1. 01/05/15 1 Introduction to Elasticity Part II SOLO HERMELIN Updated: 07. 1984 4.10.2013 12.02.2014 http://www.solohermelin.com

2. 01/05/15 2 Introduction to Elasticity SOLO Table of Content Boundary Conditions Change of Coordinates Determination of the Principal Stresses MOHRs Circles Strain Physical Meaning of Elongation Equation - First Physical Meaning of Elongation Equation - Second Stress Strain Relationship - HOOKEs Law Compatibility Equations Elastic Waves Equations Summary Stress-Strain Introduction Stress Body Forces and Moments P a r t I 3. 01/05/15 3 Introduction to Elasticity SOLO Table of Content Torsion of a Circular Bar Shear Force and Bending Moments in a Beam Bending of Unsymmetrical Beams Shear-Stress in Beams of Thin-Walled, Open Cross-Sections Deflection of Beams Double Integration Method Deflection of Beams Moment Area Method Torsion Bar of Narrow Rectangular Section Narrow Profiles Closed Sections Energy Equations Energy Methods Narrow Profiles Open Sections P a r t I 4. 01/05/15 4 Introduction to Elasticity SOLO Table of Content History of Plate Theories Plate Theories Kirchhoff-Love theory of plates (Classical Plate Theory) Naviers Analytic Solution (1823) Symmetric Bending on Cylindrical Plates Poissons Solution for Cylindrical Plates (1829) MindlinReissner plate theory Membrane Theory Vibration Pure Torsion Vibration Vibration of Euler-Bernoulli Bending Beam Vibration of Kirchhoff Plate (Classical Plate Theory) Vibration of Rectangular Plate Vibration of Cylindrical Plate Vibrations of a Circular Membrane Vibration Modes of a Free-Free Beam 5. 01/05/15 5 Introduction to Elasticity SOLO Table of Content Numerical Methods in Elasticity RayleighRitz Method Rayleigh Principle Ritz Method Weighted Residual Methods Galerkin Method. References Finite Element Method 6. 01/05/15 6 SOLO Introduction to Elasticity Continue from Part I 7. 01/05/15 7 History of Plate Theories Euler performed free vibration analyses of plate problems (Euler, 1766). Chladni, a German physicist, performed experiments on horizontal plates to quantify their vibratory modes. He sprinkled sand on the plates, struck them with a hammer, and noted the regular patterns that formed along the nodal lines (Chladni, 1802). Daniel Bernoulli then attempted to theoretically justify the experimental results of Chladni using the previously developed Euler-Bernoulli bending beam theory, but his results did not capture the full dynamics (Bernoulli, 1705). Marie-Sophie Germain (1776 1831) Joseph-Louis Lagrange (1736 1813) Ernst Florens Friedrich Chladni (1756 1827) In 1809 the French Academy invited Chladni to give a demonstration of his experiments. Napoleon Bonaparte, who attended the meeting, was very impressed and presented a sum of 3,000 francs to the Academy, to be awarded to the first person to give a satisfactory mathematical theory of the vibration of the plates. There where only two contestants, Denis Poisson and Marie-Sophie Germain. Then Poisson was elected to the Academy, thus becoming a judge instead of a contestant, and leaving Germain as the only entrant to the competition.[ In 1809 Germain began work. Legendre assisted by giving her equations, references, and current research. She submitted her paper early in the fall of 1811, and did not win the prize. The judging commission felt that the true equations of the movement were not established, even though the experiments presented ingenious results.[37] Lagrange was able to use Germain's work to derive an equation that was correct under special assumptions. SOLO 8. 