TALAT Lecture 2710: Static Design Example

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Example of structure design of the industrial building developed with the Mathcad software.

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TALAT Lecture 2710: Static Design Example

  1. 1. TALAT Lectures 2710 Static Design Example 82 pages Advanced Level prepared during the TAS project: TAS Leonardo da Vinci Training in Aluminium Alloy Structural Design Example developed with the “Mathcad” Software Date of Issue: 1999  EAA - European Aluminium Association
  2. 2. 2710 Static Design example Table of Contents (Active) 2710 Static Design example........................................................................................... 2 1. Introduction.......................................................................................................................6 2. Materials ............................................................................................................................7 3. Loads ..................................................................................................................................7 4. Load Combinations........................................................................................................... 9 5. Loads Effects ...................................................................................................................11 5.1. Loads per unit length and concentrated loads...........................................................................11 5.2 Finite element calculations ........................................................................................................12 5.3. Section forces for characteristic loads ......................................................................................14 5.4. Design moments, shear forces and deflections .........................................................................16 6. Code Checking.................................................................................................................28 6.1 Column A ..................................................................................................................................28 6.2 Column B ..................................................................................................................................42 6.3 Column C ..................................................................................................................................55 6.4 Floor Beam D ............................................................................................................................56 6.5 Roof Beam E .............................................................................................................................64 6.6 Roof Beam F .............................................................................................................................65 6.7 Welded Connections..................................................................................................................74 Table of Contents (Complete) 1 INTRODUCTION 1.1 Description 1.2 Sketches 1.3 References 1.4 S.I. units 2 MATERIALS 2.1 Aluminium 2.2 Other materials 3 LOADS 3.1 Permanent loads 3.2 Imposed loads 3.3 Environmental loads 3.3.1 Snow loads 3.3.2 Wind loads TALAT 2710 2
  3. 3. 4 LOAD COMBINATIONS 4.1 Ultimate limit state 4.2 Serviceability limit state 5 LOAD EFFECTS 5.1 Loads per unit length and concentrated loads 5.1.1 Permanent loads 5.1.2 Imposed loads, uniform distributed 5.1.3 Imposed loads, concentrated 5.1.4 Snow loads 5.1.5 Wind loads 5.2 Finite element calculations 5.2.0 Nodes and elements 5.2.1 Permanent loads 5.2.2 Imposed loads, uniformly distributed 5.2.3 Imposed loads, concentrated 5.2.4 Snow loads 5.2.4 Wind loads 5.3 Section forces for characteristic loads 5.3.1 Column A 5.3.2 Column B 5.3.3 Column C 5.3.4 Floor beam D 5.3.5 Floor beam E 5.3.5 Floor beam F 5.4 Design moments, shear forces and deflections 5.4.1 Column A 5.4.2 Column B 5.4.3 Column C 5.4.4 Floor beam D 5.4.5 Floor beam E 5.4.6 Floor beam F 5.4.7 Joint A-D 5.4.8 Joint B-D 5.4.9 Joint A-E 5.4.10 Joint B-E 5.4.11 Joint B-F 5.4.12 Joint F-C 5.4.13 Column base TALAT 2710 3
