.4.4.4(b)) to alulate (EI) eff is provided in the table below for illustration and omparison purposes. No. Table 4 - Fatored Axial loads and Magnified Moments along Exterior Column Length Using ACI 6.6.4.4.4(a) Using ACI 6.6.4.4.4(b) Axial Load, Load Combination kip M δ 1, M 2, M δ 1, M 2, ft-kip ft-kip ft-kip ft-kip 1 1.4D 871.4 1.00 91.5 91.5 1.00 91.5 91.5 2 1.2D + 1.6L + 0.5L r 869.4 1.00 91.3 91.3 1.00 91.3 91.3 3 1.2D + 0.5L + 1.6 L r 797.6 1.00 83.7 83.7 1.00 83.7 83.7 4 1.2D + 1.6L r + 0.8W 722.0 1.00 75.8 172.3 1.00 75.8 189.7 5 1.2D + 1.6L r - 0.8W 799.3 1.00 83.9-130.1 1.00 83.9-147.5 6 1.2D + 0.5L + 0.5L r + 1.6W 710.9 1.00 87.5 330.9 1.00 92.0 367.9 7 1.2D + 0.5L + 0.5L r - 1.6W 865.4 1.00 90.9-280.9 1.00 90.9-317.9 8 0.9D + 1.6W 482.9 1.00 65.5 291.2 1.00 68.0 311.6 9 0.9D - 1.6W 637.4 1.00 66.9-259.6 1.00 66.9-280.0 For olumn design ACI 318 requires the seond-order moment to first-order moment ratios should not exeed 1.40. If this value is exeeded, the olumn design needs to be revised. ACI 318-14 (6.2.6) 14
No. Table 5 - Seond-Order Moment to First-Order Moment Ratios Using ACI 6.6.4.4.4(a) Using ACI 6.6.4.4.4(b) Load Combination 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 * 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 + 1.6 L r 1.00 * 1.00 * 1.00 * 1.00 * 4 1.2D + 1.6L r + 0.8W 1.00 * 1.31 1.00 * 1.40 < 1.44 5 1.2D + 1.6L r - 0.8W 1.00 * 1.40 < 1.46 1.00 * 1.40 < 1.65 6 1.2D + 0.5L + 0.5L r + 1.6W 1.14 1.35 1.20 1.40 < 1.50 7 1.2D + 0.5L + 0.5L r - 1.6W 1.00 * 1.40 < 1.43 1.00 * 1.40 < 1.62 8 0.9D + 1.6W 1.12 1.23 1.16 1.32 9 0.9D - 1.6W 1.00 * 1.27 1.00 * 1.37 * Cutoff value of Mmin is applied to M1(1st) and M2(1st) in order to avoid unduly large ratios in ases where M1(1st) and M2(1st) moments are smaller than Mmin. 15
6. Column Design Based on the fatored axial loads and magnified moments onsidering slenderness effets, the apaity of the assumed olumn setion (22 in. x 22 in. with 8-#8 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 #4 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 4) 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 12.75 in. Where is the distane from the fiber of maximum ompressive strain to the neutral axis. ACI 318-14 (22.2.2.4.2) a 1 0.75 12.75 9.563 in. ACI 318-14 (22.2.2.4.1) Where: ' f 0.05 4000 0.05 60004000 1 0.85 0.85 0.75 ACI 318-14 (Table 22.2.2.4.3) 1000 1000 0.003 ACI 318-14 (22.2.2.1) u 16
f y 60 y 0.00207 E 29, 000 s 0.003 0.003 s ( d1 ) (19.625 12.75) 0.00162 (Tension) < y 12.75 tension reinforement has not yielded 0.65 ACI 318-14 (Table 21.2.2) ' 0.003 0.003 s 1 ( d2) (12.75 2.375) 0.00244 (Compression) > y 12.75 ' h 0.003 0.003 s2 (12.75 11) 0.00041(Compression) < y 2 12.75 6.2. Fores in the onrete and steel C f ab ACI 318-14 (22.2.2.4.1) ' 0.85 0.85 6, 000 9.563 22 1073 kip f E 0.00162 29, 000, 000 46,912 psi s s s T f A 46,912 30.79 111.2 kip s y s1 Sine > ompression reinforement has yielded ' s1 y ' fs 1 fy 60,000 psi Sine < ompression reinforement has not yielded ' s2 y ' ' fs2 s2 Es 0.00041 29, 000, 000 11,941 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 0.85 60, 000 0.85 6, 000 3 0.79 130.1 kip s1 s1 s1 ' ' ' f f A C 0.85 11,941 0.85 6, 000 2 0.79 18.9 kip s2 s2 s2 6.3. ϕp n and ϕm n Pn C Cs 1Cs 2 Ts 1, 073 130.1 18.9 111.2 1,111kip P 0.651,111 722 kip = P n The assumption that = 12.75 in. is orret u 17