01/05/15 8 http://physics.stackexchange.com/questions/90021/theory-behind-patterns-formed-on-chladni-plates Chladni Plates http://www.youtube.com/watch?v=wvJAgrUBF4w Ernst Florens Friedrich Chladni (1756 1827) SOLO 9. 9 History of Membrane Theory In the field of membrane vibrations, Euler (1766) published equations for a rectangular membrane that were incorrect for the general case but reduce to the correct equation for the uniform tension case. It is interesting to note that the first membrane vibration case investigated analytically was not that dealing with the circular membrane, even though the latter, in the form of a drumhead, would have been the more obvious shape. The reason is that Euler was able to picture the rectangular membrane as a superposition of a number of crossing strings. In 1828, Poisson read a paper to the French Academy of Science on the special case of uniform tension. Poisson (1829) showed the circular membrane equation and solved it for the special case of axisymmetric vibration. One year later, Pagani (1829) furnished a nonaxisymmetric solution. Lam (17951870) published lectures that gave a summary of the work on rectangular and circular membranes and contained an investigation of triangular membranes (Lam, 1852). Leonhard Euler (1707 1783) Simon Denis Poisson ( 1781 1840), Gabriel Lon Jean Baptiste Lam (1795 1870) SOLO 10. 10 History of Plate Theories The contest was extended by two years, and Germain decided to try again for the prize. At first Legendre continued to offer support, but then he refused all help.Germain's anonymous 1813 submission was still littered with mathematical errors, especially involving double integrals, and it received only an honorable mention because the fundamental base of the theory of elastic surfaces was not established. The contest was extended once more, and Germain began work on her third attempt. This time she consulted with Poisson. In 1814 he published his own work on elasticity, and did not acknowledge Germain's help (although he had worked with her on the subject and, as a judge on the Academy commission, had had access to her work).[36] Germain submitted her third paper, Recherches sur la thorie des surfaces lastiques under her own name, and on 8 January 1816 she became the first woman to win a prize from the Paris Academy of Sciences. She did not appear at the ceremony to receive her award. Although Germain had at last been awarded the prix extraordinaire, the Academy was still not fully satisfied.[41] Sophie had derived the correct differential equation, but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler, which led to incorrect boundary conditions.[42] Germain published her prize-winning essay at her own expense in 1821, mostly because she wanted to present her work in opposition to that of Poisson. In the essay she pointed out some of the errors in her method.[ In 1826 she submitted a revised version of her 1821 essay to the Academy. According to Andrea del Centina, a math professor at the University of Ferrara in Italy, the revision included attempts to clarify her work by introducing certain simplifying hypotheses. This put the Academy in an awkward position, as they felt the paper to be inadequate and trivial, but they did not want to treat her as a professional colleague, as they would any man, by simply rejecting the work. So Augustin-Louis Cauchy, who had been appointed to review her work, recommended she publish it, and she followed his advice Marie-Sophie Germain (1776 1831) SOLO 11. 