TALAT Lecture 2301: Design of Members Examples 6.1 - 6.6: Shear resistance of webs without and with stiffeners

1,601 views
1,498 views

Published on

This example provides calculations on shear force of members based on Eurocode 9.

Published in: Education
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
1,601
On SlideShare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
63
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

TALAT Lecture 2301: Design of Members Examples 6.1 - 6.6: Shear resistance of webs without and with stiffeners

  1. 1. TALAT Lecture 2301 Design of Members Shear Force Example 6.1 – 6.6 : Shear resistance of webs without and with stiffeners 11 pages Advanced Level prepared by Torsten Höglund, Royal Institute of Technology, Stockholm Date of Issue: 1999  EAA - European Aluminium Association TALAT 2301 – Examples 6.1 – 6.6 1
  2. 2. Example 6.1 - 6.6 Shear resistance of webs without and with stiffeners Dimensions and strength of material Data input in highlighted regions Web depth hw 2000 . mm MPa 1000000 . Pa Web thickness tw 15 . mm kN 1000 . newton Total length of web panel lw 4000 . mm kg spv 2.4 . (Mass) 355 . MPa ( 0.1 . m ) 3 0,2 proof strength of web plate material f ow Ultimate strength of web plate material f uw 470 . MPa Elastic modulus E 70000 . MPa Partial coefficient γ M1 1.1 f uw Table 5.12 η 0.4 0.2 . η = 0.665 f ow 1) No intermediate stiffener If "rigid end post" thenendpost = 1 else endpost = 0 endpost 1 a Length of web panel a lw a = 4000 mm =2 hw Ref. to Eurocode 9 2 2 a hw hw (5.97) kτ if > 1.00 , 5.34 4.00 . , 4.00 5.34 . hw a a k τ = 6.34 0.81 . h w . f ow (5.96) λ w tw E λ w 3.055 = kτ 1.32 0.48 Table 5.12 ρ v if λ w > 0.949 , , 1.66 λ w λ w ρ v= 0.28 Table 5.12 ρ v if ρ v > η , η , ρ v ρ v= 0.28 0.48 0.48 Table 5.12 ρ v if endpost 0 , if ρ v > , ,ρ v ,ρ v λ w λ w ρ v= 0.28 f ow ρ .t w .h w . V w.Rd = 2.711 . 10 kN 3 (5.95) V w.Rd v γ M1 TALAT 2301 – Mass1 h w6.1 .–w . spv Weight of web Examples . t w l 6.6 2 Mass1 = 288 kg V Rd V w.Rd 1
  3. 3. 2) Equally spaced flexible transverse stiffeners If "rigid end post" thenendpost = 1 else endpost = 0 endpost 1 Number of stiffeners n st 3 Single plate stiffener t st 15 . mm b st b st 120 . mm =8 t st Whole web panel. Buckling of stiffener t st . b st 30 . t w A st = 8.55 . 10 mm 2 3 2 Figure 5.21 A st t st . b st 2 C.G. e st e st = 12.632 mm 2 . A st t st . b st Second 3 A st . e st I st = 7.276 . 10 mm moment 2 6 4 I st of area 3 a = 4 . 10 mm 3 Panel length a lw 3 2 4 hw I st (5.99) k τ st 9. . = k τ st 2.38 t w .h w a 3 1 3 2.1 . I st (5.99) k τ stmin = k τ stmin 2.153 tw hw k τ st if k τ st < k τ stmin , k τ stmin , k τ st = k τ st 2.38 (5.97) 2 2 a hw hw (5.98) kτ if > 1.00 , 5.34 4.00 . , 4.00 5.34 . k τ = 6.34 hw a a kτ kτ k τ st kτ = 8.72 0.81 . h w . f ow (5.96) λ w λ w 2.605 = tw E kτ λ wtot λ w TALAT 2301 – Examples 6.1 – 6.6 3
  4. 4. Sub panel lw a a = 1 . 10 mm 3 Length a = 0.5 n st 1 hw 2 2 a hw hw (5.97 and 5.98) k τ if > 1.00 , 5.34 4.00 . , 4.00 5.34 . k τ = 25.36 hw a a 0.81 . h w . f ow (5.96) λ w λ wsub λ w λ wsub = 1.527 tw E kτ λ wtot 2.605 = λ λ The larger of the slenderness parameter wtot for the total panel and wsub for the sub panels is used. If λwtot > λwsub then the stiffeners are flexible else the stiffeners are rigid. λ w if λ wtot > λ wsub , λ wtot , λ wsub λ w 2.605 = 1.32 0.48 Table 5.12 ρ v if λ w > 0.949 , , ρ v if ρ v > η , η , ρ ρ v= 0.31 v 1.66 λ w λ w 0.48 0.48 Table 5.12 ρ v if endpost 0 , if ρ v > , ,ρ v ,ρ v ρ v= 0.31 λ w λ w f ow ρ .t w .h w . V wRd = 2.997 . 10 kN 3 (5.95) V wRd v γ M1 Alternative 2 = transverse stiffeners V Rd V wRd 2 Weight Mass2 h w .t w .l w n st . b st . t st . h w . spv Mass2 = 314 kg TALAT 2301 – Examples 6.1 – 6.6 4