  4. 4. 6 CODE CHECKING 6.1 Column A 6.1.1 Dimensions and material properties 6.1.2 Internal moments and forces 6.1.3 Classification of the cross section in y-y-axis bending 6.1.4 Classification of the cross section in z-z-axis bending 6.1,5 Classification of the cross section in axial compression 6.1.6 Welds 6.1.7 Design resistance, y-y-axis bending 6.1.8 Design resistance, z-z-axis bending 6.1.9 Axial force resistance, y-y buckling 6.1.10 Axial force resistance, z-z axis buckling 6.1.11 Flexural buckling of beam-column 6.1.12 Lateral-torsional buckling between purlins 6.1.13 Design moment at column base 6.1.14 Deflections 6.1.15 Summary 6.2 Column B 6.2.1 Dimensions and material properties 6.2.2 Internal moments and forces 6.2.3 Classification of the cross section in y-y-axis bending 6.2.4 Classification of the cross section in z-z-axis bending 6.2,5 Classification of the cross section in axial compression 6.2.6 Welds 6.2.7 Design resistance, y-y-axis bending 6.2.8 Design resistance, z-z-axis bending 6.2.9 Axial force resistance, y-y buckling 6.2.10 Axial force resistance, z-z axis buckling 6.2.11 Flexural buckling of beam-column 6.2.12 Lateral-torsional buckling between purlins 6.2.13 Design moment at column base 6.2.14 Deflections 6.2.14 Summary 6.3 Column C 6.4 Floor Beam D 6.4.1 Dimensions and material properties 6.4.2 Internal moments and forces 6.4.3 Classification of the cross section 6.4.4 Welds 6.4.5 Bending moment resistance 6.4.6 Bending resistance in a section with holes 6.4.7 Shear force resistance 6.4.8 Deflections 6.4.8 Summary TALAT 2710 4
  5. 5. 6.5 Roof Beam E 6.6 Roof Beam F 6.6.1 Dimensions and material properties 6.6.2 Internal moments and forces 6.6.3 Classification of the cross section 6.6.4 Welds 6.6.5 Bending moment resistance 6.6.6 Lateral-torsional buckling between purlins 6.6.7 Bending resistance in a section with holes 6.6.8 Shear force resistance 6.6.9 Concentrated transverse force 6.6.10 Deflections 6.6.11 Summary 6.7 Welded connections 6.7.1 Weld properties 6.7.2 Longitudinal weld of floor beam D 6.7.3 Base of column B 6.7.4 Connection between floor beam D and column B Software The example is worked out using the MathCad software in which some symbols have special meanin according to the following x 50.6 . mm Assign value y 2.5 . mm Global assignment x y = 53.1 mm Evaluate expression a b Boolean equals 0.5 Decimal point c (1 3 2 ) Vector d (2 4 3 ) Vector a ( c .d ) Vectorize (multiply the elements in vector c with corresponding elements in d) a = ( 2 12 6 ) Result Structure The structure was proposed by Steinar Lundberg, who also contributed with valuable suggestions. Part 1 to 6.6 was worked out by Torsten Höglund and 6.7 by Myriam Bouet-Griffon. TALAT 2710 5
  6. 6. 1. Introduction 1.1 Description The industrial building contains an administration part with offices, wardrobe, meeting rooms etc. and a fabrication hall. The load bearing system consist of frames standing at a distance of 5000 m In the serviceability limit state max. allowable deflection is 1/250 of span. 1.2 Sketches 1.3 References [1] ENV 1999-1-1. Eurocode 9 - Design of aluminium structures - Part 1-1: General rules. 1997 [2] ENV 1991-2-1. Eurocode 1 - Basis of design and actions on structures - Part 2-1: Action on structures - Densities, self-weight and imposed loads. 1995 [3] ENV 1991-2-3. Eurocode 1 - Basis of design and actions on structures - Part 2-3: Action on structures - Snow loads. 1995 [4] ENV 1991-2-4. Eurocode 1 - Basis of design and actions on structures - Part 2-4: Action on structures - Wind loads. 1995 [5] ENV 1991-1. Eurocode 1 - Basis of design and actions on structures - Part 1: Basis of design. 1994 1.4 S.I. units N kN 1000 . N MPa 1000000 . Pa kNm kN . m MPa = 1 2 mm TALAT 2710 6
  7. 7. 2. Materials 2.1 Aluminium [1], 3.2.2 The extrusions are alloy EN AW-6082, temper T6 The plates are EN AW-5083 temper H24 Strength of aluminium alloys EN AW-6082 T6 fo 260 . MPa fu 310 . MPa EN AW-5083 H24 fo 250 . MPa fu 340 . MPa [1], 5.1.1 The partial safety factor for the members γ M1 1.10 γ M2 1.25 [1], 6.1.1 The partial safety factor for welded connections γ Mw 1.25 Design values of material coefficients Modulus of elasticity E 700000 . MPa Shear modulus G 27000 . MPa Poisson´s ratio ν 0.3 α T 23 . 10 6 Coefficient of linear thermal expansion ρ 2700 . kg . m 3 Density 2.2 Other materials Comment: Properties of any other materials to be filled in 3. Loads 3.1 Permanent loads [3], ?? Permanent loads are self-weight of structure, insulation, surface materials and fixed equipment 0.5 . kN . m 2 Permanent load on roof q´ p.roof 0.7 . kN . m Permanent load on floor 2 q´ p.floor 3.2 Imposed loads [2], 6.3.1 Office area => Category B 3 . kN . m 2 q´ k.floor Uniform distributed load Q k.floor 2 . kN Concentrated load [2], 6.3.4 Roofs not accessible except for normal maintenance etc. => Category H => 0.75 . kN . m 2 Uniform distributed load q´ k.roof Concentrated load Q k.roof 1.5 . kN TALAT 2710 7
