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

2. 01/05/15 2 Introduction to Elasticity SOLO Table of Content Introduction Stress Body Forces and Moments 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 Summary Stress-Strain Compatibility Equations Elastic Waves Equations 3. 01/05/15 3 Introduction to Elasticity SOLO Table of Content Torsion of a Circular Bar Torsion 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 Bar of Narrow Rectangular Section Timoshenko Beam Theory Thick Beam Theory Narrow Profiles Open Sections Narrow Profiles Closed Sections Energy Equations Energy Methods 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 P a r t I I 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 P a r t I I 6. 01/05/15 6 SOLO The base of this Presentation (Part I) is a Summary of the Courses at Stanford University AA240A (C.R. Steele) and AA240B (G.S. Springer), Analysis of Structures in 1983-1984. They were part of my preparation for Qualification Examination in 1984. Additional material (Part II) was added using Wikipedia and the given references. Introduction to Elasticity Charles R. Steele Stanford University George S. Springer Stanford University 7. 01/05/15 7 SOLO Introduction to Elasticity Prevailing consensus is that Galileo Galilei made the first attempts at developing a theory of beams, but recent studies argue that Leonardo da Vinci was the first to make the crucial observations. Da Vinci lacked Hooke's law and calculus to complete the theory, whereas Galileo was held back by an incorrect assumption he made Elasticity History Leonardo da Vinci (1452 1519) Galileo Galilei (1564[4] 1642)[ Robert Hooke (1635 - 1703) In 1678 Robert Hooke formulated the relation between Force and Deformation, known today as Hookes Law. Jacob Bernoulli (1655 -1705) Leonhard Euler (1707 1783) Daniel Bernoulli (1700 1782) The Differential Equations of the Thin Beam were developed by Jacob Bernoulli (1705), Daniel Bernoulli (1742) and Leonhard Euler (1744), known as Bernoulli-Euler Equations. 8. 01/05/15 SOLO Introduction to Elasticity Elasticity History (continue 1) Charles-Augustin de Coulomb (1736 1806) The first comprehensive discussion of the fiber stresses in a laterally loaded beam was presented in 1776 by Charles- Augustin Coulomb. In 1785 appeared his Recherches thoriques et exprimentales sur la force de torsion eti sur llasticit des fils de mtal, etc. In this work he laid the foundations of the theory for the Bar Torsion. 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, 17xx). Ernst Florens Friedrich Chladni (1756 1827) 9. 01/05/15 9 Elasticity History (continue 2) Thomas Young (1773 1829) Youngs Modulus it is named after Thomas Young, however, the concept was developed in 1727 by Leonhard Euler, and the first experiments that used the concept of Young's modulus in its current form were performed by the Italian scientist Giordano Riccati in 1782, predating Young's work (1807) by 25 years Leonhard Euler (1707 1783) Giordano Riccati ( 1709 1790) 10. 