APPENDIX 1 MODEL CALCULATION OF VARIOUS CODES

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Rudolph Cannon
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1 163 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

2 164 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

3 165 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 > 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

4 166 - Lip angle h o - Out-to-out web depth A1.2 NUMERICAL EXAMPLE FOR SPECIMEN TCDW AS PER AISI S100: Based on Yield Strength Effective yield moment, M n S e F y Effective section modulus, S e mm 3 Yield stress, F c 247N/mm 2 M n M n N.mm 2. Lateral -Torsional Buckling Strength M n S c F c Sc mm 3 F e N/mm 2 F e N/mm F y >F e >0.56F y F c F y - * N/mm 2 M n S c F c N.mm

5 Distortional Buckling Strength F d - K d L cr 1.2h o o b 0 D mm 15 mm 90 h o 200 mm K d (0.5< 1.315< 8) F d N/mm 2 M crd S e f y * N.mm M y S fy F y * N.mm d

6 168 M n M n M y 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 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

7 169 b< 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.

8 170 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 ( 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

9 171 A1.4 NUMERICAL EXAMPLE FOR SPECIMEN TCDW AS PER AS/NZS 4600: Based on initiation of Yielding: Z e mm 3 (Calculated as per code) fy 247 N/mm 2 (Extreme flange material yield stress) M s Z e f y * Nmm 2. Based on Lateral buckling: Zc mm 3 Z f mm 3 f c M y Z f f y * Nmm C b 1 A mm 2 r x r y x 0 y o mm mm 40mm 100 mm r o mm E 2.11 *10 5 N/mm 2 G N/mm 2 J mm 4

10 172 l ez l2000 mm I w mm 6 f oz N/mm 2 f oy N/mm 2 M o C b Ar o1 1* * N.mm b b < 1.336M c 1.11 M y * N.mm f c N/mm 2

11 173 M b Z c f c * N.mm 3. Based on Distortional Buckling: f od elastic distortional buckling stress calculated as per Appendix D of AS/NZS 4600: N/mm 2 M od Z f f od * N.mm M y Z f f y * *10 6 N.mm d < 0.59 Hence M c M y

12 174 f c N/mm 2 The nomial member capacity(m b ) Z c f c * 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 N.mm A.1.5 DESIGN AS PER INDIAN STANDARD FOR COLD FORMED STEEL IS 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

13 175 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 as per IS 801: Based on Yielding F y 247 N/mm 2 S mm 3 Nominal Moment M n S F y x 247 M n N.mm 2. Based on Lateral Torsional Buckling L 2000 mm I yc mm 4 S xc mm 3 E 2.11* 10 5 N/mm 2

14 176 d 200 mm C b 1 When > & < F b When F b 2 E C b F B F b N/mm 2 Mn F b I yc / Y c N.mm * / 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 x 10 6 N.mm

15 177 APPENDIX 2 MODEL CALCULATION OF PROPOSED EQUATION Specimen TCA A.2.1 CALCULATION OF UNREDUCED SECTION MODULUS I XX mm 4 Z f I xx / y / mm 3 A.2.2 CALCULATION OF YIELDING MOMENT M Y CONSIDERING FULL SECTION M y Z f f y * X 10 6 Nmm

16 178 A.2.3 CALCULATION OF ELASTIC BUCKLING MOMENT M O C b 1 A mm 2 r x r y mm mm x 0 40mm y o 100 mm r o mm E 2.11 *10 5 N/mm 2 G N/mm 2 J mm 4 l ez l3600mm I w mm 6 f oz f oy N/mm N/mm 2 M o C b Ar o1 1* * N.mm

17 179 b b lies between & 1.310, as per the proposed equation 6.1, Critical moment M c M y ( ) *10 6 ( * ) N.mm f c N/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/ < 60 OK S check effective width of flange Compute k of the flange based on stiffener lip properties

18 180 I a 399 * t 4 4 I a 399 * mm 4 mm 4 I a mm 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)/ mm 4 R I < 1 Hence OK. n n < 1/3 n D depth of lip 15mm D/ w < 0.8 OK K - (R I ) n K ( )

19 <4 OK F cr k N/mm > flange is subject to local buckling (1- (1-0.22/ 1.199) / b w 0.979* mm 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) N/mm 2 f 2 f y * (N.A- D) / N.A 247 * [200-15] / N/mm / k

20 182 F cr k * N/mm <0.673 lip is not subjected to local buckling s d11mm. d s d s I ) 11 * mm iii. Calculation of effective width of Web: w/t f f 1 f y * ((N.A - r) / N.A) f * ((200-3) / 200) N/mm 2 overall depth of section mm f 2 f y - N.A - r) / N.A 247 * ( ) / N/mm / K 3 4+2( ) 3 +2( )

21 183 F cr k F cr N/mm > web may be subjected to local buckling (1- (1-0.22/ )/ a depth of web 400 mm b e b mm h o /b o <4.0 b 1 b e (3+y) / ( ) mm B 2 b e / / mm B 1 +b < 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 - ( ) mm Its centroidal location below the top fibre: y t/2 + r +b 1 + b neg /2 1.2/ / mm

22 184 Element t L Top flange(right) Top flange(left) Bottom flange(right) Bottom flange(left) Y from top fiber A Ay Ay 2 Ix about own axis Web Negative web element Top lip (left) Bottom lip (right) Sum Y Ay/ A / mm below top fibre I x [ I x Ay -y 2 A] [4.028* * ( ) 2 *(1020.2)] mm 4 The calculated neutral axis location ( mm) does not equal the assumed neutral axis location (200 mm); therefore, iteration is required.after further iterations, the solution converges to: I x mm 4 Y mm Effective section modulus S e mm 3 Predicted moment M Pr Z c f c * N.mm

Accordingly, the nominal section strength [resistance] for initiation of yielding is calculated by using Equation C-C3.1.

Accordingly, the nominal section strength [resistance] for initiation of yielding is calculated by using Equation C-C3.1. C3 Flexural Members C3.1 Bending The nominal flexural strength [moment resistance], Mn, shall be the smallest of the values calculated for the limit states of yielding, lateral-torsional buckling and distortional