  8. 8. [?], ?? Load from crane, the crane located in the middle of the roof beam in the production hall Concentrated load Q crane 50 . kN 3.3 Environmental loads 3.3.1 Snow loads [3] Comment: The characteristic values of the snow loads vary from nation to nation. For simplicity for this design example, a snow load is chosen including shape coefficient, exposure coefficient and thermal coefficient. In a design report the calculation of the snow loads have to be shown. 2 kN . m 2 Snow load q´ snow 3.3.2 Wind loads [3] Comment: The characteristic values of the wind loads vary from nation to nation. For simplicity, for this design example, a wind load is chosen including all coefficients. In a design report the calculation of the wind loads including all coefficients have to be shown. 0.70 kN . m 2 Maximum wind load on the external walls q´ w.wall 0.27 kN . m Wind suction on leeward side of walls 2 q´ w.lee 0.70 kN . m Wind suction on the roof 2 q´ w.roof Max. wind suction at the lower edge of the 1.0 kN . m 2 roof and 1.6 m upwards q´ w.edge TALAT 2710 8
  9. 9. 4. Load Combinations 4.1 Ultimate limit state [3] To decide the section forces on the different members, the following load combinations to . be calculated in the ultimate limit state LC 1: Permanent + imposed + crane + snow loads imposed load dominant LC 2: Permanent + imposed + crane + snow loads crane load dominant LC 3: Permanent + imposed + crane + snow loads snow load dominant LC 4: Reduced permanent + wind loads wind load dominant LC 5: Permanent + imposed + crane + snow + wind imposed load dominant LC 6: Permanent + imposed + crane + snow + wind wind load dominant [3] Comment: All possible load combinations to be calculated [5], 9.4 ultimate limit state Partial load factors for different load combinations in the [5] Table 9.2 Partial factor for permanent action, unfavourable γ Gsup 1.35 Partial factor for permanent action, favourable γ Ginf 1.0 Partial factor for variable action, unfavourable γ Q 1.5 [5] Table 9.3 ψ factor for imposed loads ψ 0i 0.7 ψ factor for snow loads ψ 0s 0.6 ψ factor for wind loads ψ 0w 0.6 [5] Eq In load combinations where the imposed load is (9.10b) dominating ξ in Eq (9.10b) is less than 1.0, say ξ 0.9 Load combinations ξ . γ Gsup ξ . γ Gsup ξ . γ Gsup γ Ginf ξ . γ Gsup ξ . γ Gsup Load case 1 permanent loads γ Q ψ 0i . γ Q ψ 0i . γ Q 0 γ Q ψ 0i . γ Q 2 distributed loads ψ u ψ 0i . γ Q γ Q ψ 0i . γ Q 0 ψ 0i . γ Q ψ 0i . γ Q 3 crane load ψ 0s . γ Q ψ 0s . γ Q γ Q 0 ψ 0s . γ Q ψ 0s . γ Q4 snow loads 0 0 0 γ Q ψ 0w . γ Q γ Q 5 wind loads Resulting load factors in the ultimate limit state 1.215 1.215 1.215 1 1.215 1.215 1.5 1.05 1.05 0 1.5 1.05 ψ u = 1.05 1.5 1.05 0 1.05 1.05 0.9 0.9 1.5 0 0.9 0.9 0 0 0 1.5 0.9 1.5 TALAT 2710 9
  10. 10. 4.2 Serviceability limit state [5], 9.5.2 Partial load factors forfrequent load combinations in the serviceability limit state and 9.5.5 LC 1: imposed load dominant LC 2: crane load dominant LC 3: snow load dominant LC 4: wind load dominant LC 5: wind load only (for comparison) LC 6: simplified, [5] (9.20) (for comparison) [5] Table 9.3 ψ factor for imposed loads ψ 1i 0.5 ψ 2i 0.3 ( = 0 for roof) ψ factor for crane loads ψ 1c 0.5 ψ 2c 0.3 ψ factor for snow loads ψ 1s 0.2 ψ 2s 0 ψ factor for wind loads ψ 1w 0.5 ψ 2w 0 Load combination 1 2 3 4 5 6 Load case 1 1 1 1 0 1 permanent loads ψ 1i 0 0 0 0 0.9 imposed distributed loads ψ s ψ 2c ψ 1c ψ 2c 0 0 0.9 imposed crane load ψ 2s ψ 2s ψ 1s ψ 2s 0 0.9 snow loads ψ 2w ψ 2w ψ 2w ψ 1w 1 0 wind loads Resulting partial load factors in theserviceability limit state Load combination 1 2 3 4 5 6 1 1 1 1 0 1 0.5 0 0 0 0 0.9 ψ s = 0.3 0.5 0.3 0 0 0.9 0 0 0.2 0 0 0.9 0 0 0 0.5 1 0 For the floor, the load combination 1 is valid For roofs, the load combinations 2, 3 and 4 are valid Comment: Load combinations 5 and 6 for comparison only TALAT 2710 10