01/05/15 SOLO Introduction to Elasticity Elasticity History (continue 3) 1821 Navier, developed special case of linear elasticity via molecular model 1822 Cauchy, stress, nonlinear and linear elasticity Claude-Louis Navier 1785 1836) Augustin Louis Cauchy (1789-1857) 1811 Simon Denis Poisson introduced the Poissons Ratio in Trait de Mcanique Simon Denis Poisson ( 1781 1840),The longitudinal vibration of rods was investigated experimentally by Chladni (1787) and Biot (1816). The published analytical equation and solutions of longitudinal vibration of rods done by Navier in 1824. 11. 01/05/15 History of Plate Theories The French mathematician Germain developed a plate differential equation that lacked a warping term (Germain, 1811), but one of the reviewers of her work, Lagrange (1813 ), corrected Germains results; thus, he was the first person to present the general plate equation properly . Marie-Sophie Germain (1776 1831) Joseph-Louis Lagrange (1736 1813) Elasticity History (continue 4) 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 . Navier (1823) considered the plate thickness in the general plate equation as a function of rigidity, D. Augustin Louis Cauchy (1789-1857) Simon Denis Poisson ( 1781 1840), Claude-Louis Navier 1785 1836) Lagrange derived the Biharmonic Equation for Thin Plates but never published, found posthumously between his papers (1813) 12. 01/05/15 Elasticity History (continue 5) In 1855, the French elasticity theorist Adhmar Jean Claude Barr de Saint-Venant can be stated the Saint-Venant's Principle saying that "... the difference between the effects of two different but statically equivalent loads becomes very small at sufficiently large distances from load." Adhmar Jean Claude Barr de Saint-Venant (1797 1886) The credit for deriving the complete torsional wave equation and giving some rigorous results belongs to Saint-Venant , who published on this subject (de Saint-Venant, 1849). 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 SOLO Introduction to Elasticity Return to Table of Content Elasticity History (continue 7) In 1877, Rayleigh proposed an improvement to the dynamic Euler-Bernoulli beam theory by including the effect of rotational inertia of the cross-section of the beam. Timoshenko improved upon that theory in 1922 by adding the effect of shear into the beam equation. Shear deformations of the normal to the mid-surface of the beam are allowed in the Timoshenko-Rayleigh theory. John William Strutt, 3rd Baron Rayleigh (1842 1919) Stepan Prokopovych Tymoshenko (1878 1973) 15. 01/05/15 15 The Ritz method was immediately recognized by Russian mathematicians as a fundamental contribution, and put to use in the computational simulation of beams and plates, which led to the seminal paper of Galerkin in 1915. In Europe however, especially in the mathematical center of that time in Goettingen, it received very little attention, even though Ritz obtained a price from the French Academy of Sciences, after having lost in the official competition for the Vaillant price in 1907 to Hadamard. It was only during the second world war, long after Ritz's death, in an address of Courant in front of the AMS, that the potential of Ritz's invention was fully recognized, and Courant presented what we now call the finite element method. This name was given to the method after Clough reinvented it in a seminal paper, working for Boeing. SOLO Introduction to Elasticity Elasticity History (continue 6) 16. 