163 APPENDIX 1 MODEL CALCULATION OF VARIOUS CODES A1.1 DESIGN AS PER NORTH AMERICAN SPECIFICATION OF COLD FORMED STEEL (AISI S100: 2007) 1. Based on Initiation of Yielding: Effective yield moment, M n S e F y S e Elastic section modulus of effective section calculated relative to extreme compression or tension fiber at F y F y Yield stress 2. Based on Lateral Torsional Buckling Strength: M n S c F c Sc Elastic section modulus of effective section calculated relative to extreme compression fiber at F c F c shall be determined as follows: F e >2.78F y no lateral buckling at bending moments less than or equal to M y 2.78F y >F e >0.56F y F c F y - F e <0.56F y F c F e where,f e
164 C b is conservatively taken as unity for all cases d - Depth of section I yc - Moment of inertia of compression portion of section about centroidal axis of entire section parallel to web, using full unreduced section. I yc S f - Elastic section modulus of full unreduced section relative to extreme compression fiber K y - Effective length factor for bending about y axis L y -Unbraced length of member for bending about y axis F e A area of the full cross-section r o polar radius of gyration of the cross section about the shear centre r x, r y radii of gyration of the cross section about the x- and y- axes respectively x o distance from shear centre to centroidal along principal x-axis taken as negative. E Modulus of elasticity of steel G Shear modulus J Saint- Venant torsion constant for a cross section Torsional warping constant of cross section Kt Effective length factors for twisting
165 Lt Unbraced length of member for twisting. Ky Effective length factors for bending about y-axis. Ly Unbraced length of member for bending about y-axis. 1. Based on Distortional Buckling Strength: Distortional Buckling Strength (moment of resistance) M n is given by, d n M y d > 0.673 M n - * M y d M y S fy xf y Where, S fy Elastic section modulus of full unreduced section relative to extreme fiber in first yield. M crd S f F d S d Elastic section modulus of full unreduced section relative to extreme compression fiber. F d - Elastic distortional buckling stress F d - K d A value accounting for moment gradient, which is permitted to be conservatively taken as 1.0 E Modulus of elasticity t Base steel thickness b 0 - Out- to-out flange width D - Out-to-out lip dimension
166 - Lip angle h o - Out-to-out web depth A1.2 NUMERICAL EXAMPLE FOR SPECIMEN TCDW-2000-200-100-1 AS PER AISI S100: 2007 1. Based on Yield Strength Effective yield moment, M n S e F y Effective section modulus, S e 55877.2186mm 3 Yield stress, F c 247N/mm 2 M n 55877.2186 247 M n 13.80 10 6 N.mm 2. Lateral -Torsional Buckling Strength M n S c F c Sc 57146.53438mm 3 F e 587.494 N/mm 2 F e 620.08N/mm 2 2.78F y >F e >0.56F y F c F y - *247-244.89 N/mm 2 M n S c F c 55877.2186 244.89 13.69 10 6 N.mm
167 3. Distortional Buckling Strength F d - K d L cr 1.2h o o b 0 D 71.42 mm 15 mm 90 h o 200 mm K d 1.709 (0.5< 1.315< 8) F d 1081.423696N/mm 2 M crd S e f y 57146.53438*1081.423696 61799616.42N.mm M y S fy F y 57146.53438*247 14286633.6 N.mm d 0.48080
168 M n M n M y 14.28 10 6 N.mm The least of the above will be the nominal moment capacity of the section. Hence the governing mode of failure is lateral torsional buckling and the moment capacity is Mn 13.69 10 6 N.mm A.1.3 DESIGN AS PER AUSTRALIAN/NEW ZEALAND STANDARD FOR COLD FORMED STEEL (AS/NZS 4600:2005) 1. Based on initiation of Yielding: M s Z e f y Z e is the effective section modulus calculated with the extreme compression or tension fibre at f y fy is the yield stress 2. Based on Lateral Torsional Buckling: M b Z c f c Where Zc effective section modulus calculated at a stress fc in the extreme f c M c critical moment Z f full unreduced section modulus for the extreme compression fibre The critical moment (M c ) shall be calculated as follows: b M c M y
169 b< 1.336 M c 1.11 M y - b M c M y Where b non-dimensional slenderness ratio used to determine M c for members subjected to lateral buckling b M y moment causing initial yield at the extreme compression fibre of the full section Z f f y M o elastic buckling moment Where M o C b Ar o1 r o1 polar radius of gyration of the cross section about the shear centre. C b is permitted to be taken as unity for all cases. A r o1 area of the full cross-section polar radius of gyration of the cross section about the shear centre. r x, r y radii of gyration of the cross section about the x- and y- axes respectively x o, y o coordinates of the shear sentre of the cross section f oy elastic buckling stress in an axially loaded compression member for the flexural buckling about the y- axis.