  11. 11. 5. Loads Effects 5.1. Loads per unit length and concentrated loads Distance between all frames c frame 5000 . mm Because of continuous purlins and secondary floor beams the load on a beam in a frame is mo than the distance between the beam times the load per area. Therefore, for the second frame, t load is increased with a factor ofkf where k f 1.1 5.1.1 Permanent loads k f . c frame . q´ p.floor q p.floor = 3.85 kN . m 1 Permanent load on floor q p.floor k f . c frame . q´ p.roof q p.roof = 2.75 kN . m 1 Permanent load on roof q p.roof 5.1.2 Imposed loads, uniform distributed k f . c frame . q´ k.floor q k.floor = 16.5 kN . m 1 Distributed load on floor q k.floor k f . c frame . q´ k.roof q k.roof = 4.125 kN . m 1 Distributed load on roof q k.roof 5.1.3 Imposed loads, concentrated Concentrated load on floor Q k.floor 2 . kN Concentrated load on roof Q k.roof 1.5 . kN Concentrated load from crane P crane 50 . kN 5.1.4 Snow loads k f . c frame . q´ snow q snow = 11 kN . m 1 Snow load on roof q snow TALAT 2710 11
  12. 12. 5.1.5 Wind loads k f . c frame . q´ w.wall q w.wall = 3.85 kN . m 1 Maximum wind load on the q w.wall external walls k f . c frame . q´ w.lee q w.lee = 1.485 kN . m 1 Wind suction on leeward q w.lee side of walls k f . c frame . q´ w.roof q w.roof = 3.85 kN . m 1 Wind suction on the roof q w.roof Max. wind suction at the k f . c frame . q´ w.edge q w.edge = 5.5 kN . m 1 q w.edge lower edge of the roof and 1.6 m upwards 5.2 Finite element calculations Nodes and elements TALAT 2710 12
  13. 13. Moment diagrams 5.2.1 Permanent loads Values of moments and shear forces for separate columns and beams are given in 5.3 5.2.2 Imposed loads, uniformly distributed 5.2.3 Imposed loads, concentrated 5.2.4 Snow loads 5.2.5 Wind loads TALAT 2710 13
  14. 14. 5.3. Section forces for characteristic loads 5.3.1 Column A (FE-cal- Bending moments, section 1, 2, 3 and 4 2.90 4.40 2.14 1.88 culation) Sections in columns, load cases in rows 9.43 15.6 12.1 5.96 row 1, permanent loads MA 3.65 4.60 5.22 5.81 . kNm row 2, distributed loads row 3, crane load 4.28 4.63 4.22 3.04 row 4, snow loads 12.8 6.45 2.52 1.95 row 5, wind loads 18.8 7.32 Axial force, part 1-2 and 3-4 60.5 12.0 Parts in columns, load cases in rows NA 1.42 2.84 . kN 0.56 28.5 27.8 0.50 15.6 12.0 Deflection in section 4 δ A 2.48 . mm 4.19 8.36 5.3.2 Column B (FE-cal- Bending moments, section 1, 2, 3, 4,5 and 6 0.07 2.73 4.51 4.35 7.46 9.07 culation) 4.26 15.4 18.2 4.69 8.29 15.5 Sections in columns, load cases in rows MB 3.27 0.96 2.15 19.27 15.7 11.2 . kNm row 1, permanent loads row 2, distributed loads 6.57 5.24 1.59 20.0 34.6 33.5 row 3, crane load row 4, snow loads 17.0 13.5 0.49 10.95 11.4 12.7 row 5, wind loads 35.1 23.5 9.18 Sections in columns, load cases in rows 84.4 33.9 12.8 Axial force, parts 1-2, 3-4 and 5-6 NB 31.7 31.1 4.34 kN 0.57 95.0 95.7 38.2 0.53 31.2 34.8 13.8 Deflection in section 6 δ B 2.57 . mm 4.30 8.32 TALAT 2710 14
  15. 15. 5.3.3 Column C Comment: To reduce the extent of this example, this column is left out. It can be given, conservatively the same section as column B 5.3.4 Floor beam D (FE-cal- Bending moments, section 1, 2 and 3 6.56 10.4 7.25 culation) 27.8 43.6 33.6 Sections in columns, load cases in rows MD 0.62 4.86 3.11 . kNm row 1, permanent loads row 2, distributed loads 0.40 1.62 3.65 row 3, crane load row 4, snow loads 8.97 2.00 13.0 row 5, wind loads 11.7 11.4 Sections in columns, load cases in rows 48.5 50.5 Shear force, section 1 and 3 VD 1.4 0.6 . kN 3.21 0.67 0.67 13.1 3.66 3.66 Deflection in section 2 δ D 1.63 . mm 0.92 0.91 5.3.5 Roof beam E Comment: To reduce the extent of this example, this column is left out 5.3.6 Roof beam F (FE-cal- Bending moments, section 1, 2 and 3 11.8 26.4 4.08 culation) 13.0 42.4 5.28 Sections in columns, load cases in rows MF 35.0 102.0 10.94 kNm row 1, permanent loads row 2, distributed loads 54.1 101.7 17.5 row 3, crane load row 4, snow loads 22.4 38.0 0.26 row 5, wind loads TALAT 2710 15
  16. 16. Shear force, section 1 and 3 14.3 13.2 Sections in columns, load cases in rows 21.1 20.1 VF 43.4 23.3 . kN 7.71 57.6 52.5 12.6 21.1 17.4 Deflection in section 2 δ F 22.2 . mm 29.3 11.1 5.4. Design moments, shear forces and deflections 5.4.1 Column A Bending moments For the load cases 1 - 5 the bending moment in section 1 of column B is: 2.9 9.43 <1> (5.3.1) MA = 3.65 kNm 4.28 12.8 The load factor matrix is 1.215 1.215 1.215 1 1.215 1.215 1.5 1.05 1.05 0 1.5 1.05 (4.1) ψ u 1.05 = 1.5 1.05 0 1.05 1.05 0.9 0.9 1.5 0 0.9 0.9 0 0 0 1.5 0.9 1.5 TALAT 2710 16
