Explain shortly the meaning of the following eight words in relation to shells structures.

Transcription

1 Delft University of Technology Fculty of Civil Engineering nd Geosciences Structurl Mechnics Section Write your nme nd study number t the top right-hnd of your work. Exm CIE4143 Shell Anlysis Tuesdy 15 August 017, 13:30 16:30 hours Problem 1 Explin shortly the mening of the following eight words in reltion to shells structures. Membrne forces Edge bem Snders-Koiter equtions Knockdown fctor Gussin curvture Elpr Love Abutment force Problem Figure 1 shows prt of shell nd edge bem with coordinte system. Drw in this figure the shell loding, membrne forces, moments nd sher forces in the positive directions. (You cn hnd in this pge with your nswers.) Figure 1. Prt of shell b Does k xx hve positive vlue or negtive vlue? (fig. 1) Does k yy hve positive vlue or negtive vlue? (fig. 1) Does k G hve positive vlue or negtive vlue? (fig. 1) 1

2 Problem 3 Which vribles re in the Snders-Koiter equtions? Choose A, B, C or D. A shpe xuv (, ), y( uv, ), zuv (, ) B slopes z z ( uv, ), ( uv, ) x y C curvtures kxx ( uv, ), kyy ( uv, ), kxy ( uv, ) D Lme prmeters x( uv, ), y( uv, ) Consider Tylor expnsion of k xx in u = v = 0. k = b + bu+ bv+ bu + buv+ bv + bu + buv+ buv + b v xx b Which constnts will end up in the solution of p z? Choose A, B, C or D. A b 1 to b 10 B b to b 10 C b 4 to b 10 D b 7 to b 10 c We wnt to mke the expnsion of kxx s short s possible. We lso wnt to compute the exct vlue of p z in point u = v = 0. Which constnts should be included in the expnsion? Choose A, B, C or D. A b 1 B b 1 to b 3 C b 1 to b 6 D b 7 to b 10 d Consider two shell prts tht re connected in the line u = 0. The curvtures of the prts hve Tylor expnsions with the sme vlues of b 1 nd b. The other constnts in the Tylor expnsions re different. Wht is the level of continuity of the connection? Choose A, B, C or D. A B C D C 0 continuity C 1 continuity C continuity C 3 continuity e Wht level of continuity in the shpe is required to prevent ny edge disturbnce? A B C D C 0 continuity C 1 continuity C continuity C 5 continuity

3 f Wht is the level of continuity between shell finite elements? A B C D C 0 continuity C 1 continuity C continuity C 5 continuity Consider Tylor expnsion of x in u = v = x = b1+ bu + bv 3 + bu 4 + buv 5 + bv 6 + bu 7 + buv 8 + buv 9 + b10v. g Which constnts will end up in the solution of p z? Choose A, B, C or D. A b 1 to b 10 B b to b 10 C b 4 to b 10 D b 7 to b 10 h We wnt to mke the expnsion of x s short s possible. We lso wnt to determine the exct vlue of p z in point u = v = 0. Which constnts should be included? Choose A, B, C or D. A b 1 B b 1 to b 3 C b 1 to b 6 D b 7 to b 10 i Consider improving the Snders-Koiter equtions with geometricl nonlinerity (lrge deformtions). In theory, how cn this be done? Choose A, B, C or D. A Replce xyz,, by x + ux, y + uy, z + u z. B Replce kxx, kyy, k xy by κ κ κ 1 xx xx, κyy yy, κ xy ρ xy. C Replce kxx, kyy, k xy by κxx +κ xx, κyy +κ yy, κ xy +ρxy. D Replce kxx, kyy, k xy by κxx, κyy, ρxy. 3

4 Problem 4 A shell structure hs the shpe of hlve torus (fig. ). The dimeter is m. The dimeter of the torus centre line is 8 m. The shell wll is 4 mm thick. One edge is fixed. A point lod of 75 kn is pplied to the free edge in the positive z direction. The principl membrne forces re shown in figure 3. Sher deformtion hs not been included in the computtion (Kirchhoff insted of Mindlin-Reissner). dimeter of the torus centre line 8000 mm smll dimeter of the torus 000 mm thickness t = 4 mm Young s modulus E = 10 5 N/mm Poisson s rtio ν = 0. mss density ρ = 7850 kg/m 3 yield strength 500 N/mm grvittionl ccelertion g = 9.8 m/s element size 100 mm x 100 mm lod F = 75 kn theoreticl buckling force n cr = 0.6 E t / knockdown fctor C = 1/6 Is this shell thin? Explin your nswer. b Clculte the influence length. c Are the elements sufficiently smll? Explin your nswer. d Will this shell buckle? Explin your nswer. e Wht other wys exist to check the buckling strength? Figure. Hlf torus with fixed support nd point lod 4

5 Figure 3. Principl membrne forces in the torus y x z Figure 4. Shell shpe, = 3, -1 < u < 1, -1 < v < 1, 5

6 Problem 5 Consider the prmeteristion (fig. 4) v x = u(1 ) 1+ v + 1+ v y = v xy z = Wht is the nme of this shpe? b Wht is the physicl mening of prmeter? Wht is the unit of? Wht re the units of u nd v? c Is this prmeteristion orthogonl? Explin your nswer. (You cn use Mple or nother mthemticl progrm.) Mple hs been used to clculte Lme s prmeters nd the curvtures of this shpe. These equtions re not provided becuse they re lrge. Insted the Tylor expnsions t u = v = 0 re provided. = = x y (1 u uv u +...) kxx = 0 kyy = ( uv +...) kxy = (1 u v +...) 1 1 k1= (1 u uv v +...) k = ( 1 + u uv + v +...) 1 1 km = ( uv +...) kg = ( 1+ u + v...) d Is the prmeteristion in the principl curvture directions? Explin your nswer. e Check the provided Gussin curvture. f Is the deformtion of this shpe extensionl? Choose A, B, C or D. A Yes becuse the Gussin curvture chnges; it depends on u nd v. B Yes becuse the distnce between the prmeter lines vries. C No becuse flt sheet cn be twisted in this shpe without stretching. D There is no deformtion; it is just shpe. g Check the eqution of Guß for this shpe t u = v = 0. 6