170 f oz elastic buckling stress in an axially loaded compression member for torsional buckling l ex, l ey, l ez, effective length for buckling about the x-axis and y-axes, and for twisting, respectively G shear modulus of elasticity (80 10 3 Mpa) J torsion constant for a cross section I w warping constant for a cross section 3. Based on Distortional Buckling: The critical moment (M c ) shall be calculated as follows: d < 0.59: M c M y d M c M y d M c M y Where M y moment causing initial yield at the extreme compression fibre of the full section d non-dimensional slenderness used to determine M c for the member subjected to distortional buckling M od elastic buckling moment in the distortional mode Z f f od Minimum of above moment is taken as Moment capacity of the section
171 A1.4 NUMERICAL EXAMPLE FOR SPECIMEN TCDW-2000-200-100-1 AS PER AS/NZS 4600:2005 1. Based on initiation of Yielding: Z e 55877.219 mm 3 (Calculated as per code) fy 247 N/mm 2 (Extreme flange material yield stress) M s Z e f y 55877.219 * 247 13.801 10 6 Nmm 2. Based on Lateral buckling: Zc 55877.219 mm 3 Z f 57146.534 mm 3 f c M y Z f f y 57146.534 *247 14.28 10 6 Nmm C b 1 A 759.998 mm 2 r x r y x 0 y o 87.577 mm 29.908mm 40mm 100 mm r o1 94.058 mm E 2.11 *10 5 N/mm 2 G 76923.077 N/mm 2 J 808.530mm 4
172 l ez l2000 mm I w 6400821135.242 mm 6 f oz 215.678 N/mm 2 f oy 3787.886 N/mm 2 M o C b Ar o1 1*759.998*94.058 32883664.012N.mm b 0.659 b < 1.336M c 1.11 M y - 1.11* 1.14286633.5-13945135.69 N.mm f c 244.024 N/mm 2
173 M b Z c f c 55877.219*244.024 13635382.49N.mm 3. Based on Distortional Buckling: f od elastic distortional buckling stress calculated as per Appendix D of AS/NZS 4600:2005 - - 7386.865 N/mm 2 M od Z f f od 57146.534*7386.865 422133731.9 N.mm M y Z f f y 57146.534*247 14.28*10 6 N.mm d 0.184 < 0.59 Hence M c M y
174 f c 249.88N/mm 2 The nomial member capacity(m b ) Z c f c 55877.219*249.88 13962599.48 N/mm 2 The least of the above will be the nominal moment capacity of the section, Hence the governing mode of failure is lateral torsional buckling and the moment capacity is M n 13.63 10 6 N.mm A.1.5 DESIGN AS PER INDIAN STANDARD FOR COLD FORMED STEEL IS 801-1975 1. Based on Yielding Nominal Moment S F y F y Specified minimum yield point. S unreduced Elastic section modulus 2. Based on Lateral Torsional Buckling When < < F b When F b 2 E C b
175 Mn F B I yc / Y c N.mm L the unbraced length of the member I yc the moment of inertia of the compression portion of a section about the gravity axis of the entire section parallel to the web S xc Compression section modulus of entire section about major axis, I x divided by distance to extreme compression fibre E d modulus of elasticity depth of section. C b bending coefficient which can conservatively be taken asunity. Numerical example for specimen TCDW-2000-200-100-1 as per IS 801:1975 1. Based on Yielding F y 247 N/mm 2 S 68502.02 mm 3 Nominal Moment M n S F y 68502.02 x 247 M n 16.92 10 6 N.mm 2. Based on Lateral Torsional Buckling L 2000 mm I yc 339894.582 mm 4 S xc 55877.2186 mm 3 E 2.11* 10 5 N/mm 2
176 d 200 mm C b 1 When > & < 3793.7727 3032.126 14370.26 F b When F b 2 E C b F B F b 146.8043N/mm 2 Mn F b I yc / Y c N.mm 146.8043*6242528/100 16118534 N.mm The least of the above will be the nominal moment capacity of the section, Hence the governing mode of failure is lateral torsional buckling and the moment capacity is Mn 16.11 x 10 6 N.mm
177 APPENDIX 2 MODEL CALCULATION OF PROPOSED EQUATION Specimen TCA-3600-400-150-6 A.2.1 CALCULATION OF UNREDUCED SECTION MODULUS I XX + 2 + 4 + 2 + 4-6209913.6 + 23760800 + 3815378.667 33786092.27 mm 4 Z f I xx / y 33786092.27 / 200 168930.4614 mm 3 A.2.2 CALCULATION OF YIELDING MOMENT M Y CONSIDERING FULL SECTION M y Z f f y 168930.4614 *247 41.725 X 10 6 Nmm