01/05/15 11 History of Plate Theories Cauchy (1828) and Poisson (1829) developed the problem of plate bending using general theory of elasticity. Then, in 1829, Poisson successfully expanded the Germain-Lagrange plate equation to the solution of a plate under static loading. In this solution, however, the plate flexural rigidity D was set equal to a constant term (Ventsel and Krauthammer, 2001). Navier (1823) considered the plate thickness in the general plate equation as a function of rigidity, D. Simon Denis Poisson ( 1781 1840), Claude-Louis Navier 1785 1836) Augustin Louis Cauchy (1789-1857) SOLO 12. 01/05/15 12 History of Plate Theories (continues 1) Some of the greatest contributions toward thin plate theory came from Kirchhoffs thesis in 1850 (Kirchhoff, 1850). Kirchhoff declared some basic assumptions that are now referred to as Kirchhoffs hypotheses. Using these assumptions, Kirchhoff: simplified the energy functional for 3D plates; demonstrated, under certain conditions, the Germain-Lagrange equation as the Euler equation; and declared that plate edges can only support two boundary conditions. Lord Kelvin (Thompson) and Tait (1883) showed that plate edges are subject to only shear and moment forces. Gustav Robert Kirchhoff (1824 1887) William Thomson, 1st Baron Kelvin (1824 1907) Peter Guthrie Tait (1831 1901) SOLO 13. 01/05/15 13 RayleighRitz method John William Strutt, 3rd Baron Rayleigh, (1842 1919) Lord Rayleigh published in the Philosophical Transactions of the Royal Society, London, A, 161, 77 (1870) that the Potential and Kinetic Energies in an Elastic System are distributed such that the frequencies (eigenvalues) of the components are a minimum. His discovery is now called the Rayleigh Principle An extension of Rayleighs principle, which enables us to determine the higher frequencies also, is the Rayleigh-Ritz method. This method was proposed by Walter Ritz in his paper Ueber eine neue Methode zur Loesung gewisser Variationsprobleme der Mathematishen Physik , [On a new method for the solution of certain variational problems of mathematical physics], Journal fr reine und angewandte Mathematik vol. 135 pp. 1 - 61 (1909).. Walther Ritz (1878 1909) SOLO Introduction to Elasticity Elasticity History (continue 6) 14. 01/05/15 14 History of Plate Theories (continues 7) Levy (1899) successfully solved the rectangular plate problem of two parallel edges simply-supported with the other two edges of arbitrary boundary condition. Meanwhile, in Russia, Bubnov (1914) investigated the theory of flexible plates, and was the first to introduce a plate classification system. Bubnov worked at the Polytechnical Institute of St. Petersburg (with Galerkin, Krylov, Timoshenko). Bubnov composed tables of maximum deflections and maximum bending moments for plates of various properties . Galerkin (1933) then further developed Bubnovs theory and applied it to various bending problems for plates of arbitrary geometries. Timoshenko (1913, 1915) provided a further boost to the theory of plate bending analysis; most notably, his solutions to problems considering large deflections in circular plates and his development of elastic stability problems. Timoshenko and Woinowsky-Krieger (1959) wrote a textbook that is fundamental to most plate bending analysis performed today. Hencky (1921) worked rigorously on the theory of large deformations and the general theory of elastic stability of thin plates. Fppl (1951) simplified the general equations for the large deflections of very thin plates. The final form of the large deflection thin plate theory was stated by von Karman, who had performed extensive research in this area previously (1910). Boris Grigoryevich Galerkin (1871 1945) Ivan Grigoryevich Bubnov (1872 - 1919) Stepan Prokopovych Tymoshenko (1878 1973) SOLO Introduction to Elasticity Return to Table of Content 15. 