1、薄壳理论全册配套最完整薄壳理论全册配套最完整 精品课件精品课件(英文版）英文版） Elasticity Purpose : Physical introduction to elasticity, - Stress - Strain - Relationship - Methods of resolution 2 Elasticity - CAUC - Tian Jin - 2011 Stress vector 0 lim dS n n dF t dS Remark : nn tt 3 Elasticity - CAUC - Tian Jin - 2011 Stress Tensor Conc
2、lusion 1 : In a point M of a structure, we can define as many stress vectors that there are normal vector issued of the point M Conclusion 2 : We need a tool to represent the state of Stresses in a single point of a solid body 4 Elasticity - CAUC - Tian Jin - 2011 Equilibrium Question : How many ind
3、ependent stress vectors are in a point ? Tool for answer : Equilibrium of a very small volume. 5 Elasticity - CAUC - Tian Jin - 2011 Equilibrium of a tetrahedron Surface of triangle ABC Surface of triangle OBC Surface of triangle OAC Surface of triangle OAB dS dSxdS dSydS dSzdS xx yy zz dFtdSx dFtdS
4、y dFtdSz xx yy zz tt tt tt 0 xyznv dFdFdFdFF dV nxyzV nxyz tdStdSxtdSytdSzFdV tdStdSxtdSytdSz nxyz dSx t dStttdSy dSz 6 Elasticity - CAUC - Tian Jin - 2011 Equilibrium of a tetrahedron nxyzxyz xyxzx nxyyzy xzyzz n tttttttn tn tn 7 Elasticity - CAUC - Tian Jin - 2011 Representation of a tensor by a M
5、ohrs Circle There is an axis system such as the tensor is a diagonal tensor , , 00 00 00 xyxzxI xyyzyII xyzI II III xzyzzIII I II III xyz 0, xyxzx xyyzyIIIIII xzyzz iii VV 8 Elasticity - CAUC - Tian Jin - 2011 Mohrs Circle We shall study the stresses when the output normal to a little surface is ins
6、ide the principal plane for stresses (I, II) 00coscos 00sinsin 0000 II nIIII III tn Conclusion : When the output normal is inside a principal plane for stresses, then, the stress vector is inside the same plane 9 Elasticity - CAUC - Tian Jin - 2011 Mohrs Circle Remark : In a principal plane for stre
7、sses, we define the Mohrs axis such as : ,/2 cos sin 0 I nII tn 10 Elasticity - CAUC - Tian Jin - 2011 Mohrs Circle cos2 22 sin2 2 IIIIII n III n nnn t 11 Elasticity - CAUC - Tian Jin - 2011 Full Mohrs Circle xyxzx xyyzy xzyzy 12 Elasticity - CAUC - Tian Jin - 2011 Mohrs Circle Remark 1 : On a small
8、 surface whose output normal is y the shear stress is represented on the Mohrs circle by a positive value 13 Elasticity - CAUC - Tian Jin - 2011 Mohrs Circle Remark 2 : On a small surface whose output normal is x the shear stress is represented on the Mohrs circle by a negative value 14 Elasticity -
9、 CAUC - Tian Jin - 2011 Remark How many unknown quantity in the stress tensor ? How many equations ? Equilibrium of the forces, Equilibrium of the moments. 15 Elasticity - CAUC - Tian Jin - 2011 Equilibrium of the forces , , , , , , , yy yyyy tx y ztx y z tx ydy z dxdztx y z dxdztx y zdytx y zdxdzdV
10、 yy , , , , 0 yxz Vol tx y ztx y ztx y z dVdVdVfdV xyz 0 y xz Vol t tt f xyz 16 Elasticity - CAUC - Tian Jin - 2011 Equilibrium Equations 0 0 0 yxvol xzx x xyyzyvol y yzvol xzz z f xyz f xyz f xyz 0 vol divf 17 Elasticity - CAUC - Tian Jin - 2011 Equilibrium Equations 1 0 1 20 1 0 vol rrrrz r vol rz
11、r vol zzrzrz z f rrzr f rrzr f rrzr 18 Elasticity - CAUC - Tian Jin - 2011 Equilibrium of the moments By writing the equilibrium of the moments, we can demonstrate that the tensor of stress is a symmetric tensor. 1 22 1 22 1 22 1 22 xy xyxy xy xyxy xy yxxy xy yxxy dx xdx x dx xdx x dy ydy x dy ydy x
12、 19 Elasticity - CAUC - Tian Jin - 2011 Remark How many unknown quantity in the stress tensor ? 9 How many equations ? Equilibrium of the forces, 3 Equilibrium of the moments. 3 Conclusion : There is an infinity of solutions 20 Elasticity - CAUC - Tian Jin - 2011 Remark Stresses are very important t
13、o find for the engineer. Because the knowledge of the stresses allows the engineer to design structures that will never breakdown. 21 Elasticity - CAUC - Tian Jin - 2011 Displacement Displacements are very important to find for the engineer. Because the knowledge of the displacement allows the engin