h a h h h h M n C Cs 1 d2 Cs2 Ts d1 2 2 2 2 2 2 M n 22 9.563 22 22 22 22 1,073 130.1 2.375 18.9 111.2 19.625 729 kip.ft 2 2 2 2 2 2 M 0.65 729 474kip.ft M M 189.7 kip.ft n u 2 Table 6 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 1 871.4 91.5 14.85 0.00096 0.65 871.4 459.4 2 869.4 91.3 14.85 0.00097 0.65 869.4 459.7 3 797.6 83.7 13.75 0.00128 0.65 797.6 468.2 4 722.0 189.7 12.75 0.00162 0.65 722.0 474.1 5 799.3-147.5 13.78 0.00127 0.65 799.3 468.0 6 710.9 367.9 12.61 0.00167 0.65 710.9 474.8 7 865.4-317.9 14.76 0.00099 0.65 865.4 460.2 8 482.9 311.6 7.36 0.005 0.9 482.9 557.2 9 637.4-280.0 11.68 0.00204 0.65 637.4 478.8 Therefore, sine ϕm n > M u for all ϕp n = P u, use 22 x 22 in. olumn with 8-#8 bars. 18
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 318-14. 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) 19
Figure 9 spcolumn Model Input Wizard Windows 20
Figure 10 Column Setion Interation Diagram about the X-Axis Design Chek for Load Combination 4 (spcolumn) 21
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8. Summary and Comparison of Design Results No. Table 7 - Fatored Axial loads and Magnified Moments at Column Ends Comparison P u, kip δ s M 1(2nd), ft-kip M 2(2nd), ft-kip Hand Referene spcolumn Hand Referene spcolumn Hand Referene spcolumn Hand Referene * spcolumn 1 871.4 871.4 871.4 N/A N/A N/A 24.6 24.6 24.6 48.7 48.7 48.7 2 869.4 869.4 869.4 N/A N/A N/A 33.4 33.4 33.4 66.4 66.4 66.4 3 797.6 797.6 797.6 N/A N/A N/A 25.0 25.0 25.0 49.5 49.5 49.5 4 722.0 722.0 722.0 1.53 1.38 1.53 62.7 60.6 62.7 189.7 173.5 189.7 5 799.3 799.3 799.3 1.53 1.38 1.53 20.9 23.0 20.9-147.5-131.3-147.4 6 710.9 710.9 7110.9 1.55 1.39 1.55 92.0 87.5 92.0 367.9 331.9 367.8 7 865.4 865.4 865.4 1.55 1.39 1.55 7.0 11.5 7.0-317.9-281.9-317.9 8 482.9 482.9 482.9 1.34 1.25 1.34 68.0 65.5 68.0 311.6 292.0 311.7 9 637.4 637.4 637.4 1.34 1.25 1.34-5.4-2.9-5.3-280.0-260.3 280.0 No. Table 8 - Magnified Moments along Column Length to First-Order Moment Ratios Comparison δ M 1, ft-kip M 2, ft-kip M 1/M 1(1st) M 2/M 2(1st) Hand Referene * spcolumn Hand Referene * spcolumn Hand Referene * spcolumn Hand Referene * spcolumn Hand Referene * spcolumn 1 1.00 --- 1.00 91.5 --- 91.5 91.5 --- 91.5 1.00 --- 1.00 1.00 --- 1.00 2 1.00 --- 1.00 91.3 --- 91.3 91.3 --- 91.3 1.00 --- 1.00 1.00 --- 1.00 3 1.00 --- 1.00 83.7 --- 83.7 83.7 --- 83.7 1.00 --- 1.00 1.00 --- 1.00 4 1.00 --- 1.00 75.8 --- 75.8 189.7 --- 189.7 1.00 --- 1.00 1.44 --- 1.44 5 1.00 --- 1.00 83.9 --- 83.9-147.5 --- -147.4 1.00 --- 1.00 1.65 --- 1.65 6 1.00 --- 1.00 92.0 --- 92.0 367.9 --- 367.8 1.20 --- 1.20 1.50 --- 1.50 7 1.00 --- 1.00 90.9 --- 90.9-317.9 --- -317.9 1.00 --- 1.00 1.62 --- 1.62 8 1.00 --- 1.00 68.0 --- 68.0 311.6 --- 311.7 1.16 --- 1.16 1.32 --- 1.32 9 1.00 --- 1.00-66.9 --- -66.9-280.0 --- 280.0 1.00 --- 1.00 1.37 --- 1.37 * Moment magnifiation along the length of the olumn is not overed by the referene 28