  17. 17. The values in the moment vector shall be multiplied with the corresponding load factor for every lo combination i 1 .. cols ψ u (cols (ψu) is the number of columns in the matrix ψu) <i > <1> . <i > M MA ψ u 3.524 3.524 3.524 2.9 3.524 3.524 14.145 9.901 9.901 0 14.145 9.901 M = 3.832 5.475 3.832 0 3.832 3.832 kNm 3.852 3.852 6.42 0 3.852 3.852 0 0 0 19.2 11.52 19.2 M The moments in the columns of the matrix (= load combination) are added <i > M sum M i T M sum = ( 25.353 22.752 23.677 16.3 13.833 1.909 ) kNm Maximum and minimum of moment are M Amax max M sum M Amax = 25.353 kNm 1 1 M Amin min M sum M Amin = 16.3 kNm 1 1 <i > <s> . <i > <i > (5.3.1) Moments in section 2 s 2 M MA ψ u M sum M i T M sum = ( 37.743 32.793 33.501 5.275 31.938 21.048 ) kNm M Amax max M sum s M Amin min M sum s <i > <s> . <i > <i > (5.3.1) Moments in section 3 s 3 M MA ψ u M sum M i T M sum = ( 11.471 3.677 3.494 1.64 9.203 2.246 ) kNm M Amax max M sum s M Amin min M sum s <i > <s> . <i > <i > (5.3.1) Moments in section 4 s 4 M MA ψ u M sum M i T M sum = ( 7.86 2.563 7.002 1.045 6.105 2.253 ) kNm M Amax max M sum s M Amin min M sum s TALAT 2710 17
  18. 18. Resulting maximum moments and minimum moments in section 1 to 4 are T M Amax = ( 25.353 5.275 11.471 1.045 ) kNm T M Amin = ( 16.3 37.743 1.64 7.86 ) kNm Axial force Axial force in part 1-2 s 1 18.8 60.5 <s> (5.3.1) NA = 1.42 kN <i > <s> . <i > <i > N NA ψ u N sum N 28.5 i 15.6 N Amax max N sum s T N sum = ( 137.751 109.887 127.626 4.6 123.711 87.126 ) kN N Amin min N sum s <i > <s> . <i > <i > (5.3.1) Axial force in part 3-4 s 2 N NA ψ u N sum N i T N sum = ( 48.932 42.254 60.212 10.68 38.132 25.532 ) kN N Amax max N sum s N Amin min N sum s Resulting maximum and minimum axial forces in part 1, 2 and 3 are: T T N Amax = ( 4.6 10.68 ) kN N Amin = ( 137.8 60.2 ) kN Deflection Deflection in section 6 s 1 0.56 0.5 <s> <i > <s> . <i > <i > (5.3.1) δ A = 2.48 mm δ δ A ψ s δ sum δ i 4.19 δ Amax max δ sum δ Amin min δ sum 8.36 s s T δ sum = ( 1.054 1.8 2.142 4.74 8.36 6.113 ) mm Resulting maximum and minimum deflection in section 6 δ Amax= ( 8.36 ) mm δ Amin ( 1.054 ) mm = TALAT 2710 18
  19. 19. 5.4.2 Column B Bending moments For the load cases 1 - 5 the bending moment in section 1 of column B is: 0.07 4.26 <1> (5.3.2) MB = 3.27 kNm 6.57 17 The load factor matrix is 1.215 1.215 1.215 1 1.215 1.215 1.5 1.05 1.05 0 1.5 1.05 (4.1) ψ u 1.05 = 1.5 1.05 0 1.05 1.05 0.9 0.9 1.5 0 0.9 0.9 0 0 0 1.5 0.9 1.5 The values in the moment vector shall be multiplied with the corresponding load factor for every l combination i 1 .. cols ψ u (cols (ψu) is the number of columns in the matrix ψu) <i > <1> . <i > M MB ψ u 0.085 0.085 0.085 0.07 0.085 0.085 6.39 4.473 4.473 0 6.39 4.473 M= 3.433 4.905 3.433 0 3.433 3.433 kNm 5.913 5.913 9.855 0 5.913 5.913 0 0 0 25.5 15.3 25.5 M The moments in the columns of the matrix (= load combination) are added <i > M sum M i T M sum = ( 3.042 6.43 8.901 25.43 12.258 20.541 ) kNm Maximum and minimum of moment are M Bmax max M sum M Bmax = 25.43 kNm 1 1 M Bmin min M sum M Bmin = 8.901 kNm 1 1 TALAT 2710 19
  20. 20. <i > <s> . <i > <i > (5.3.2) Moments in section 2 s 2 M MB ψ u M sum M i T M sum = ( 20.693 13.331 10.619 22.98 32.843 34.013 ) kNm M Bmax max M sum s M Bmin min M sum s <i > <s> . <i > <i > (5.3.2) Moments in section 3 s 3 M MB ψ u M sum M i T M sum = ( 31.953 22.796 24.717 5.245 32.394 24.498 ) kNm M Bmax max M sum s M Bmin min M sum s <i > <s> . <i > <i > (5.3.2) Moments in section 4 s 4 M MB ψ u M sum M i T M sum = ( 36.484 47.266 50.594 12.075 26.629 22.169 ) kNm M Bmax max M sum s M Bmin min M sum s <i > <s> . <i > <i > (5.3.2) Moments in section 5 s 5 M MB ψ u M sum M i T M sum = ( 69.124 72.458 86.153 9.64 58.864 48.293 ) kNm M Bmax max M sum s M Bmin min M sum s <i > <s> . <i > <i > (5.3.2) Moments in section 6 s 6 M MB ψ u M sum M i T M sum = ( 76.18 74.245 89.305 9.98 64.75 50.155 ) kNm M Bmax max M sum s M Bmin min M sum s Resulting maximum moments and minimum moments in section 1 to 6 are T M Bmax = ( 25.43 10.619 32.394 50.594 9.64 9.98 ) kNm T M Bmin = ( 8.901 34.013 5.245 12.075 86.153 89.305 ) kNm Axial force Axial force in part 1-2 s 1 35.1 84.4 <s> (5.3.2) NB = 31.7 kN <i > <s> . <i > <i > N NB ψ u N sum N 95 i 31.2 N Bmax max N sum s T N sum = ( 288.031 264.317 307.051 11.7 259.951 203.251 ) kN N Bmin min N sum s TALAT 2710 20
