Lecture 9. Stability of Elastic Structures. Lecture 10. Advanced Topic in Column Buckling

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1 Lecture 9 Stabiity of Eastic Structures Lecture 1 Advanced Topic in Coumn Bucking robem 9-1: A camped-free coumn is oaded at its tip by a oad. The issue here is to find the itica bucking oad. a) Suggest a simpe form of the bucked of the coumn, satisfying kinematic boundary conditions. b) Use the Rayeigh-Ritz quotient to find the approximate vaue of the bucking oad. c) Come up with another bucking shape which woud give you a ower vaue for the bucking oad. d) Find the exact soution of the probem and show the convergence of the approximate soution to the exact soution. Foow the exampe of a pin-pin coumn, which is presented in the notes of Lecture 9. 1

2 robem 9-1 Soution: a) Kinematic boundary condition, in term of shape function I( x ), for a camped-free coumn is () '() Choose a bucking shape ( x) x '( x ) x ''( x ) b) Use Rayeigh-Ritz Quotient, the itica bucking oad is N c I'' I'' dx II ' 'dx u dx x u xdx N c L

3 c) Choose a bucking shape simiar to a cantiever beam I( x ) x Lx I '( x ) x 6Lx I ''( x ) 6x 6L N c I'' I'' dx II ' 'dx ³ 6x 6L dx x 6Lx dx ³ 1L 4 L 5 5 Compare to the resut in b), N c.5 L N c, this bucking shape givers a ower vaue L d) Choose bucking shape I( x) S 1 cos x L

4 S S I '(x) sin x L L S I ''( x ) cos x L L N c I'' I''dx II ' 'dx ª S S º «cos x» dx «L L ¼» S sin x dx L L ³ S 4L N c.47 L Check for oca equiibrium of the soution ª S 4 S S w IV N c w '' A «cos x cos x «L L 4L L L»¼ This is the exact soution to the camped-free bucking 4

5 robem 9-: Consider a camped-free coumn oaded by a compressive force at the free end. a) Determine the itica senderness ratio E it distinguishing between the eastic and pastic bucking response. What is the bucking stress and strain? b) Cacuate the itica pastic bucking oad for E.5E it and the corresponding stress and strain. c) Cacuate the itica eastic bucking oad for E E it and the corresponding stress and strain. d) Compare a three resuts. robem 9- Soution: a) First, find the bending oad For Camed-Free coumn S L S 4L Second, find the bucking stress and strain Reca that S A 4AL V bucking 5

6 I I r r A A Then Reca that Er E V bucking 4L 4L r L r Er E V 4L 4 V S E 4E H Third, find when V bucking V yied E E it when V y, which is S E V y 4E it S E 4V y E it E V y E it b) E.5E it, the coumn yieds + hits pastic bucking 6

7 S E S t E t p (From Lecture Note, equation 9.7) 1 E it 4 Ei t ¹ p S S n n 1 E it 4 Eit c) E it, the coumn wi bucke easticay E E E V 4 4E 16E it it V S E H 16E it d) Compare the three resuts Yied y y Eastic Bucking E.67.7 y.67 it it astic Bucking E 4 it y 4 it it E To simpy our comparison, assume n., E t.5e (*) and reca (*)In order to compare pastic bucking to eastic+yied, we need to make future assumption about the materia properties. V y E it 7

8 robem 9-: Consider the pin-pin coumn. a) Suggest a poynomia bucking shape function ( x) to improve the approximate soution derived in ecture note. Note that the one used in cass was the paraboic shape. b) Determine the accuracy reative to the exact soution. E I L x robem 9- Soution: S x x a) The exact soution is w sin, use the none-dimensioned vaue F, the Tayor L L series expansion is So we know the shape function must be SF sin SF SF 6 IF ( ) C 1 F C F For x L, the boundary conditions are The first boundary condition gives this doesn t hep. The second boundary condition gives I 1 I' F ¹ I C 1 C 8