14、eer to design the structures with the good stiffness. Each point P of the structure will have a displacement when the load will be applied. This displacement is a vector with 3 unknown components. , , , , , , u x y z PPv x y z w x y z Therefore, we need now 6 equations ! 22 Elasticity - CAUC - Tian
15、Jin - 2011 Strains The strains are very usefull because : There is a linear relation between stresses and strains By integration of the strains its possible to find the displacements. Equations of compatiblity are very easy to demonstrate in term of strain Strain allows us to find the 6 equations we
16、 need to solve a problem of structure 23 Elasticity - CAUC - Tian Jin - 2011 Strains We can define a strain tensor with 6 components like this : xyxzx xyyzy xzyzy E 24 Elasticity - CAUC - Tian Jin - 2011 Strains 25 Elasticity - CAUC - Tian Jin - 2011 Strains : Definition 2half diminution of the righ
17、t angle xy x y xy M PMP MP M QMQ MQ 26 Elasticity - CAUC - Tian Jin - 2011 Strains , , , , , , , , , , , , , , , , , , , , , , u x y z MM v x y z u x y z u xdx y zu x y zdx x PP v x y z v xdx y zv x y zdx x u x y z u x ydy zu x y zdy y QQ v x y z v x ydy zv x y zdy y 27 Elasticity - CAUC - Tian Jin
18、- 2011 Strains 22 1 2 x x uuv xxx u x 22 , , , , , x dxu xdx y zu x y zv xdx y zv x y zdx dx 28 Elasticity - CAUC - Tian Jin - 2011 Strains 2angle x,MPangle y,MQ 2tan x,MPtan y,MQ 11 22 xy xy xy u v dy dx vuy x dxdyxy 29 Elasticity - CAUC - Tian Jin - 2011 Relation between strain and displacement 1
19、2 1 2 1 2 xy xz yz uv yx uw zx wv yz x y z u x v y w z 30 Elasticity - CAUC - Tian Jin - 2011 Compatibility Equations , 1 2 xy xy uv xy uv yx 22 2 22 2 xyy x x yyx 2222 2222 222 , 2 xy xy uv yyxxxy uv x yx yyx yx 31 Elasticity - CAUC - Tian Jin - 2011 Equation for Compatibility 22 2 22 22 2 22 222 2
20、2 2 2 2 yxy x yxxy yyz z zyyz xxzz xzxz 2 2 2 yzxy xxz yyxyz xz xyzy zxz y zxxyz z xyyzx x yzzyx 32 Elasticity - CAUC - Tian Jin - 2011 Rotation 11 22 wv yz uw rot MM zx vu xy 1 2 1 2 1 2 x x x wv yz uw zx vu xy 33 Elasticity - CAUC - Tian Jin - 2011 Mohr Circle for the Strains xyxzx xyyzy xyz xzyzz
21、 xyz E , , 00 00 00 I II I II III III I II III E 34 Elasticity - CAUC - Tian Jin - 2011 Memorize what we will need ! 11 22 wv yz uw rot PP zx vu xy 1 2 1 2 1 2 x y z xy yz zx u x v y w z uv yx vw zy wu xz , , , , , , u x y z PPv x y z w x y z uvw trace E xyz 35 Elasticity - CAUC - Tian Jin - 2011 Me
22、morize what we will need ! 1 11 22 1 wv rz uw rot PP zr vur rrr 1 1 1 2 11 2 1 2 r z r z zr u r v u r w z uvv rrr vw zr wu rz , , , , , , u rz PPv rz w rz 1uuvw trace E rrrz 36 Elasticity - CAUC - Tian Jin - 2011 Memorize what we will not need ! 1 11 22 1 wv rz uw rot PP zr vur rrr 1 1cos sinsin 1 1
23、 2 111cos 2sinsin 11 2sin r r r u r v u r wu v rrr uvv rrr vw w rrr wwu rrr , , , , , , u r PPv r w r 112cos sinsin uvw trace Euv rrrrr 37 Elasticity - CAUC - Tian Jin - 2011 Relationship Stress / Strain This relation ship is due to : Robert Hooke Denis Poisson Thomas Young Gabriel Lam 38 Elasticity
24、 - CAUC - Tian Jin - 2011 Robert Hooke Robert Hooke, n le 18 juillet 1635 Freshwater (le de Wight) et mort le 3 mars 1703 Londres, est un des plus grands scientifiques exprimentaux du XVIIe sicle et donc une des figures cls de la rvolution scientifique de lpoque moderne. 39 Elasticity - CAUC - Tian
25、Jin - 2011 Denis Poisson Simon Denis Poisson (21 juin 1781 Pithiviers - 25 avril 1840 Sceaux) est un mathmaticien, gomtre et physicien franais. 40 Elasticity - CAUC - Tian Jin - 2011 Thomas Young Thomas Young (13 juin 1773- 10 mai 1829), est un physicien, mdecin et gyptologue britannique. 41 Elastic
