Column Buckling.

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1 Column Buckling

2 Contents Stability and Buckling( 稳定性与失稳 ) Examples of Columns( 压杆应用示例 ) Conventional Design of Columns( 压杆的常规设计方法 ) Euler s Formula for in-ended Columns( 端部铰接压杆欧拉公式 ) Buckling Modes( 失稳模态 ) Extension of Euler s Formula( 欧拉公式的扩展 ) Buckling in Orthogonal lanes( 相互垂直平面内的失稳 ) Ways to Improve Column Stability( 提高压杆稳定性的途径 ) Applicability of Euler s Formula( 欧拉公式的适用范围 ) Failure Diagram of Columns( 压杆的失效总图 ) Critical Stress of Columns( 压杆临界应力的确定 ) Design of Columns( 压杆的稳定性设计方法 ) Eccentrically oaded Columns( 偏心压杆 )

$(fracture/yielding), stiffness$ (deformation) and fatigue iteria.

3 Stability and Buckling Stability is characterized as the ability of a structure to maintain its (stable) equilibrium under working conditions. Buckling is the behavior of a structure losing its equilibrium under working conditions. This is another type of failure iterion, in addition to strength (fracture/yielding), stiffness (deformation) and fatigue iteria. Buckling occurs suddenly and results in catastrophic accident. 3

4 Examples of Columns 4

within specifications AE After these design calculations, many discover that the

5 Conventional Design of Columns In the design of columns, osssectional area is selected such that - allowable stress is not exceeded A - deformation falls within specifications AE After these design calculations, many discover that the column is unstable under loading and that it suddenly becomes sharply curved or buckles. 5

6 Euler s Formula for in-ended Columns w w w w Consider an axially loaded beam. After a small perturbation, the system reaches a neutral equilibrium configuration such that d w M w dx EI EI dw w 0 dx EI w" k w 0, k EI w Asin kx B cos kx 0w0 w B 0; EIn k n 6

7 Buckling Modes EI 4 9 ; EI ; EI 7

8 Cantilevered Columns A A A column with one fixed and one free end, will behave as the upperhalf of a pin-connected column. B e B The itical loading is calculated from Euler s formula, EI e EI 4 A e E i 4 i e r E equivalent r length 8

Columns with Two Fixed Ends The symmetry of the supports and of the loading requires that the shear at C and the horizontal reactions at both ends be zero.

9 Columns with Two Fixed Ends The symmetry of the supports and of the loading requires that the shear at C and the horizontal reactions at both ends be zero. The equation of the deflection curve involves sine and cosine functions. oint D must be a point of inflection, where the bending moment is zero. It follows that the portion DE of the column must behave as a pin ended column. EI e EI e 9

10 Columns with One Fixed End and One Free End w w w The differential equation dw dx w Vx EI d w Vx w dx EI EI Vx w Asin kx B cos kx, k EI V 0 w0 w B 0; Asin k V 0 w Ak cos k tan k k k EI EI EIk Equivalent length: e

11 Extension of Euler s Formula e EI e EI E E ; e r r i i equivalent length; length coefficien t 11

12 Sample roblem Assuming the same material and oss-sectional dimension, which of the following four columns is most susceptible to buckling? 5l 7l 9l l (a) (b) (c) (d) EI Euler Formula l 1

13 Sample roblem For the truss shown, F, β, and AC are given. Find 0 < < /, under which column AB and BC reach itical stability simultaneously. Solution: 1. Euler s equation for column AB: F EI EI FAB Fcos lab l cos. Euler s equation for column BC: F BC EI Fsin l F cos F sin EI l cos cot EI l sin EI l sin CB 3. If column AB & BC reach itical stability simultaneously: tan A B C 13

14 Sample roblem Which of the following best desibes the relationship between itical 1 and? (a) 1 = ; (b) 1 < ; (c) 1 > ; (d) not sure. A 0 B A B a a C 1 D C 0 D a a Solution: EI a : FN AD a 1 1 EI EI b : FN AB a a 1 EI a 14

15 Sample roblem For the column shown, find the relationship between the itical loads corresponding to oss-sections desibed by (a), (b) and (c). r r r r l Solution: a b c (a) (b) (c) 4 4 EI a E r E r 64 4 l l l : : 4 EIb E r 1 1 : : l l : 1 : r 4 EI 16 3 c E E r l l 1 l 1 a b c 15

16 Buckling in Orthogonal lanes For the column shown, find the relationship between the itical loads corresponding to buckling in x-y and x-z plane respectively. Solution: l b b y z EI z z zl EI y y yl 3 bb 1 z I z 3 I bb 1 y y 4 What if the oss-section is ineased to b b? 16

17 Buckling in Orthogonal lanes Buckling in x-z: μ y = 1 y EI y z x y y Buckling in x-y: μ z = 0.5 z EI 0.5 z 17

18 Ways to Improve Column Stability Selection of materials. Deease effective column length (μ). Inease moment of inertia for a given oss-sectional area. EI e E i r EI E Coordination of end conditions and moment of inertia in orthogonal planes: For a group of columns, make each one equally stable (avoid the last straw). i i z rz y ry 18