01/05/15 15 Plate Theories Plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions There are several theories that attempt to describe the deformation and stress in a plate under applied loads two of which have been used widely. These are The Kirchhoff-Love theory of plates (also called classical plate theory) The Mindlin-Reissner plate theory (also called the first-order shear theory of plates) Membrane Shell Model: for extremely thin plates dominated by membrane effects, such as inflatable structures and fabrics (parachutes, sails, balloon walls, tents, inflatable masts, etc) von-Krmn model: for very thin bent plates in which membrane and bending effects interact strongly on account of finite lateral deflections. Proposed by von Krmn in 1910 . Important model for post-buckling analysis. SOLO Introduction to Elasticity 16. 01/05/15 16 Plate and Membrane Theories The distinguishing limits separating thick plate, thin plate, and membrane theory. The characterization of each stems from the ratio between a given side of length a and the elements thickness http://scholar.lib.vt.edu/theses/available/etd-08022005 145837/unrestricted/Chapter4ThinPlates.pdf SOLO Introduction to Elasticity Return to Table of Content 17. 01/05/15 17 Plate Theories Kirchhoff-Love theory of plates (Classical Plate Theory) The assumptions of Kirchhoff-Love theory are straight lines normal to the mid-surface remain straight after deformation straight lines normal to the mid-surface remain normal to the mid-surface after deformation the thickness of the plate does not change during a deformation. The KirchhoffLove theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and was developed in 1888 by Love using assumptions proposed by Kirchhoff in 1850. The theory assumes that a mid-surface plane can be used to represent a three-dimensional plate in two-dimensional form. Gustav Robert Kirchhoff (1824 1887) Augustus Edward Hough Love (1863 1940) SOLO 18. 01/05/15 18 Plate Theories Kirchhoff-Love theory of plates (Classical Plate Theory) Gustav Robert Kirchhoff (1824 1887) Augustus Edward Hough Love (1863 1940) 1. The material of the plate is elastic, homogenous, and isotropic. 2. The plate is initially flat. 3. The deflection (the normal component of the displacement vector) of the midplane is small compared with the thickness of the plate. The slope of the deflected surface is therefore very small and the square of the slope is a negligible quantity in comparison with unity. The assumptions of Kirchhoff theory are 4. The straight lines, initially normal to the middle plane before bending, remain straight and normal to the middle surface during the deformation, and the length of such elements is not altered. This means that the vertical shear strains xy and yz are negligible and the normal strain z may also be omitted. This assumption is referred to as the hypothesis of straight normals. 5. The stress normal to the middle plane, z, is small compared with the other stress components and may be neglected in the stress-strain relations. 6. Since the displacements of the plate are small, it is assumed that the middle surface remains unstrained after bending. SOLO Introduction to Elasticity 19. 01/05/15 19 SOLO Plate Theories Kirchhoff Plate Theory (Classical Plate Theory) Introduction to Elasticity 20. 01/05/15 20 SOLO Deformed Midsurface Original Midsurface ydxdyd y w xd x w wd xy xy += + = x w y w yx = = , Deformed Midsurface Original Midsurface Deformed Midsurface Original Midsurface0 0 :,22 0 :, :, 22 2 2 2 2 2 2 2 2 2 2 = + = + = = + = + = == = + = = = = == = = == = = y w y w y u z u x w x w x u z u yx w kkz yx w z x u y u z w z z u y w kkz y w z y u x w kkz x w z x u zy yz zx xz xyxy yx xy z zz yyyy y yy xxxx x xx wuz y w zuz x w zu zxyyx == == = ,, Small Displacements Strain Plate Theories Kirchhoff Plate Theory (Classical Plate Theory) Introduction to Elasticity 21. 