7 Problem 6 In engineering we often cn choose between clculting something nd using tble. Let us pply this to finite element nlysis of shell structures. A computer cn compute n element stiffness mtrix or it cn look it up in tble. The question is; which costs less? Consider liner elstic shell element with 3 nodes nd 6 dofs per node. The following dt pplies. Computtion time costs 0.30 euro/hour (mintennce, electricity nd write off). Hrd disk spce costs 0.0 euro/gb/yer. In one yer, on one computer, times shell element is computed or looked up. Finding one number in list tkes log n evlutions of <, where n is the number of steps in the domin of the input vrible. The domin of ech input vrible of the tble is divided in 10 steps. A computer cn perform evlutions of < per second. Computing 3 node shell element stiffness mtrix tkes 4000 multiplictions or dditions. A computer cn perform multiplictions nd dditions per second. Which input vribles should the tble hve? For exmple shell thickness, How mny input vribles hve you selected? b How mny output vribles (numbers) should the tble hve? Explin your nswer. c Give smrt wys to reduce the number of input nd output vribles of the tble. d Suppose tht the thicknesses of shell elements vry between 0.01 mm nd 1 m. For the tble we need to divide this domin in steps. (Proposed re 10 steps. This is n estimtion.) Wht determines the ctul step size? How cn computer proceed to determine the correct step size? e How mny numbers need be stored in the tble? How much GB is this? f How much time does it tke to find the right element stiffness mtrix in the tble? g How much time does it tke to compute n element stiffness mtrix? h Which costs less; computing or looking up in tble? Explin your nswer. i Wht hs most influence on nswer h? 7

8 Answers to problem 1 Membrne forces.. in plne norml forces nd sher forces nxx, nyy, nxy, n yx Edge bem. bem t the edge of thin shell Snders-Koiter equtions 1 equtions tht describe the behviour of thin shells Knockdown fctor. reduction of the theoreticl liner buckling lod to include the effect of imperfections Gussin curvture.. kk 1 Elpr ellipticl prboloid Love. Augustus Love ws the first to propose shell theory. Abutment force.. horizontl lod on the support of n rch Answers to problem b kxx is positive. kyy is negtive. k G is negtive. Answers to problem 3 C b A c C d D e D f A g A h C i B Answer to problem 4 / t = 1000 / 4 = 50 > 30 So it is thin. b.4 t = = 15 mm c The elements re too lrge t the edges. About 6 should be in the influence length. d Et ncr = 0.6 = 0.6 = 190 N/mm = 190 kn/m 1000 n n = cr ult = 30 kn/m < 1557 kn/m 6 It will buckle. 8

9 e - Perform liner buck buckling nlysis with the softwre (nd pply the knock down fctor). - Add imperfections nd perform sttic nonliner nlyses. Answers to problem 5 Hypr (The figure ppers strnge becuse of the edges creted by this prticulr prmeteristion nd the very lrge domin of u nd v. Nevertheless, the equtions show tht this is hypr.) b Rdius of curvture in the origin. hs the unit length, for exmple m. u nd v hve no unit. c Substitution in x x + y y + z z u v u v u v gives zero. So it is orthogonl. d No becuse k xy 0. e 1 1 kg = k1k = (1 u uv v +...) ( 1 + u uv + v +...) = 1 ( u uv+ v + u u + uv uv + uv uv+ uv uv + v uv + uv v +...) = 1 ( u + v u v u v +...) = 1 ( 1 + u + v...) correct f D g 1 1 y 1 k = ( ) + ( x G ) xy u x u v y v 1 ( 1 + u + v +...) = = 1 ( (1 u uv u...)) 8 1 ( ) + ( ) (1 u uv u...) u u v (1 u uv u...) v = ( ( u uv u...)) + ( 0) 4 (1 + 1u uv 1u +...) u v (1 1u uv 1u...) 8 1 = v u... = ( 1+ u + v +...) (1 u uv u...) 8 9

10 Answers to problem 6 l1, l, l3, kxx, kyy, kxy, t, E, ν These re 9 input vribles. (If you considered flt element, thus 6 input vribles, then this would be correct too. ) b Output vribles: A stiffness mtrix hs (3 x 6) = 34 elements. c Key words: - Stiffness is liner in E. - Dimensionless prmeters - Eliminte element rigid body modes - Symmetry of the stiffness mtrix d Keywords: - Accurcy of the output - Interpoltion For exmple, computer cn compute for smll steps nd disregrd the step results for which the difference with n interpoltion is smll. e (10+1) 9 x 34 = x 8 = byte = 6000 GB f log 10 x 9 / = s g 4000 / = s h Computing: x / 3600 x 0.30 = 1 euro/yer Looking up: 6000 x x / 3600 x 0.30 = = 100 euro/yer Computing costs less. i Hrd disk spce 10