  21. 21. <i > <s> . <i > <i > (5.3.2) Axial force in part 3-4 s 2 N NB ψ u N sum N i T N sum = ( 198.187 196.927 240.352 28.7 166.868 130.732 ) kN N Bmax max N sum s N Bmin min N sum s <i > <s> . <i > <i > (5.3.2) Axial force in part 5-6 s 3 N NB ψ u N sum N i T N sum = ( 69.291 65.484 86.451 11.52 56.871 42.831 ) kN N Bmin min N sum s N Bmax max N sum s Resulting maximum and minimum axial forces in part 1, 2 and 3 are: T T N Bmax = ( 11.7 28.7 11.52 ) kN N Bmin = ( 307.1 240.4 86.5 ) kN Deflection Deflection in section 6 s 1 0.57 0.53 <s> <i > <s> . <i > <i > (5.3.2) δ B = 2.57 mm δ δ B ψ s δ sum δ i 4.3 δ Bmax max δ sum δ Bmin min δ sum 8.32 s s T δ sum = ( 1.076 1.855 2.201 4.73 8.32 6.276 ) mm Resulting maximum and minimum deflection in section 6 δ Bmax ( 8.32 ) mm = δ Bmin ( 1.076 ) mm = 5.4.3 Column C Comment: To reduce the extent of this example, calculation of this column is left out. It can, conservatively, be given the same dimensions as column B TALAT 2710 21
  22. 22. 5.4.4 Floor beam D Bending moments For the load cases 1 - 5 the bending moment in section 1 of beam D is: 6.56 27.8 <1> (5.3.4) MD = 0.62 kNm 0.4 8.97 The load factor matrix is 1.215 1.215 1.215 1 1.215 1.215 1.5 1.05 1.05 0 1.5 1.05 (4.1) ψ u 1.05 = 1.5 1.05 0 1.05 1.05 0.9 0.9 1.5 0 0.9 0.9 0 0 0 1.5 0.9 1.5 The values in the moment vector shall be multiplied with the corresponding load factor for every l combination i 1 .. cols ψ u (cols (ψu) is the number of columns in the matrix ψu) <i > <1> . <i > M MD ψ u 7.97 7.97 7.97 6.56 7.97 7.97 41.7 29.19 29.19 0 41.7 29.19 M = 0.651 0.93 0.651 0 0.651 0.651 kNm 0.36 0.36 0.6 0 0.36 0.36 0 0 0 13.455 8.073 13.455 M The moments in the columns of the matrix (= load combination) are added <i > M sum M i T M sum = ( 49.379 36.59 37.109 6.895 41.306 23.414 ) kNm Maximum and minimum of moment are M Dmax max M sum M Dmax = 6.895 kNm 1 1 M Dmin min M sum M Dmin = 49.379 kNm 1 1 TALAT 2710 22
  23. 23. <i > <s> . <i > <i > (5.3.4) Moments in section 2 s 2 M MD ψ u M sum M i T M sum = ( 84.597 67.164 65.949 13.4 86.397 67.977 ) m kN M Dmin min M sum s M Dmax max M sum s <i > <s> . <i > <i > (5.3.4) Moments in section 3 s 3 M MD ψ u M sum M i T M sum = ( 52.658 36.139 35.348 26.75 64.358 57.038 ) m kN M Dmin min M sum s M Dmax max M sum s Resulting maximum moments and minimum moments in section 1 to 3 are 6.895 49.379 M Dmax = 86.397 kNm M Dmin = 13.4 kNm 26.75 64.358 Shear force Shear force in section 1 s 1 11.7 48.5 <s> (5.3.4) VD = 1.4 kN <i > <s> . <i > <i > V VD ψ u V sum V 0.67 i 3.66 V Dmax max V sum V Dmin min V sum s s T V sum = ( 89.038 67.844 67.615 6.21 85.745 61.723 ) kN Shear force in section 2 s 2 11.4 50.5 <s> (5.3.4) VD = 0.6 kN <i > <s> . <i > <i > V VD ψ u V sum V 0.67 i 3.66 V Dmax max V sum V Dmin min V sum s s T V sum = ( 88.368 65.373 65.241 16.89 91.662 71.133 ) kN Resulting maximum and minimum shear forces in section 1 and 3 are 89.038 6.21 V Dmax = kN V Dmin = kN 91.662 16.89 Deflection TALAT 2710 23
  24. 24. 89.038 6.21 V Dmax = kN V Dmin = kN 91.662 16.89 Deflection Deflection in section 2 s 1 3.21 13.1 <s> (5.3.4) δ D = 1.63 mm <i > <s> . <i > <i > δ δ D ψ s δ sum δ 0.92 i 0.91 δ Dmax max δ sum δ Dmin min δ sum s s T δ sum = ( 10.249 4.025 3.883 2.755 0.91 17.295 ) mm Resulting maximum and minimum deflection in section 2 δ Dmax= ( 17.295 ) mm δ Dmin ( 0.91 ) mm = 5.4.5 Roof beam E Comment: To reduce the extent of this example, calculation of this beam is left out. It can be given the same dimensions as floor beam D 5.4.6 Roof beam F Moment For the load cases 1 - 5 the bending moments in section 1 to 3 of the beam F are <i > <s> . <i > <i > (5.3.6) Moments in section 1 s 1 M MF ψ u M sum M i T M sum = ( 119.277 129.177 145.887 21.8 99.117 79.827 ) m kN M Fmin min M sum s M Fmax max M sum s <i > <s> . <i > <i > (5.3.6) Moments in section 2 s 2 M MF ψ u M sum M i T M sum = ( 294.306 321.126 336.246 30.6 260.106 218.226 ) m kN M Fmin min M sum s M Fmax max M sum s <i > <s> . <i > <i > (5.3.6) Moments in section 3 s 3 M MF ψ u M sum M i T M sum = ( 40.114 42.661 48.238 3.69 39.88 37.348 ) m kN M Fmin min M sum s M Fmax max M sum s TALAT 2710 24
  25. 25. Resulting maximum moments and minimum moments in section 1 to 3 are 21.8 145.887 M Fmax = 336.246 kNm M Fmin = 30.6 kNm 3.69 48.238 Shear force Shear force in section 1 s 1 14.3 21.1 <s> (5.3.4) VF = 43.4 kN <i > <s> . <i > <i > V VF ψ u V sum V 57.6 i 21.1 V Fmax max V sum V Fmin min V sum s s T V sum = ( 146.435 156.47 171.5 17.35 127.444 105.289 ) kN Shear force in section 3 s 2 13.2 20.1 <s> (5.3.4) VF = 23.3 kN <i > <s> . <i > <i > V VF ψ u V sum V 52.5 i 17.4 V Fmax max V sum V Fmin min V sum s s T V sum = ( 117.903 119.343 140.358 12.9 102.243 82.758 ) kN Resulting maximum and minimum shear force in section 1 and 3 are 171.5 17.35 V Fmax = kN V Fmin = kN 140.358 12.9 Deflection Deflection in section 2 s 1 <s>T <i > <s> . <i > <i > (5.3.4) δ F = ( 7.71 12.6 22.2 29.3 11.1 ) mm δ δ F ψ s δ sum δ i δ Fmin min δ sum s T δ sum = ( 20.67 18.81 20.23 2.16 11.1 65.4 ) mm δ Fmax max δ sum s [5] (9.20) Simplified verification δ Fmax = ( 65.4 ) mm δ Fmin ( 11.1 ) mm = [5] (9.16) Load combination 3 δ sum = 20.2 mm 3 TALAT 2710 25