01/05/15 21 SOLO ( ) ( ) ( ) ( ) ( ) xyxy yyxxzzyyxxyyyy yyxxzzyyxxxxxx E EEE EEE zz zz + = +=++ + = =++ + = = = 12 11 11 0 0 ( ) ( ) ( ) yx wzE yx w zGG E y w x wzEE y w x wzEE xyxyxy yyxxyy yyxxxx + = == + = + =+ = + =+ = 22 2 2 2 2 22 2 2 2 2 22 1 2 12 11 11 0 :,22 :, :, 22 2 2 2 2 2 2 2 2 === == = + = == = = == = = zzyzxz xyxy yx xy yyyy y yy xxxx x xx yx w kkz yx w z x u y u y w kkz y w z y u x w kkz x w z x u Strain Stress-Strain Plate Theories Kirchhoff Plate Theory (Classical Plate Theory) Introduction to Elasticity 22. 01/05/15 22 SOLO Deformation Energy Plate Theories Kirchhoff Plate Theory (Classical Plate Theory) ( ) ( ) ( ) ( ) ( ) ydxd yx w y w x w y w y w x w x whE ydxdzdz yx w y w x w y w y w x w x wE ydxdzd y w z y w x w z yx w z x w z y w x w z E zdydxdzdydxdU S h hS S h h V yyyyxyxyxxxx V T + + + + = + + + + = + + + = ++== + + 22 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2/ 2/ 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2/ 2/ 2 2 2 2 2 222 2 2 2 2 2 2 2 2 12 1122 1 12 12 1 12 12 1 2 2 1~~ 2 1 The Virtual Work due to External Loads q [N/m2 ] and Discrete Forces Fi [N] is ( ) ( ) ( ) ( ) ydxdyyxxtyxwFydxdtyxwqW i S iii S += ,,,, Kinetic Energy ( ) = S ydxd t tyxwh K 2 ,, 2 Total Energy ( ) ( ) ( ) ( ) ( ) ( ) ( ) ++ + + + + =+= i S iii S S D S ydxdyyxxtyxwFydxdtyxwq ydxd yx w y w x w y w y w x w x whE ydxd t tyxwh WUKL ,,,, 12 1122 1,, 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 32 Introduction to Elasticity 23. 01/05/15 23 Top Surface Normal Stresses In plane Shear Stresses Bending Stresses 2 D View ( ) + = + = + = + == 2 2 2 2 2 2 2 2 2 32/ 2/ 2 2 2 2 2 2 2/ 2/ 2 2 2 2 2 2/ 2/ 11211 y w x w D y w x whE zdz y w x wE zdz y w x wzE zdzM h h h h h h xxxx ( ) + = + = + = + == 2 2 2 2 2 2 2 2 2 32/ 2/ 2 2 2 2 2 2 2/ 2/ 2 2 2 2 2 2/ 2/ 11211 y w x w D y w x whE zdz y w x wE zdz y w x wzE zdzM h h h h h h yyyy ( ) ( ) ( ) yx w D yx whE zdz yx wE zdz yx wzE zdzM h h h h h h xyxy = = + = + == 22 2 32/ 2/ 2 22/ 2/ 22/ 2/ 11 11211 ( )2 3 112 : = hE D is called the Isotropic Plate Rigidity or Flexural Rigidity Plate Theories Kirchhoff Plate Theory (Classical Plate Theory) SOLO Introduction to Elasticity 24. 01/05/15 24 ( ) = = + + = xy yy xx xy yy xx k k k D yx w y w x w D yx w y w x w y w x w D M M M 100 01 01 100 01 01 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ( )2 3 112 : = hE D is called the Isotropic Plate Rigidity or Flexural Rigidity Moments Plate Theories Kirchhoff Plate Theory (Classical Plate Theory) SOLO Introduction to Elasticity 25. 25 SOLO Top Surface Transverse Shear Stresses Bending Stresses 2 D View Parabolic Distribution across thickness Transverse Shear Forces (as shown) Associated with the Shear Forces are Transverse Shear Stress xz and yz. For a homogeneous plate and using an equilibrium argument, the stress may be shown to vary parabolically over the thickness = = 2 2 max 2 2 max 4 1, 4 1 h z h z yzyzxzxz max 2/ 2/ 2 3 max 2/ 2/ 2 2 max 2/ 2/ max 2/ 2/ 2 3 max 2/ 2/ 2 2 max 2/ 2/ 3 2 3 44 1 3 2 3 44 1 yz h h yz h h yz h h yzy xz h h xz h h xz h h xzx h h z zzd h z zdQ h h z zzd h z zdQ = = == = = == + + + + + + Plate Theories Kirchhoff Plate Theory (Classical Plate Theory) Introduction to Elasticity 26. 01/05/15 26 SOLO Equilibrium Equations 0=+ ++ += ydxdqxdQxdyd y Q QydQydxd x Q QF y y yx x xz ( ) 0=++ ++ += ydxdQxdMxdyd y M MydMydxd x M MM yyy yy yyxy xy xyx ( ) 0= ++ += xdydQydMydxd x M MxdMxdyd y M MM xxx xx xxyx xy yxy Plate Theories Kirchhoff Plate Theory (Classical Plate Theory) x xxxy y yyxyyx Q x M y M Q y M x M q y Q x Q = + = + = + ,, Introduction to Elasticity 27. 