19 Applicability of Euler s Formula Critical buckling stress EI r E Ai E E A A ir A i slenderness ratio Applicability of Euler s formula r E E σ : proportion limit; λ p : itical slenderness ratio. E 06 Ga Q35: Ma Y 35 Ma 19

20 Failure Diagram of Columns Y E i revious analyses assumed stresses below the proportional limit and initially straight, homogeneous columns e Experimental data demonstrate r - for large e /i r, follows Euler s formula and depends upon E but not Y. - for small e /i r, is determined by the yield strength Y and not E. Empirical formulas for intermediate columns - for intermediate e /i r, depends on both Y and E. 0

21 Critical Stress of Columns Y a b or a b E Y Strength analysis Stability analysis O E 1. ong columns: p 3. Short columns: Y Y Y E. Intermediate columns: a Y ab Y p Y b λ 1

22 Sample roblem For the wood column shown, E = 10 Ga, σ p = 9 Ma, σ Y = 13 Ma, = λ for intermediate columns. Find the itical loads for rectangular oss-sections: (1) h = 10 mm, b = 90 mm; () h = b = 104 mm Solution: E p z p 3m y a Y Y 8.6 b l i r p Slender column. EI 79.9 kn l. 100 Intermediate column. = A = (a bλ)a = kn y z

23 Sample roblem For the composite beam and column structure shown, E AB = E BD = E, d AB = d BD = d, :d = 30, λ.bd = 100. Find, under which BD reaches the itical condition. A EI C B EA D Deformation compatibility: 3 Ed Solution: BD : 10 i d 4 F BD EI EI 3EI EA 3 FBD FBD 3

24 Design of Columns Stability analysis of columns - Stability check - Cross-section design - Allowable load/stress 1. Method of safety factor F F F st nst F st A n st n st : safety factor [F st ]: allowable load [σ st ]: allowable stress. Method of discount factor st F A :discount/stability factor 4

25 l z =800 mm l y =770 mm Sample roblem A Q75 steel column is cylindrically pinned at both ends. p = 96, σ Y = 75 Ma, = λ for intermediate columns. Analyze the column stability for F = 60 kn and n st = 3.5. x x F F h =45mm b=0mm l z l y y z Solution: - In x-y plane, both ends are pinned. i z a Y p 96, Y 4 b Iz h zlz mm, z 61.9 A 3 i 1.99 z 5

26 - In x-z plane, both ends are fixed. i y Iy b yly mm, y 66.7 A 3 i 5.77 The column buckling first happens in the x-z plane. F Y z y p y 41 Ma 6 6 A kn 6 kn F F F F n 60 kn < 6 kn The column is stable. y 6

27 Eccentric oading: The Secant Formula Eccentric loading is equivalent to a centric load and a moment. Bending occurs for any nonzero eccentricity. Question of buckling becomes whether the resulting deflection is excessive. The deflection become infinite when = d w w e d w e w dx EI dx EI EI w Asinkx B cos kx e, k EI max 0 w 0 w tan sin cos 1 B e; Asink e 1 cos k A e tan k w e k kx kx w w e tan k sin k cos k 1 eseck 1 e sec 1 EI EI e sec 1., EI e

28 Eccentric oading: The Secant Formula ec i 0 r ec i 1 r max i For small slenderness ratio: e r ec max 1. A ir Maximum stress M c wmax A I A I ec 1 sec A i r max max ec 1 sec A i r EI ec 1 1 sec A i r ir EA Secant Formula A max ec 1 e 1 sec ir ir EA For large slenderness ratio, the curves get very close to Euler s curve, and thus that the effect of the eccentricity of the loading becomes negligible. The secant formula is chiefly useful for intermediate values of slenderness ratio. e c 8

29 Contents Stability and Buckling( 稳定性与失稳 ) Examples of Columns( 压杆应用示例 ) Conventional Design of Columns( 压杆的常规设计方法 ) Euler s Formula for in-ended Columns( 端部铰接压杆欧拉公式 ) Buckling Modes( 失稳模态 ) Extension of Euler s Formula( 欧拉公式的扩展 ) Buckling in Orthogonal lanes( 相互垂直平面内的失稳 ) Ways to Improve Column Stability( 提高压杆稳定性的途径 ) Applicability of Euler s Formula( 欧拉公式的适用范围 ) Failure Diagram of Columns( 压杆的失效总图 ) Critical Stress of Columns( 压杆临界应力的确定 ) Design of Columns( 压杆的稳定性设计方法 ) Eccentrically oaded Columns( 偏心压杆 ) 9

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS CHTER MECHNICS OF MTERILS 10 Ferdinand. Beer E. Russell Johnston, Jr. Columns John T. DeWolf cture Notes: J. Walt Oler Texas Tech University 006 The McGraw-Hill Companies, Inc. ll rights reserved. Columns