  26. 26. 5.4.7 Joint A-D (5.4.1) and Moment M Amax = 11.5 kNm (5.5.4) 3 M Dmin = 49.4 kNm 1 M Amin = 37.7 kNm 2 Shear V Dmax = 89 kN 1 Check: M Amax M Dmin M Amin = 0.17 kNm 3 1 2 5.4.8 Joint B-D (5.4.2) and Moment M Bmax = 32.4 kNm M Bmin = 34 kNm (5.5.4) 3 2 M Dmin = 64.4 kNm V 3 Check: M Bmax M Dmin M Bmin = 2.05 kNm V 3 3 2 Shear V Dmax = 91.7 kN 2 N V B3 = 2.306 kN V B2 = 18.184 kN Axial N B3 = 240.4 kN N B2 = 307.1 kN 5.4.9 Joint A-E and joint B-E 5.4.10 Comment: To reduce the extent of this example, calculation of this joint is left out 5.4.11 Joint B-F (5.4.2) and Moment M Bmin = 86.153 kNm (5.5.6) 5 M Fmin = 145.9 kNm 1 M Bmax = 50.6 kNm 4 Shear V Fmax = 171.5 kN 1 Check: M Bmin M Fmin M Bmax = 9.14 kNm 5 1 4 (The reason why the sum of the moments is not = 0 is the fact that all the moments does not belong to the same load combination) TALAT 2710 26
  27. 27. 5.4.12 Joint F-C (5.4.6) Moment M Fmin = 48.238 kNm 3 M Fmax = 3.7 kNm 3 Shear V Fmin = 12.9 kN 2 V Fmax = 140.4 kN 2 5.4.13 Column bases See 6.1.13 and 6.2.13 TALAT 2710 27
  28. 28. 6. Code Checking 6.1 Column A 6.1.1 Dimensions and material properties Section height: h 160 . mm Flange depth: b 150 . mm Web thickness: tw 5 . mm Flange thickness: tf 14 . mm Overall length: L1 3 .m Distance between purlins: cp 1 .m [1] Table 3.2b Alloy: EN AW-6082 T6 EP/O t > 5 mm newton newton f 0.2 260 . fu 310 . 2 2 mm mm heat_treated 1 (if heat-treated then 1 else 0) [1] (5.4), (5.5) f o f 0.2 fa fu fo newton newton newton [1] (5.6) fv f v = 150 E 70000 . G 27000 . 2 2 2 3 mm mm mm Partial safety factors: γ M11.10 γ M2 1.25 Inner radius: r 5 . mm Web height: bw h 2 .t f 2 .r b w = 122 mm S.I. units: kN 1000 . newton kNm kN . m MPa 1000000 . Pa TALAT 2710 28
  29. 29. 6.1.2 Internal moments and forces (5.4.2) Bending moments and axial forces for load case LC1, LC3 and LC6 in section 1, 2, 3 and 4 0 25.4 23.8 1.91 3.0 37.7 33.5 21.1 x M LC1 . kNm M LC3 . kNm M LC6 . kNm 3.1 11.5 3.49 2.24 5.5 7.86 7.00 2.25 138 127 87 138 127 87 N LC1 . kN N LC3 . kN N LC6 . kN 49 60 26 49 60 26 Bending moment kNm Axial force kN 4 4 2 2 0 0 40 20 0 20 40 0 50 100 150 Load case 1 Load case 1 Load case 3 Load case 3 Load case 6 Load case 6 Load case 1 Load case 3 Load case 6 Moment in section 2 M LC1 = 37.7 kNm M LC3 = 33.5 kNm M LC6 = 21.1 kNm 2 2 2 Moment at column base 1 M LC1 = 25.4 kNm M LC3 = 23.8 kNm M LC6 = 1.91 kNm 1 1 1 Axial force in part 1-2 N LC1 = 138 kN N LC3 = 127 kN N LC6 = 87 kN 1 1 1 Preliminary calculations show that load case 1 is governing. Study part 1-2 from column base to floor beam. Moment in top of part 1-2 (section 2) is larger than at column base (section 1) why M 1.Ed (below) correspond to section 2 andM 2.Ed to section 1 of the column. TALAT 2710 29
  30. 30. Load case 1 Bending moment in section 2 M 1.Ed M LC1 M 1.Ed = 37.7 kNm 2 Bending moment at column base (1) M 2.Ed M LC1 M 2.Ed = 25.4 kNm 1 Axial force in part 1-2 (compression) N Ed N LC1 N Ed = 138 kN 1 6.1.3 Classification of the cross section in y-y-axis bending β w bending a) Web b1 [1] 5.4.3 b1 bw t1 tw β w 0.40 . β w 9.76 = t1 250 . newton [1] Tab. 5.1 ε β 1w 11 . ε β 1w 10.786 = fo 2 mm Heat treated, unwelded = no β 2w 16 . ε β 2w 15.689 = longitudinal weld β 3w 22 . ε β 3w 21.573 = class w if β w β 1w , if β w > β 2w , if β w > β 3w , 4 , 3 , 2 , 1 > class w = 1 [1] 5.4.5 Local buckling β w 32 220 ρ cw if 22 , 1.0 , ρ cw 1 = ε β w β w 2 ε ε t w.ef.b if class w 4 , t w . ρ cwt w , ( b = bending) t w.ef.b = 5.0 mm b) Flanges [1] 5.4.3 ψ 1 0.8 [1] (5.7.),(5.8.) g if ψ > 1 , 0.7 0.3 . ψ , g=1 1 ψ b tw 2 .r b2 b2 t2 tf β f g. β f 4.821 = 2 t2 [1] Tab. 5.1 ε = 0.981 β 1f 3 . ε β 1f 2.942 = β 2f 4.5 . ε β 2f 4.413 = β 3f 6 . ε β 3f 5.883 = class f if β > β 1f , if β f > β 2f , if β f > β 3f , 4 , 3 , 2 , 1 f class f = 3 TALAT 2710 30