01/05/15 27 SOLO Equi;ibrium Equation (continue 1) + + = = 2 2 2 2 22 y w x w yx D y Q x Q q yx ( ) + = + + = + = 2 2 2 2 2 2 2 22 1 y w x w y D y w x w y D yx w x D y M x M Q yyxy y ( ) + = + + = + = 2 2 2 2 2 2 2 22 1 y w x w x D y w x w x D yx w D yx M y M Q xxxy x Plate Theories Kirchhoff Plate Theory (Classical Plate Theory) + + = yx w y w x w y w x w D M M M xy yy xx 2 2 2 2 2 2 2 2 2 2 1 Introduction to Elasticity 28. 28 SOLO Introduction to Elasticity 29. 29Joseph-Louis Lagrange (1736 1813) Marie-Sophie Germain (1776 1831) Consider a Homogeneous Isotropic Plate of Constant Rigidity D. Elimination of the Bending Moments and Curvatures from the Field Equations yields the famous equation for Thin Plates, first derived by Lagrange in 1913. He never published it, and was found posthumously in his Notes. Because of the previous contribution of Germain this is called Germain-Lagrange Equation qwDwD == 224 Biharmonic Operator 4 4 22 4 4 4 2 2 2 2 2 2 2 2 224 2 yyxxyxyx + + = + + == SOLO Return to Table of Content Introduction to Elasticity 30. Naviers Analytic Solution (1823) Claude-Louis Navier 1785 1836) SOLO Introduction to Elasticity 31. Naviers Analytic Solution SOLO Introduction to Elasticity 32. Naviers Analytic Solution SOLO Introduction to Elasticity Return to Table of Content 33. 01/05/15 33 Symmetric Bending on Cylindrical Plates The only unknown is the Plate deflection w which depends on coordinates r only (w = w (r)) and determinates the forces, moments, stresses, strains and displacements in the Plate: (1) Axial Symmetry r, , r, (r =0), Mr, M, Qr, (Q=0) ( )( ) ( ) ( )00 2 22 2 2 == == + = === = rr r rd wd r z r u r rur ruu rd wd z rd ud ntDisplaceme rd wd zu Displacement Introduction to ElasticitySOLO 34. 01/05/15 34 Symmetric Bending on Cylindrical Plates ( ) ( ) + =+ = + =+ = 2 2 22 2 2 22 1 11 11 rd wd rd wd r zEE rd wd rrd wdzEE r rr (2) Hookes Law expressed in terms of w Introduction to ElasticitySOLO 35. 01/05/15 35 Symmetric Bending on Cylindrical Plates (3) Bending Moments and Shear Force ( ) += + == = += + == + + + + 2 22/ 2/ 2 2 2 2 2/ 2/ 2 3 2 22/ 2/ 2 2 2 2 2/ 2/ 11 1 112 : 1 rd wd rd wd r Dzdz rd wd rd wd r E zdzM hE D rd wd rrd wd Dzdz rd wd rrd wdE zdzM h h h h h h h h rr = r r rdrq r Q 0 1 + + = = 2/ 2/ 2/ 2/ h h h h rr zdzrdrdM zdzrdrdM ( )( ) 0=+++= drQdrdrQdQrddrqF rrrz d/1 ( ) 0=++++ rQrdQdrQdrdQrQrdrq rrrrr 0 ( ) rdrqrQdrQdrdQ rrr ==+ Introduction to ElasticitySOLO 36. 01/05/15 36 Symmetric Bending on Cylindrical Plates (4) Moments Equilibrium ( ) ( ) ( ) 0 2 sin2 2/ = ++ d rrrr d rdMrddrQrdMdrdrMdM 0=+++ rdMrdrQrMrdMdrMdrdMrM rrrrrr 0 0=+ MrQr rd Md M r r r 0 1 2 2 2 2 2 2 = + ++ + rd wd rd wd r DrQ rd wd rrd wd D rd d r rd wd rrd wd D r 0 1 2 2 22 2 2 2 2 2 = + ++ + rd wd rd wd r DrQ rd wd rrd wd rrd wd rd d rD rd wd rrd wd D r ( )d/1 rd/1 Introduction to ElasticitySOLO 37. 01/05/15 37 Symmetric Bending on Cylindrical Plates (4) Moments Equilibrium (continue - 1) 0 1 2 2 2 2 = + rd wd r DrQ rd wd rd d rD rd wd D r D Q rd wd r rd d rrd d rd wd rrd wd rd d rd wd rrd wd rd d rd wd r r = = += + 1111 2 2 22 2 2 2 D Q rd wd r rd d rrd d rd wd rrd wd rd d r = = + 11 2 2 Introduction to ElasticitySOLO 38. 01/05/15 38 Symmetric Bending on Cylindrical Plates (4) Moments Equilibrium (continue - 2) D Q rd wd r rd d rrd d rd wd rrd wd rd d r = = + 11 2 2 = r r rdrq r Q 0 1 D rq rd wd r rd d rrd d rd wd rrd wd rd d r rd d = = + 11 2 2 ( ) rqQr rd d r = D q rd wd rrd wd rd d rd d r r = + + 1 1 1 2 2 D q rd wd rrd wd rd d rrd d = + + 11 2 2 2 2 Governing Equation Introduction to ElasticitySOLO 39. 