No. Table 9 - Design Parameters Comparison, in. ε t = ε s φ φp n, kip φm n, kip.ft Hand Referene * spcolumn Hand Referene * spcolumn Hand Referene * spcolumn Hand Referene * spcolumn Hand Referene * spcolumn 1 14.85 14.85 14.85 0.00096 0.00096 0.00096 0.65 0.65 0.65 871.4 871.4 871.4 459.4 459.4 459.4 2 14.85 14.82 14.85 0.00097 0.00097 0.00097 0.65 0.65 0.65 869.4 869.4 869.4 459.7 459.7 459.7 3 13.75 13.75 13.75 0.00128 0.00128 0.00128 0.65 0.65 0.65 797.6 797.6 797.6 468.2 468.2 468.2 4 12.75 12.75 12.75 0.00162 0.00162 0.00162 0.65 0.65 0.65 722.0 722.0 722.0 474.1 474.1 474.1 5 13.78 13.78 13.78 0.00127 0.00127 0.00127 0.65 0.65 0.65 799.3 799.3 799.3 468.0 468.0 468.0 6 12.61 12.61 12.61 0.00167 0.00167 0.00167 0.65 0.65 0.65 710.9 710.9 7110.9 474.8 474.8 474.8 7 14.76 14.76 14.76 0.00099 0.00099 0.00099 0.65 0.65 0.65 865.4 865.4 865.4 460.2 460.2 460.2 8 7.36 7.36 7.36 0.00500 0.00500 0.00500 0.90 0.90 0.90 482.9 482.9 482.9 557.2 557.2 557.2 9 11.68 11.68 11.68 0.00204 0.00204 0.00204 0.65 0.65 0.65 637.4 637.4 637.4 478.8 478.8 478.8 * Notes on ACI 318-11 Building Code Requirements for Strutural Conrete, Twelfth Edition, 2013 Portland Cement Assoiation, Example 11-2 In all of the hand alulations and the referene used illustrated above, the results are in preise agreement with the automated exat results obtained from the spcolumn program. 29
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 4 to 7, M u inluding seond-order effets exeeds 1.4 M u due to first-order effets (see Table 5). 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 1.25. 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 0.25. The value 0.25 is equivalent to a seondary-to-primary moment ratio of 1.33. Hene, the upper limit of 1.4 on the seondary-to-primary moment ratio was seleted by the ACI 318. The moment magnifiation fator values δ s alulated in this doument and spcolumn are different from the values alulated by the referene. ACI 318 provides three equation options to alulate the effetive stiffness modulus (EI) eff as was disussed previously in this doument. Equation 6.6.4.4.4(b) is more aurate than equation 6.6.4.4(a) but is more diffiult to use beause I se is not known until reinforement is hosen. The referene used equation 6.6.4.4(a) due to its simpliity while spcolumn uses equation 6.6.4.4.4(b) sine an iterative proedure is used to selet the optimum reinforement onfiguration. As an be seen in Table 5 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 4-7 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. 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. 30