01/05/15 16 History of Plate Theories (continues 1) 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, 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. Bubnov (1914) following Timoshenkos work, 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 Return to Table of Content 17. 01/05/15 17 SOLO Stress The Oxyz coordinates are in the center of the small volume dx dy dz. Let write the Force and Moment equations around the point O (the center of mass of the small volume) [ ] [ ]{ } [ ]( )0=+=+ =+ OOOBS OBS IIIMM amFF where S indicates Surface and B Body forces and moments. xx xx x dx+ 1 2 xx xx x dx 1 2 yx yx y dy+ 1 2 yz yz y dy 1 2 zx zx z dz+ 1 2 zx zx z dz 1 2 xy xy x dx+ 1 2 xy xy x dx 1 2 yy yy y dy+ 1 2 yy yy y dy 1 2 zy zy z dz+ 1 2 zy zy z dz 1 2 xz xz x dx 1 2 yz yz y dy+ 1 2 yx yx y dy 1 2 zz zz z dz+ 1 2 zz zz z dz 1 2 z y x dy dx dz O xz xz x dx+ 1 2 dzdxdy y dy y dzdydx x dx x yx yx yx yx xx xx xx xx / +/+ / +/ 2 1 2 1 2 1 2 1 dzdydxadzdydxGdydxdz z dz z xxB zx zx zx zx =+ / +/+ 2 1 2 1 dzdxdy y dy y dzdydx x dx x yy yy yy yy xy xy xy xy / +/+ / +/ 2 1 2 1 2 1 2 1 dzdxdy y dy y dzdydx x dx x yz yz yz yz xz xz xz xz ++ / +/ 2 1 2 1 2 1 2 1 dzdydxadzdydxGdydxdz z dz z zzB zz zz zz zz =+ / +/+ 2 1 2 1 Introduction to Elasticity 18. 01/05/15 18 SOLO Stress (continue 1) OBS amFF =+ xx xx x dx+ 1 2 xx xx x dx 1 2 yx yx y dy+ 1 2 yz yz y dy 1 2 zx zx z dz+ 1 2 zx zx z dz 1 2 xy xy x dx+ 1 2 xy xy x dx 1 2 yy yy y dy+ 1 2 yy yy y dy 1 2 zy zy z dz+ 1 2 zy zy z dz 1 2 xz xz x dx 1 2 yz yz y dy+ 1 2 yx yx y dy 1 2 zz zz z dz+ 1 2 zz zz z dz 1 2 z y x dy dx dz O xz xz x dx+ 1 2 zzB zzyzxz yyB zyyyxy xxB zxyxxx aG zyx aG zyx aG zyx =+++ =+++ =+++ VECTOR NOTATION CARTESIAN TENSOR NOTATION aG =+ ~ aG x i j ij =+CAUCHYs FIRST LAW OF MOTION (1827) Force Equation Introduction to Elasticity Augustin Louis Cauchy (1789-1857) 19. 01/05/15 19 SOLO Stress (continue 3) xx xx x dx+ 1 2 xx xx x dx 1 2 yx yx y dy+ 1 2 yz yz y dy 1 2 zx zx z dz+ 1 2 zx zx z dz 1 2 xy xy x dx+ 1 2 xy xy x dx 1 2 yy yy y dy+ 1 2 yy yy y dy 1 2 zy zy z dz+ 1 2 zy zy z dz 1 2 xz xz x dx 1 2 yz yz y dy+ 1 2 yx yx y dy 1 2 zz zz z dz+ 1 2 zz zz z dz 1 2 z y x dy dx dz O xz xz x dx+ 1 2 Moment Equation[ ] [ ]{ } [ ]( )0=+=+ OOOBS IIIMM ( ) 2/ 2/ 33 2/ 2/ 2 2 2 32 2 2 2 2 2 22 3123 dzz dzz dyy dyy dz dz dx dx dy dy dz dz O z z dy dydxzyz y dxxyzzyxxI = = = = += +=+= ( )22 12 dzdy dzdydx xxI O += or 0 0 2 2 2 2 2 2 2 2 2 2 2 2 === dz dz dy dy dx dx dx dx dy dy dz dz O zyyxxxyzxyxyI Hence the Moments of Inertia are: ( ) ( ) ( ) I xx dx dy dz dy dz I xy I xz I yy dx dy dz dx dz I yx I yz I zz dx dy dz dx dy I zx I zy O O O O O O O O O = + = = = + = = = + = = 12 0 12 0 12 0 2 2 2 2 2 2 The Moment Equations are: ( ) ( ) ( ) = = = yxOOzOzO zxOOyOyO zyOOxOxO yyIxxIzzxIM xxIzzIyyIM zzIyyIxxIM Introduction to Elasticity 20. 