01/05/15 39 Introduction to ElasticitySOLO Return to Table of Content 40. Poissons Solution for Cylindrical Plates (1829) The bending of circular plates can be examined by solving the governing equation with appropriate boundary conditions. These solutions were first found by Poisson in 1829. Cylindrical coordinates are convenient for such problems. Simon Denis Poisson ( 1781 1840), The governing equation in coordinate-free form is In cylindrical coordinates (r,,z) For symmetrically loaded circular plates, w = w (r), we have Therefore, the governing equation is If q and D are constant, direct integration of the governing equation gives us where Ci are constants. The slope of the deflection surface is For a circular plate, the requirement that the deflection and the slope of the deflection are finite at r = 0 implies that C = C = 0. SOLO D q w = 22 2 2 2 2 2 2 11 z ww rr w r rr w + + = = r w r rr w 12 D q r w r rrr r rr w = = 1122 Introduction to Elasticity 41. Poissons Solution for Cylindrical Plates (1829) Simon Denis Poisson ( 1781 1840), Clamped edges For a circular plate with clamped edges, we have w (a) = 0, (a) = 0 at the edge of the plate (radius ). Using these boundary conditions we get The in-plane displacements in the plate are The in-plane strains in the plate are For a plate of thickness 2h the bending stiffness is D=2Eh3 /[3(1-2 )] and we have The moment resultants (bending moments) are SOLO Introduction to Elasticity Return to Table of Content 42. 01/05/15 42 Plate Theories MindlinReissner plate theory Raymond David Mindlin (1906- 1987) Eric Reissner (1913 - 1996) The Mindlin-Reissner theory of plates is an extension of KirchhoffLove plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1951 by Raymond Mindlin. A similar, but not identical, theory had been proposed earlier by Eric Reissner in 1945. Both theories are intended for thick plates in which the normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface. The Mindlin-Reissner theory is used to calculate the deformations and stresses in a plate whose thickness is of the order of one tenth the planar dimensions while the Kirchhoff-Love theory is applicable to thinner plates. Both theories include in-plane shear strains and both are extensions of Kirchhoff-Love plate theory incorporating first- order shear effects. SOLO Introduction to Elasticity 43. 01/05/15 43 MindlinReissner plate theoryKirchhoffLove plate theory Equilibrium equations Constitutive relations Therefore the only non-zero strains are in the in-plane directions. Unlike Kirchhoff-Love plate theory where are directly related to , Mindlin's theory requires that SOLO Deformed Midsurface Original Midsurface Deformed Midsurface Original Midsurface x w y w yx = = , wuz y w zuz x w zu zxyyx == == = ,, wuz y w zuz x w zu zxyyx == == = ,, x w y w yx , ( ) + = xy yy xx xy yy xx E EE EE 12 00 0 11 0 11 22 22 yx w kkz yx w z x u y u y w kkz y w z y u x w kkz x w z x u xyxy yx xy yyyy y yy xxxx x xx == = + = == = = == = = 22 2 2 2 2 2 2 2 2 :,22 :, :, 0=== zzyzxz 0,0, = zzyzxz ( ) ( ) ( ) + + + = yz xz xy yy xx yz xz xy yy xx E E E EE EE 12 0000 12 000 00 12 00 000 11 000 11 22 22 44. 01/05/15 44 MindlinReissner plate theory Constitutive relations SOLO 45. 01/05/15 MindlinReissner plate theory Governing equations Relationship to Reissner theory Reissner's theory Mindlin's theory SOLO Return to Table of Content 46. 46 Let consider an arbitrary Membrane Surface Element S, encompassed by a closed curve , and its projection on x-y plane is the Surface Element A. A Membrane is an Elastic Skin (h