26、ity - CAUC - Tian Jin - 2011 Relationship Stress / Strain 42 Elasticity - CAUC - Tian Jin - 2011 Relationship Stress / Strain 0 0 x x yzx F S L L 43 Elasticity - CAUC - Tian Jin - 2011 Relationship Stress / Strain 44 Elasticity - CAUC - Tian Jin - 2011 Relationship Stress/ Strain 2 1 trace EIE Etrac
27、eI EE 45 Elasticity - CAUC - Tian Jin - 2011 Relationship Stress/ Strain 2 1 11 2 E G E 2 32 E 46 Elasticity - CAUC - Tian Jin - 2011 Method of resolution Eugenio Beltrami, , est un mathmaticien et physicien italien n le 16 novembre 1835 Crmone, mort le 18 fvrier 1900 Rome. 47 Elasticity - CAUC - Ti
28、an Jin - 2011 Method of resolution Gabriel Lam (22 juillet 1795, Tours - 1er mai 1870, Paris) est un mathmaticien franais 48 Elasticity - CAUC - Tian Jin - 2011 Methods of resolution Benot Paul mile Clapeyron, n 26 fvrier 1799 Paris et mort le 28 janvier 1864 dans cette mme ville, est un ingnieur et
29、 physicien franais. 49 Elasticity - CAUC - Tian Jin - 2011 Internal Energy The internal energy in a small volume dV is equal to : 1 222 2 xxyyzzxyxyxzxzyzyz dW dV 1 2 dWtraceEdV 50 Elasticity - CAUC - Tian Jin - 2011 Grandeur mesurer : x dl l Rn l s R R l l S S l Principe of the strain gauge 51 Elas
30、ticity - CAUC - Tian Jin - 2011 x y z l r Hookes Law x yzx y z l l r r r r 2 2 2 2 Sr Sr Sr Sl Sl Rll Rll Principe of the strain gauge 52 Elasticity - CAUC - Tian Jin - 2011 Influence of the metal gauge 53 Elasticity - CAUC - Tian Jin - 2011 Influence of the metal gauge 54 Elasticity - CAUC - Tian J
31、in - 2011 Influence of the metal gauge 55 Elasticity - CAUC - Tian Jin - 2011 Influence of the metal gauge 56 Elasticity - CAUC - Tian Jin - 2011 R R k l l kGauge factor (2) 120 R 350 lot de fabrication kfacteur transverse t l Strain gauge : results 57 Elasticity - CAUC - Tian Jin - 2011 Wheatstones
32、 Bridge vE R RR vE R RR vE R RR R RRRR C D 4 14 23 3 41 2 142 23()() Le pont est quilibr si v 0 vR RR R0 1234 ABv R1R4 R3R2 C D E Elasticity - CAUC - Tian Jin - 2011 Wheatstones Bridge Quarter bridge Application Measurement of Strain Equilibrium of the bridge R1=R2=R3=R4=R E v 59 Elasticity - CAUC -
33、 Tian Jin - 2011 Wheatstones Bridge Half bridge Insensitivity to temperature R R v r r E 60 Elasticity - CAUC - Tian Jin - 2011 Wheatstones Bridge full bridge Application : EFFORT SENSOR RR RR v E 61 Elasticity - CAUC - Tian Jin - 2011 Equation de la courbe : 01T2T23T34T4 EA06 240 LZ 120 0= -3. 49 1
34、 = 2. 92 2 = -6. 70E-2 3 = 3. 71E-4 4 =-4. 70E-7 Correction thermique des jauges auto compenses 62 Elasticity - CAUC - Tian Jin - 2011 v ERr Rr ER Rr Er Rr v ER Rr R E R R Er Rr R phnomne 4 2 242 2 42 4 1 1 2 4 2 1 1 2 () E 2 r R R r r R E v Influence of connection wires 63 Elasticity - CAUC - Tian
35、Jin - 2011 temperature lenght effect onof the wire (2 ) 424222 11 4222 11 22 11 42 2 442 strain RrEEREr v RrRrRr EREr v RRrr RR ERrErr v RRRR ERER rEr v RR RR the wire R rR r r E v Temperature effects on the lead wires can be cancelled by using a 3-wire bridge 64 Elasticity - CAUC - Tian Jin - 2011
36、Jauges simples 65 Elasticity - CAUC - Tian Jin - 2011 Rosettes de 2 jauges 66 Elasticity - CAUC - Tian Jin - 2011 Rosettes de 3 jauges 67 Elasticity - CAUC - Tian Jin - 2011 Rosettes superposes (2 ou 3 jauges) 68 Elasticity - CAUC - Tian Jin - 2011 1 - Static for the structures 69 Static - Energetic
37、 - CAUC- 2011 OBJECTIVES: Characterization of the links, Define the nature of the system, * Isostatic * Hyperstatic Computation of the efforts introduced by the links Static for structure 70 Static - Energetic - CAUC- 2011 A mechanical system is a set of pieces connected by links. To study this syst
38、em it is necessary to do the modeling of it. This modeling can be accomplished in several ways. The parts of the system can be considered as : Rigid Deformable The links can be modeled : with gap or without gap with or without energy dissipation . Static for structure 71 Static - Energetic - CAUC- 2