  31. 31. [1] 5.4.5 Local buckling: β f 10 24 ρ cf if 6 , 1.0 , ρ cf 1 = ε β f β f 2 ε ε t f.ef if class f 4 , t f . ρ cft f , t f.ef = 14.0 mm Classification of the cross-section in y-y axis bending class y if class f > class w , class f , class w class y = 3 6.1.4 Classification of the cross section in z-z-axis bending Cross section class of web: No bending stresses class w 1 Cross section class for flanges: According to above class f = 3 class z if class f > class w , class f , class w class z = 3 6.1.5 Classification of the cross section in axial compression β wc compression a) Web b1 b1 bw t1 tw β wc β wc = 24.4 t1 [1] Tab. 5.1 β 1w 10.786 = β 2w 15.689 = β 3w 21.573 = class wc if β wc β 1w , if β wc > β 2w , if β wc > β 3w , 4 , 3 , 2 , 1 > class wc = 4 [1] 5.4.5 Local buckling β wc 32 220 ρ cw if 22 , 1.0 , ρ cw 0.931 = ε β wc β wc 2 ε ε t w.ef if class wc 4 , t w . ρ cwt w , t w = 5 mm t w.ef = 4.7 mm b) Flanges Same as in bending t f.ef = 14.0 mm class f = 3 Classification of the total cross-section in axial compression class c if class f > class wc , class f , class wc class c = 4 TALAT 2710 31
  32. 32. 6.1.6. Welds [1] 5.5 HAZ softening at column ends [1] Tab.5.2 ρ haz 0.65 [1] Fig.5.6 Extent of HAZ (MIG-weld) t1 tf b haz if t 1 > 6 . mm , if t 1 > 12 . mm , if t 1 > 25 . mm , 40 . mm , 35 . mm , 30 . mm , 20 . mm b haz = 35 mm 6.1.7 Design resistance, y-y-axis bending [1] 5.6.1 W Elastic modulus of the gross cross section el: 2 .b .t f 2 .t f .t w A g = 4.86 . 10 mm 3 2 Ag h 1. . 3 tw . h 2 .t f 3 Ig bh b 12 I g = 2.341 . 10 mm 7 4 I g .2 W el = 2.926 . 10 mm 5 3 W el h W Plastic modulus of the gross cross section pl: 1. b .h tw . h 2 .t f W pl = 3.284 . 10 mm 2 2 5 3 W pl b 4 TALAT 2710 32
  33. 33. W Elastic modulus of the effective cross section effe: t f = 14 mm t f.ef = 14 mm bw As tf.ef = tf then bc b c = 61 mm 2 t w = 5 mm t w.ef.b = 5 mm bf 0.5 . b tw 2 .r b f = 67.5 mm 2 .b f . t f b c. t w A eff = 4.86 . 10 mm 3 2 A eff Ag t f.ef t w.ef.b Shift of gravity centre: 2 h tf bc 1 e ef 2 .b f . t f t f.ef . . t w t w.ef.b . e ef = 0 mm 2 2 2 A eff Second moment of area wiht respect to centre of gross cross section: 2 3 h tf bc 2 .b f . t f t f.ef . . t I eff = 2.341 . 10 mm 7 4 I eff Ig w t w.ef.b 2 2 3 Second moment of area wiht respect to centre of effective gross section: e ef . A eff I eff = 2.341 . 10 mm 2 7 4 I eff I eff I eff W eff = 2.926 . 10 mm 5 3 W eff h e ef 2 [1] Tab. 5.3 Shape factor α - for class 1 or 2 cross-sections: W pl α 1.2.w α 1.2.w = 1.122 W el - for welded, class 3 cross-sections: TALAT 2710 33
  34. 34. [1] Tab. 5.3 Shape factor α - for class 1 or 2 cross-sections: W pl α 1.2.w α 1.2.w = 1.122 W el - for welded, class 3 cross-sections: β 3w β w W pl W el [1] (5.16) α 3.ww 1 . α 3.ww 1.245 = β 3w β 2w W el β 3f β f W pl W el [1] (5.16) α 3.wf 1 . α 3.wf= 1.088 β 3f β 2f W el β, β2, β3 are the slenderness parameter and the limiting values for the most critical element in the cross-section, so it is the smaller value of3.ww and α3.wf α α 3.w if α 3.ww α 3.wf , α 3.ww , α 3.wf α 3.w 1.088 = W eff - for class 4 cross-sections: α 4.w α 4.w = 1 W el class y = 3 α y if class y > 2 , if class y > 3 , α 4.w , α 3.w , α 1.2.w α y 1.088 = M Design moment of resistance of the cross section c,Rd f o . α .y el W [1] (5.14) M y.Rd M y.Rd = 75.3 kNm γ M1 6.1.8 Design resistance, z-z-axis bending Cross section class class z = 3 t f .b 3 2. I z = 7.875 . 10 mm 6 4 Gross cross section: Iz 12 t f.ef . b 3 2. I z.ef = 7.875 . 10 mm 6 4 Effective cross section: I z.ef 12 I z.2 I z.ef . 2 Section moduli: Wz W z.ef b b Wz Shape factor: α z α z=1 W z.ef f o . α .z z W Bending resistance: M z.Rd M z.Rd = 24.818 kNm γ M1 TALAT 2710 34
  35. 35. 6.1.9 Axial force resistance, y-y buckling [1] 5.8.4 Cross section area of gross cross sectionAgr b .h tw . h 2 .t f A gr = 4.86 . 10 mm 3 2 A gr b Cross section area of effective cross sectionAef t w.ef = 4.653 mm 2 .b 2 . t f b w. t w A ef = 4.818 . 10 mm 3 2 A ef A gr t f.ef t w.ef ( t f = 14 mm t w = 5 mm 2 . b 2 = 135 mm t f.ef = 14 mm t w.ef = 4.653 mm ) A ef Effective cross section factor η η = 0.991 A gr Second moment of area of gross cross sectionIy: 2 2. . 3 h tf 1. 2 .b .t f . 2 .t f .t w 3 Iy btf h 12 2 12 [1] Table 5.7 Buckling length factor Ky 1.5 L1=3 m Case 5 l yc K y .L 1 l yc = 4.5 m π .E .I y 2 Buckling load N cr 2 l yc N cr = 798.639 kN A gr . η . f o [1] 5.8.4.1 Slenderness parameter λ y λ y 1.252 = N cr [1] Table 5.6 α if ( heat_treated 1 , 0.2 , 0.32 ) α = 0.2 λ o if ( heat_treated 1 , 0.1 , 0 ) λ o 0.1 = 0.5 . 1 α . λ y 2 φ λ o λ y φ = 1.399 1 χ y χ y = 0.494 2 2 φ φ λ y [1] Table 5.5 Symmetric profile k1 1 [1] Table 5.5 No longitudinal welds k2 1 fo Axial force resistance N y.Rd χ .y . k 1 . k 2 . η .A gr N y.Rd = 562.6 kN γ M1 TALAT 2710 35

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