  5. 5. 3) Transverse intermediate, rigid stiffeners Number of transverse stiffeners n st 3 endpost = 1 Web panel lw a a = 1 . 10 mm 3 length a = 0.5 n st 1 hw 2 2 a hw hw (5.97) kτ if > 1.00 , 5.34 4.00 . , 4.00 5.34 . k τ = 25.36 hw a a 0.81 . h w . f ow (5.96) λ w λ w 1.527 = tw E kτ 1.32 0.48 Table 5.12 ρ v if λ w > 0.949 , , ρ v= 0.414 1.66 λ w λ w Table 5.12 ρ v if ρ v > η , η , ρ v ρ v= 0.414 f ow ρ .t w .h w . V wRd = 4.01 . 10 kN 3 (5.95) V wRd v γ M1 t st 18 . mm b st 220 . mm Check rigidity of stiffener t st . b st 30 . t w A st = 1.071 . 10 mm 2 4 2 Figure 5.21 A st t st . b st 2 C. G. e st e st = 40.672 mm 2 . A st t st . b st Second 3 A st . e st I st = 4.617 . 10 mm moment 2 7 4 I st of area 3 h .t 3 3 (5.104) or a . w w , 0.75 . h . t 3 I limit = 4.05 . 10 mm 7 4 (5.105) I limit if < 2 , 1.5 w w hw 2 a Ist > Ilimit OK! Axial force 1 1.4 . t w 2. in stiffener N st V wRd E . f ow . γ M1 TALAT 2301.06 N st f ow (6.03h) σ st σ st = 241 MPa = 323 MPa σst < fow OK! A st γ M1 Alternative 3 = transverse intermediate, rigid stiffeners V Rd V wRd 3 TALAT 2301 – Examples 6.1 –. 6.6 . 5 Weight Mass3 hw tw lw n st . b st . t st . h w . spv Mass3 = 345 kg
  6. 6. 4) Rigid transverse stiffeners and longitudinal stiffeners Number of longitudinal stiffeners n sl 2 b sl Longitudinal plate stiffener t sl 15 . mm b sl 120 . mm =8 t sl Number of rigid transverse stiffeners n st = 3 If "rigid end post" thenendpost = 1 else endpost = 0 endpost 1 Buckling of longitudinal stiffener t sl. b sl 40 . t w A sl = 1.08 . 10 mm 2 4 2 Area A sl t sl. b sl 2 GC e sl e sl = 10 mm 2 . A sl t sl. b sl Second 3 A sl. e sl I sl = 7.56 . 10 mm moment 2 6 4 I sl of area 3 lw Panel length a a = 1 . 10 mm 3 n st 1 3 n sl. I sl 2 4 hw (5.99) k τ st 9 . . = k τ st 65.916 t w .h w a 3 1 . 3 2.1 . n sl I sl (5.99) k τ stmin = k τ stmin 2.748 tw hw (5.99) k τ st if k τ st < k τ stmin , k τ stmin , k τ st = k τ st 65.916 2 hw (5.97) kτ 5.34 4.00 . k τ st kτ = 87.256 a 0.81 . h w . f ow (5.96) λ w λ w 0.823 = tw E kτ Whole panel λ wtot λ w λ wtot = 0.823 TALAT 2301 – Examples 6.1 – 6.6 6
  7. 7. Sub panel hw Depth h1 h 1 = 667 mm 3 2 h1 (5.98) kτ 4.00 5.34 . k τ = 6.373 a 0.81 . h 1 . f ow λ w 0.823 = (5.96) λ w tw E kτ Sub-panel λ wsub λ w λ wsub = 1.016 compare λ wtot 0.823 = λ λ The larger of the slenderness parameter wtot for the total panel and wsub for the sub panels is used. If λwtot > λwsub then the stiffeners are flexible else the stiffeners are rigid. λ w if λ wtot > λ wsub , λ wtot , λ wsub λ w 1.016 = 1.32 0.48 Table 5.12 ρ v if λ w > 0.949 , , ρ v= 0.493 1.66 λ w λ w Table 5.12 ρ v if ρ v > η , η , ρ v 0.48 0.48 Table 5.12 ρ v if endpost 0 , if ρ v > , ,ρ v ,ρ v ρ v= 0.493 λ w λ w f ow ρ .t w .h w . V wRd = 4.78 . 10 kN 3 (5.95) V wRd v γ M1 Alternative 4 = transverse and longitudinal intermediate stiffeners V Rd V wRd 4 Weight Mass4 h w .t w .l w n st . b st . t st . h w n sl. t sl. b sl. l w . spv Mass4 = 380 kg TALAT 2301 – Examples 6.1 – 6.6 7