9 d x 1 where d F dx dx C C ' 1 1 ' C1 C 4 C 4 C 1 So we have 4 ( ) C1 C1 4 ( ) C1 We can us the Rayeigh-Ritz Quotient N '' dx ' dx '( ) 1 4 x C1 d dx 4 '( ) C d dx 1 ''( x) 8C d dx ''( ) 64C d dx Since we have considered the shape function for d x d L, we must adjust the imits on the integra N ' '' 64C1 d dx dx C d dx dx 1 L dx dx x 1 dx after engthy agebra 4 x x dx 4 9

10 N 1 L b) The resut are compared with the poynomia used in cass and the exact soution Exact Soution araboic Cubic C sin C C 1 C C 1 Coefficient Error N/A 1.5% 1.% Notice how we significanty reduce the error by incuding a higher order term. 1

11 robem 9-4: resent a step-by-step derivation of the bucking soution of the pin-camped coumn from the oca equiibrium equation. E I L x robem 9-4 Soution: Boundary condition for this probem Start with 4 th order ODE ww wl w' w''l w IV w'' IV w w'' We have an eigenvaue probem 4, i 1 4 Define K 4 ik wc C x C sin Kx C cos Kx

12 Use the boundary conditions to sove for constants C, 1 C, C and C 4 w w ' w L C 1 C L w L Substitute C C 1, into the above expression w'' L w C1C4 C C 1 4 w' x C KC cos KxKC sin Kx 4 w' C KC C KC C sin KL C 4 cos KL C KLsin KL C 1cos KL 4 w'' x K C sin KxK C cos Kx w'' L K C sin KLK C cos KL 4 4 det> KL sin KL 1cos KLC K sin KL K cos KL C4 K KL KL KL K KL KL cos sin sin 1cos KL cos KL sin KL sin KL KL tan KL cos KL So the equation to sove in order to find is tan KL KL The smaest roots are KL and KL 4.49, we choose KL

13 L L.7L.7L 1

14 robem 9-5: a) Derive the soution for an imperfect camped-free coumn (ike that considered in probem 9-1, foowing a simiar derivation given in the notes for a pin-pin coumn in the notes. b) Find the ratio of current defection ampitude to the ampitude of the initia imperfection such that the resuting oad is 8% of the theoretica bucking oad of a perfect coumn. robem 9-5 Soution: a) wx : shape of initia imperfection wx : actua bucked shape w o x : ampitude of initia imperfection w o : end ampitude of actua imperfection Moment equiibrium of imperfect coumn w w'' w w o erfect coumn w x Assume that the initia imperfection is in the same shape as the bucking shaper From boundary condition o 1cos w x w Ox w x w o 1 cos Ox wl 14

15 w o O cos OL From moment equiibrium of imperfect coumn erfect coumn L n 1 w wo cosx wo 1 1cosx wo w wo w o S E I O 4L Imperfect coumn w o w o w o w o 1 4L wo wo 1 w o b) When.8 w w o o 1. w w o o 5 15

b) What shoud the ratio of b 1 /b be in order for the probabiity of bucking in either of the bucking panes to be the same?

16 robem 9-6: The pin-pin eastic coumn of ength L (shown beow) is an I section can bucke in either pane. a) Determine the bucking oad in terms of L, b 1,b, t and E. Assume that t<<b. b) What shoud the ratio of b 1 /b be in order for the probabiity of bucking in either of the bucking panes to be the same? Bonus: What coud happen for very arge width to thickness ratio? b b 1 L x robem 9-6 Soution: a) The moment s of inertia for an I shape oss-section is I I zz 1 b b t 1 1 ¹ 1 tb b 1 6b 1 yy tb 1 1 tb 1 1 tb 1 tb

17 If I yy I zz, the coumn wi bucke in x-z pane S yy S E tb b 6b 1 1 If I yy I zz, the coumn wi bucke in x-y pane S S zz E tb 6 1 b) For the probabiity of bucking in either of the panes to be the same, we want I yy I zz The ony physica soution is 1 tb b 6b 1 1 tb b 1 b1 1 b b b b c) If b 1!! t, b!! t, then oca pate bucking my deveop. 17

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