Recitation 9: Buckling of Sections

Structural Mechanics 2.080 Recitation 9 Semester Yr Recitation 9 Buckling of Sections Consider the box column shown: Find the critical buckling load. Column can buckle globally (Euler) or locally (plate) Global (Euler) Buckling Pc = ⇡ 2 EI L2 Ix = 2 b 1 h3 + b1 h 12 ✓ b2 2 ◆2 ! +2 hb32 12 b1 h3 b1 b22 h b32 h + + 2 6 6 ✓ ◆2 ! 3 b2 h hb3 b1 Iy = 2 + b2 h +2 1 12 2 12 = = b2 h3 b21 b2 h b31 h + + 2 6 6 Ix < Iy ! Will buckle about x-axis ⇡2E So Pc = 2 L ✓ b1 h3 b1 b22 h b32 h + + 6 2 6 ◆ Local (Plate) Buckling Treat each side as an individual plate. 9-1 Structural Mechanics 2.080 Recitation 9 Semester Yr * Assume uniform compression ! stress in each plate is the same. 1 i1 = Plate Q: P1 hb1 2 i2 = Plate Q: P2 hb2 For a plate simply supported on the loaded edges: Pc = ✓ D= kc ⇡ 2 D b Eh3 12(1 _ ⌫ 2 ) (where kc depends on BC on other sides and dimensions) ◆ So icr,1 = Pc1 kc1 ⇡ 2 D = hb1 hb21 and icr,2 = kc2 ⇡ 2 D Pc2 = hb2 hb22 All plates buckle at the same time, so icr,1 = icr,2 ⇡2D kc1 hb12 kc2 ⇡ 2 D = ! kc2 = kc1 hb22 Total load = 2P1 + 2P2 ! Pc,tot ✓ ✓ b2 b1 ◆2 kc1 ⇡ 2 D kc2 ⇡ 2 D = 2Pc1 + 2Pc2 = 2 + b2 b1 " # ✓ ◆2 1 1 b2 = 2⇡ 2 Dkc1 + · b1 b2 b1 ✓ ◆ 2⇡ 2 Dkc1 b2 = 1+ b1 b1 9-2 ◆ Structural Mechanics 2.080 Recitation 9 Semester Yr But what is kc1 ?? - In general, the adjacent plates on the unloaded edges will cause a bending moment (some­ where between simply supported and fully clamped) b2 (Plot on page 16 gives kc1 as function of ) b1 (assumes L/b1 > 5) Example h = 2 mm b1 = 100 mm b2 = 50 mm Find the length, L, that marks transition between global and local buckling. Pc,global = = Pc,local = = = ✓ ◆ ⇡ 2 E b1 h3 b1 b22 h b32 h + + 6 2 6 L2 ✓ ◆ 2 3 ⇡ E 100(2) 100(50)2 (2) 503 (2) ⇡2E + = + (291, 800 mm4 ) 6 2 6 L2 L2 ✓ ◆ 2⇡ 2 Dkc1 b2 1+ (From plot: kc1 ' 5.2) b1 b1 ✓ ◆ 2⇡ 2 Eh3 kc1 b2 1 + b1 b1 (12)(1 _ ⌫ 2 ) 2 3 2⇡ E(2) (5.2) (1 + 0.5) = ⇡ 2 E(0.114 mm2 ) 100(12)(1 _ 0.32 ) Let Pc,global = Pc,local and solve for L: ⇡2E (291, 800) = ⇡ 2 E(0.114) L2 L ' 1600 mm = 1.6 m 9-3