WORCESTER POLYTECHNIC INSTITUTE

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1 WORCETER OLYTECHNIC INTITUTE MECHANICAL ENGINEERING DEARTMENT DGN OF MACHINE ELEMENT ME-330, B 018 Lecture November 018

2 Examples w Eccentric load Concentric load w

3 Examples Vestas V80-.0 MWatt Installing a nacelle M = 98 tons m

4 lenderness ratio: r Buckling of a column Definitions: r l k, with k I A hort columns: r 10 Calculated based on compression stress criterion: x A Intermediate/long columns: 10 r Calculated based on critical unit load criterion: cr A

5 Long columns: concentric load Buckling of a column Bending moment: M y For small deflections: (Governing ODE) d y dx M y d y dx 0 Deflection olution (deflection) indicates: y ( x) C 1 sin x C cos x

6 Long columns Deflection: y ( x) C 1 sin x C cos x C 1 and C are determined from boundary conditions (end conditions) ossible end conditions (make sure you understand BC s in terms of slope and deflection):

7 Long columns: end conditions + critical load cr Deflection: y ( x) C 1 sin x C cos x For the rounded-rounded end conditions: (1) y( x 0) 0 y( x l) 0 C 0 () C sin l 0 Indicating that: l n π; n 1,, Many solutions. Therefore, n n π l E I ; n 1,,3... Many critical loads.

8 Long columns: end conditions + critical load cr Typically, designs are based on the smallest critical load. Therefore, cr E I ; for n 1 l using: I A k and r l k cr E Critical load per unit area in ; for n 1 terms of slenderness ratio r A Corresponding deflection: y C 1 x sin l For BC s: (1) () y( x 0) 0 y( x l) 0

9 Accounting for different BC s In order to take into account other boundary conditions, the concept of effective length, l eff, is introduced: r l k eff A cr E r

10 Designing columns General procedure 1) Determine force to be supported and expected length of the column (design objective and corresponding constraint(s)). Use free-body diagram(s) and equilibrium conditions. ) Determine cross-section parameters of a proposed column: Area, A Moment of inertia, I Radius of gyration, k 3) Determine slenderness ratio, r Identify boundary conditions (BC s) and apply appropriate value for the effective length l eff -- use appropriate table (Table 4-3.) l k eff 4) Identify material to use and its corresponding compressive yield strength, yc, and elastic modulus, E.

11 Designing columns General procedure 4) Determine slenderness ratio at half-yield: ( r ) D is obtained as follows: ( ) r D (Design criterion) (a) et the load per unit area as: (b) Therefore, yc r E cr yc (Half the yield strength value in compression) A (c) olve for slenderness ratio -- using previous equation, (b). This is the slenderness ratio at half-yield in compression. ( ) r D E yc

12 Designing columns General procedure 5) Determine type of column based on proposed design: Johnson: if r (step 3) < ( r ) D (tep 4) Euler: if r (step 3) > ( r ) D (tep 4) 6) Determine critical load Johnson: cr A yc 1 E yc r Euler: cr E A r

13 Designing columns General procedure 7) Determine allowed load: allowed cr, is the critical load (step 6) F, is the security factor ( > 1) cr F 8) If allowed is > than force to be supported, then, a satisfactory design has been obtained -- not necessarily the optimal!! Go to step (1) and refine your design, if possible (e.g., weight minimization) 9) If allowed is < than force to be supported, then, go to step (1) and improve design. You can select a different section and/or material. Use of MathCad is strongly recommended!!

14 Eccentrically loaded Eccentrically loaded column Moments: M A M ( e y) 0 For small deflections: (Governing ODE) d dx M y d y dx y e Deflection For BC s: y dy dx x l x 0 0; 0 l l y x esec 1

15 Eccentrically loaded Maximum moment: M max l e y l e sec x M c M c Compressive stress: c A I A Ak with the maximum stress level at M M max which yields c e c l, max 1 sec setting c A k k 4E A, max yc A e c k 1 yc l sec k eff 4 E A olve for to obtain the critical load (ecant column formula)

16 Reading assignment Chapters 4 of textbook: ections 4.1 to 4.19 Review notes and text: E501, E50 Homework assignment Author s: 4-10, 4-11 olve: (4-33, 4-34, 4-35, 4-36) rows f and g, 4-4, 4-51 (all cases), 4-5 (all cases)