20 SOLO Stress (continue 4) xx xx x dx+ 1 2 xx xx x dx 1 2 yx yx y dy+ 1 2 yz yz y dy 1 2 zx zx z dz+ 1 2 zx zx z dz 1 2 xy xy x dx+ 1 2 xy xy x dx 1 2 yy yy y dy+ 1 2 yy yy y dy 1 2 zy zy z dz+ 1 2 zy zy z dz 1 2 xz xz x dx 1 2 yz yz y dy+ 1 2 yx yx y dy 1 2 zz zz z dz+ 1 2 zz zz z dz 1 2 z y x dy dx dz O xz xz x dx+ 1 2 Moment Equation[ ] [ ]{ } [ ]( )0=+=+ OOOBS IIIMM Assuming that the Moments of the Body Forces around O are zero we obtain: ( ) ( ) zyx xy xy xy xy yz yz yz yz dydz dzdydx dzdy dzdydx dz dydxdz z dz z dy dzdxdy y dy y 2222 1212 22 1 2 1 22 1 2 1 += + + ( ) ( ) zxy xz xz xz xz zx zx zx zx dzdx dzdydx dzdx dzdydx dx dzdydx x dx x dz dydxdz z dz z 2222 1212 22 1 2 1 22 1 2 1 += + + ( ) ( ) yxx yx yx yx yx xy xy xy xy dxdy dzdydx dydx dzdydx dy dzdxdx x dx x dx dzdydy y dy y 2222 1212 22 1 2 1 22 1 2 1 += + + ( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ] 12/ 12/ 12/ 2222 2222 2222 yxyyzxy zyyxzzx zyxzyyz dxdydydx dzdxdzdx dydzdzdy += += += or If we consider the limiting case dx dy dz 0 0 0 0 = = = yzxy xzzx zyyz CAUCHYs Second Law of Motion (1827) Introduction to Elasticity Return to Table of Content Augustin Louis Cauchy (1789-1857) 21. 21 SOLO Stress (continue 5) Boundary Conditions yA yA zA zA yA yA yA zA yA xA zA yA zA xA xA yA xA zA t txA tyA tzA n n O xA yA zA C B A d xA d yA d zA xB f f xA f yA f zAB xA yA zA= + + 1 1 1 The Oxyz coordinates are in the center of the small volume dx dy dz. O xA yA zA a cartesian system of coordinates. Consider the volume bounded by the points O(0,0,0), A(dx,0,0), B(0,dy,0), C(0,0,dz). - the unit vector normal to the surface ABC1 xB - the stress vector on ABC t The components of are: t n 1 xBa) perpendicular to ABC (in direction) n on ABC t t x t y t zxA A yA A zA A= + + 1 1 1b) Introduction to Elasticity 22. 22 SOLO Stress (continue 6) yA yA zA zA yA yA yA zA yA xA zA yA zA xA xA yA xA zA t txA tyA tzA n n O xA yA zA C B A d xA d yA d zA xB f f xA f yA f zAB xA yA zA= + + 1 1 1 t x t y t z S dy dz dx dz dx dy x dy dz dx dz dx dy y dy dz dx dz dx dy x f x f xA A yA A zA A ABC xA xA A A yA xA A A zA xA A A A xA yA A A yA yA A A zA yA A A A xA zA A A yA zA A A zA zA A A A xA A yA 1 1 1 2 2 2 1 2 2 2 1 2 2 2 1 1 1 + + + + + + + + + y f z dx dy dz a x a y a z dx dy dz A zA A A A A xA A yA A zA A A A A + = + + 1 6 1 1 1 6 ( ) ( ) ( ) dx dy S x z dx dz S x y dy dz S x x S dx dy dx dz dy dz A A ABC B A A A ABC B A A A ABC B A ABC A A A A A A 2 1 1 2 1 1 2 1 1 1 2 2 2 2 = = = = + + Boundary Conditions (continue 1) Introduction to Elasticity 23. 23 SOLO Stress (continue 7) Boundary Conditions (continue 2) yA yA zA zA yA yA yA zA yA xA zA yA zA xA xA yA xA zA t txA tyA tzA n n O xA yA zA C B A d xA d yA d zA xB f f xA f yA f zAB xA yA zA= + + 1 1 1 t x x x y x z S f dx dy dz a dx dy dz t x x x y x z xA xA xA B A yA xA B A zA yA B A ABC xA A A A xA A A A yA xA yA B A yA yA B A zA yA B A + = 1 1 1 1 1 1 6 6 1 1 1 1 1 1 + = + = S f dx dy dz a dx dy dz t x x x y x z S f dx dy dz a dx dy dz ABC yA A A A yA A A A zA xA zA B A yA zA B A zA zA B A ABC zA A A A zA A A A 6 6 1 1 1 1 1 1 6 6 When , therefore in the limitd x d y d z d x d y d z SA A A A A A ABC, ,