178 A.2.3 CALCULATION OF ELASTIC BUCKLING MOMENT M O C b 1 A 119.998 mm 2 r x r y 170.413mm 39.596mm x 0 40mm y o 100 mm r o1 175.091 mm E 2.11 *10 5 N/mm 2 G 76923.077 N/mm 2 J 1190.380mm 4 l ez l3600mm I w 1881449.290 mm 6 f oz f oy 95.827N/mm 2 238.801N/mm 2 M o C b Ar o1 1*1119.998*175.091 49792199.279N.mm
179 b 0.915 b lies between 0.879 & 1.310, as per the proposed equation 6.1, Critical moment M c M y (1.378-2 ) 41.725*10 6 (1.378-0.489*0.915 2 ) 40414708.13 N.mm f c 239.239N/mm 2 A.2.4 CALCULATION OF EFFECTIVE SECTION MODULUS AS PER CODE For the first iteration, assume a compression stress F y 247N/mm 2 in the top fibre of the section and that the neutral axis is 200mm. Below the top fibre. i. Calculation of effective width of flange: w b 99.38mm w/t 99.38/2 49.69< 60 OK S 1.28 1.28 37.41129319 check effective width of flange Compute k of the flange based on stiffener lip properties
180 I a 399 * t 4 4 I a 399 * 2 4 4 6388.194 mm 4 mm 4 I a 2523.922mm 4 R radius of corner d c lip depth ((R + ) + ( )) 15 ((2 + ) + ( )) 11mm. 90 degrees. I s (d 3 t sin 2 (11 3 *2*sin 2 90º)/12 221.833mm 4 R I 0.08789< 1 Hence OK. n n 0.2499 < 1/3 n 0.333 D depth of lip 15mm D/ w 0.1509< 0.8 OK K - (R I ) n K (0.08789) 0.1509
181 2.252<4 OK F cr k 2.252 173.75 N/mm 2 1.199> 0.673 flange is subject to local buckling (1- (1-0.22/ 1.199) / 1.199 0.979 b w 0.979*99.38 97.29mm ii. Calculation of effective width of Stiffener lip: w/t d/t 11/2 5.5 Maximum stress in lip (by similar triangles) f f 1 f y * ((N.A - - r) / N.A) 247 * ((200 - - 3) / 200) 242.06 N/mm 2 f 2 f y * (N.A- D) / N.A 247 * [200-15] / 200 228.475 N/mm 2 228.475/242.06 0.94388 k 0.4502
182 F cr k 0.4502 * - 2838.17 N/mm 2 0.29204<0.673 lip is not subjected to local buckling s d11mm. d s d s I ) 11 * 0.1509 1.66025mm iii. Calculation of effective width of Web: w/t 333.33 f f 1 f y * ((N.A - r) / N.A) f 1 247 * ((200-3) / 200) 242.552 N/mm 2 overall depth of section 400 + 2 + 2 404mm f 2 f y - N.A - r) / N.A 247 * (404-200 - 3) / 200-247.494 N/mm 2-247.494/242.5541.020 K 3 4+2(1+1.020) 3 +2(1+1.020) 24.5345
183 F cr k F cr 24.5345-42.1095 N/mm 2 2.40002 > 0.673 web may be subjected to local buckling (1- (1-0.22/ 2.40002)/2.40002 0.37847 a depth of web 400 mm b e b 151.388 mm h o /b o 5.38667<4.0 b 1 b e (3+y) 151.388/ (3+1.020) 37.6553 mm B 2 b e /2 151.388/2 75.6939 mm B 1 +b 2 37.6553+75.6939 113.349 < 196 196 web is not fully effective for this iteration. Recomputing properties by parts. Considering the ineffective portion of the web as an element with a negative length B neg - (196-113.349) -7.368491mm Its centroidal location below the top fibre: y t/2 + r +b 1 + b neg /2 1.2/2 + 3 + 37.6553+ 7.368/2 82.9806 mm
184 Element t L Top flange(right) Top flange(left) Bottom flange(right) Bottom flange(left) Y from top fiber 2 97.29 1 2 41.42 1 2 99.38 403 2 50.62 403 A 194.58 82.84 198.76 101.24 Ay Ay 2 Ix about own axis 194.58 194.58 64.86 82.84 82.84 27.61 80100.28 32280412.84 66.25 40799.72 16442287.16 33.75 Web 1.2 400 202 480 96960 19585920 6400000 Negative web element 1.2-82.65 82.98-99.18-8230.09-682938.268-56460.02 Top lip (left) 2 9 6.9 18 124.2 856.98 121.5 Bottom lip (right) 2 22 400 44 17600 7040000 1774.66 Sum 655.91 1020.2 227631.53 74666816.13 634698.38 Y Ay/ A 227631.53/1020.2 223.12 mm below top fibre I x [ I x Ay -y 2 A] [4.028*10 11 + 7.46*10 7 - (347.047) 2 *(1020.2)] 32917421.69 mm 4 The calculated neutral axis location (344.584mm) does not equal the assumed neutral axis location (200 mm); therefore, iteration is required.after further iterations, the solution converges to: I x 31516355.43mm 4 Y 344.584mm Effective section modulus S e 135095.1838mm 3 Predicted moment M Pr Z c f c 135095.1838*239.238 32319999.99N.mm