39、011 Questions : -What is the load introduced by S1 on S2 - What is the displacement of S1 according to S2 Displacement / Degree of Freedom 72 Static - Energetic - CAUC- 2011 Load introduced by S1 on S2 Rx Ry Rz Mx My Mz Displacement allowed of S1 relative to S2 Tx Ty Tz Rx Ry Rz DEFINITION : The num
40、ber of degree of freedom of two solids linked together, is the number of elementary displacement that the first solid is allowed to own relatively to the second solid. . It will be noted Nc. Static for structure 73 Static - Energetic - CAUC- 2011 This parameter (DOF) can usually be obtained relative
41、ly easily by analyzing the physical link. If the movements of S1 from S2 are small, which is usually the case in problems of structural mechanics, we note TX, TY, TZ the components of the translation vector and X, Y, Z the components of the vector rotation. We can then bring up the small displacemen
42、ts torsor Dd Link 74 Static - Energetic - CAUC- 2011 Cantilever Link 75 Static - Energetic - CAUC- 2011 Link 76 Static - Energetic - CAUC- 2011 Link : Ball and Sliping Pivot 77 Static - Energetic - CAUC- 2011 Link : Slide bar and Pivot 78 Static - Energetic - CAUC- 2011 LOAD 79 Static - Energetic -
43、CAUC- 2011 What happens in a link in terms of load is generally complicated to analyse. But its easy to speak about the resultant of the forces ans of the moment. Resultant Rx Ry Rz Moment Mx My Mz DEFINITION : The number of independent terms, non equal to zero, of the torsor of the mechanical actio
44、ns is the number of degree of linkage. It will be noted Ns. Degree of Linkage 80 Static - Energetic - CAUC- 2011 If the link is a perfect link we can write that there is no energy spent for all the displacements allowed by the join 0 XXYYZZXXYYZZ TTTTTTMMM 00 00 00 00 00 00 XX YY ZZ XX YY ZZ TT TT T
45、T M M M 0may be0But no sure 0may be0But no sure 0may be0But no sure 0may be0But no sure 0may be0But no sure 0may be0But no sure XX YY ZZ XX YY ZZ TT TT TT M M M Exemple 81 Static - Energetic - CAUC- 2011 0may be0But no sure 0may be0But no sure 0may be0But no sure 0may be0But no sure 0may be0But no s
46、ure 0may be0But no sure XX YY ZZ XX YY ZZ TT TT TT M M M Nature of a system 82 Static - Energetic - CAUC- 2011 N solids Nci DOF Nsi Reaction 1 N i i IncNs 3in 2D 6in 3D EqN EqN Nature of a system : Isostatic 83 Static - Energetic - CAUC- 2011 IncEqIf its possible to find the whole unknown reactions
47、just by writing the equilibrium of the system Nature of a system : Hyperstatic 84 Static - Energetic - CAUC- 2011 What happend if I add a join ? 0IncEq 1IncEq IncEqH Nature of a system : Hypostatic 85 Static - Energetic - CAUC- 2011 What happend if I remove a join ? 0IncEq 1IncEq IncEqM Nature of a
48、system 86 Static - Energetic - CAUC- 2011 IncEqHM Isostatic System : Exemple 87 Static - Energetic - CAUC- 2011 Inc Eq Static : Important remark 88 Static - Energetic - CAUC- 2011 All the equations that translate a state of equilibrium have been written in an intial position, before the loads have b
49、een applied 2 - Energetic for structures 89 Static - Energetic - CAUC- 2011 OBJECTIVES: Work of External Loads Stiffness and Compliance Matrix Maxwell Bettis Theorem Castiglianos Theorem Energetic for structures 90 Static - Energetic - CAUC- 2011 Assumptions All the joins will be perfect, The load w
50、ill be applied slowly, The behaviour of the structures will be a linear behaviour The work of the external loads (forces and moments) will be transformed totaly in elastic energy inside the structure Exemple of non linear structure 91 Static - Energetic - CAUC- 2011 Internal energy in a spring 0 . L