  8. 8. 5) Shear resistance contribution of the flanges added to girder 4 The panel is at the end of the plate girder. Then the bending moment is neglected Flange thickness and width tf 50 . mm bf 750 . mm Yield strength of flange plate f of 355 . MPa Design shear force V Ed 6000 . kN V Ed . l w M Ed = 2.4 . 10 kN . m 4 Design bending moment M Ed f of (5.101) M fRd b f .t f . h w tf . M Ed M fRd = 2.481 . 10 kN . m 4 γ M1 = 0.967 M fRd a = 1 . 10 mm 3 4.4 . b f . t f f of 2. (5.101) c 0.08 .a c = 217.5 mm t w . h w . f ow 2 b f . t f . f of 2 2 M Ed (5.101) V fRd if M Ed < M fRd , . 1 ,0 V fRd = 178.626 kN c . γ M1 M fRd Sum V Rd V wRd V fRd 5 V Rd = 4.955 . 10 kN 3 Example 5 = inclusive shear resistance contribution of the flanges 5 Weight Mass5 h w .t w .l w n st . b st . t st . h w n sl. t sl. b sl. l w . spv Mass5 = 380 kg TALAT 2301 – Examples 6.1 – 6.6 8
  9. 9. 6) Corrugated web Web depth hw 2000 . mm MPa 1000000 . Pa Web thickness tw 12 . mm kN 1000 . newton Total length of web panel lw 4000 . mm 0,2 proof strength of web plate material f ow 355 . MPa Ultimate strength of web plate material f uw 470 . MPa Elastic modulus E 70000 . MPa Trapezoidal web: bo 140 . mm bu 140 . mm bd 400 . mm hc 100 . mm b u . 0.5 2 2 sw bd bo hc s w = 116.619 mm Partial coefficient γ M1 1.1 f uw Table 5.12 η 0.4 0.2 . η = 0.665 f ow Local buckling of bo , bu and bu Max width bm if b o < b u , b u , b o b m = 140 mm bm if b m > s w , b m , s w b m = 140 mm b m f ow (5.96) λ w 0.35 . . tw E λ w 0.291 = Table 5.12 0.48 0.48 Non-rigid ρ v if λ w < ,η , η = 0.665 ρ v= 0.665 end post η λ w f ow 0.7 . ρ . t w . h w . V wRd = 3.604 . 10 kN 3 (5.117) V wRd v γ M1 TALAT 2301 – Examples 6.1 – 6.6 9
  10. 10. Trapezoidal web, widthbd b o .t w b u .t w 2 .s w .t w A = 6.159 . 10 mm 3 2 Area A Gravity b o .t w .h c 2 . s w . t w . 0.5 . h c centre e gc e gc = 50 mm A Second 2 hc 2 1 b o .t w .h c 2 .s w .t w . A . e gc . I x = 2.683 . 10 mm moment 2 4 3 Ix of area 3 bd 3 bd t (5.122) Iz . w I z = 123.554 mm 3 bu bo 2 . s w 10.9 1 60 . E . 3 4 I z.I x V o.cr = 1.468 . 10 kN 4 (5.121) V o.cr hw h w . t w . f ow (5.120) λ ow λ ow 0.762 = V o.cr 0.7 . ρ v= 0.465 0.60 χ o (5.119) 2 χ o if χ o > 0.7 . ρ v , 0.7 . ρ v , χ o χ o 0.435 = 0.8 λ ow f ow χ .o w . t w . V o.Rd = 3.366 . 10 kN 3 (5.118) V o.Rd h γ M1 V Rd = 3.366 . 10 kN 3 Min V Rd if V wRd > V o.Rd , V o.Rd , V wRd 6 6 bu bo 2 .s w Weight Mass6 t w .l w .h w . . spv Mass6 = 296 kg bd TALAT 2301 – Examples 6.1 – 6.6 10
  11. 11. Weight / Summary resistance Increase in V Rd resistance Massi Massi 1 i = . = = Mass1 Mass1 f i i = 1000 . kN f i = 1) No intermediate stiffeners t w1_5 = 15 mm 1 2.711 1 1 1 2) Transverse flexible stiffeners 2 2.997 1.106 1.09 0.986 3) Transverse rigid stiffeners 3 4.01 1.479 1.198 0.81 4) Transverse rigid + longitudinal stiffeners 4 4.777 1.762 1.318 0.748 5) Transverse rigid + longitudinal stiffeners 5 4.955 1.828 1.318 0.721 + contribution of flanges 6 3.366 1.242 1.026 0.827 6) Trapezoidal web t w = 12 mm f ow "No buckling" V 0.Rd η . h w . t w1_5 . V 0.Rd = 6.436 1000 . kN γ M1 Comments The second column gives the increase in shear resistance compared to girder 1 when adding stiffeners. By adding both transversal and horizontal stiffeners (5) the resistance can be almost doubled. The last column, "weight per resistance" show that stiffeners give lighter girders, but the comparison does no pay regard to the cost for welding of the stiffeners. The contribution of the flanges (girder 5) increase the resistance with = 3.7 % a compared to girder 4 Torsten Höglund TALAT 2301 – Examples 6.1 – 6.6 11

×