MT-1620 al.2002 Unit 8 Solution Procedures Readings R Ch. 4 T&G 17,Ch.3(18-26) Ch.4(27-46) Ch.6(54-73) Paul A Lagace Ph D Professor of aeronautics Astronautics and engineering systems Paul A Lagace @2001
MIT - 16.20 Fall, 2002 Unit 8 Solution Procedures Readings: R Ch. 4 T & G 17, Ch. 3 (18-26) Ch. 4 (27-46) Ch. 6 (54-73) Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001
MT-1620 al.2002 Summarizing what we've looked at in elasticity, we have 15 equations In 15 unknowns 3 equilibrium 6 strains 6 strain-displacement 3 displacements 6 stress-strain 6 stresses These must be solved for a generic body under some generic loading subject to the prescribed boundary conditions(BCs There are two types of boundary conditions 1. Normal(stress prescribed) 2. Geometric(displacement prescribed) you must have one or the other To solve this system of equations subject to such constraints over the continuum of a generic body is, in general, quite a challenge. There are basically two solution procedures 1. Exact--satisfy all the equations and the B.C.s 2. Numerical --come as close as possible"(energy methods, etc. Paul A Lagace @2001 Unit 8-p. 2
MIT - 16.20 Fall, 2002 Summarizing what we’ve looked at in elasticity, we have: 15 equations in 15 unknowns - 3 equilibrium - 6 strains - 6 strain-displacement - 3 displacements - 6 stress-strain - 6 stresses These must be solved for a generic body under some generic loading subject to the prescribed boundary conditions (B.C.’s) There are two types of boundary conditions: 1. Normal (stress prescribed) 2. Geometric (displacement prescribed) --> you must have one or the other To solve this system of equations subject to such constraints over the continuum of a generic body is, in general, quite a challenge. There are basically two solution procedures: 1. Exact -- satisfy all the equations and the B.C.’s 2. Numerical -- come as “close as possible” (energy methods, etc.) Paul A. Lagace © 2001 Unit 8 - p. 2
MT-1620 Fall 2002 Let's consider"exact techniques. a common and classic, one is Stress functions Relate six stresses to (fewer) functions defined in such a manner at they identically satisfy the equilibrium conditon Can be done for 3-d case Can be done for anisotropic(most often orthotropic)case See: Lekhnitskii, Anisotropic Plates, Gordan breach. 1968 >Lets consider plane stress (eventually) isotropic 8 equations in 8 unknowns 2 equilibrium 3 strains 3 strain-displacement 2 displacements 3 stress-strain 3 stresses Paul A Lagace @2001 Unit 8-p.3
MIT - 16.20 Fall, 2002 Let’s consider “exact” techniques. A common, and classic, one is: Stress Functions • Relate six stresses to (fewer) functions defined in such a manner that they identically satisfy the equilibrium conditon • Can be done for 3-D case • Can be done for anisotropic (most often orthotropic) case --> See: Lekhnitskii, Anisotropic Plates, Gordan & Breach, 1968. --> Let’s consider • plane stress • (eventually) isotropic 8 equations in 8 unknowns - 2 equilibrium - 3 strains - 3 strain-displacement - 2 displacements - 3 stress-strain - 3 stresses Paul A. Lagace © 2001 Unit 8 - p. 3
MT-1620 al.2002 Define the" Airy"Stress Function = o(x, y) English a scalar mathematician + v 8-2) yy v (8-3) aXd where: V=potential function for body forces fx and ty dV V exists if VXf=0 (CI that is of Paul A Lagace @2001 Unit 8-p. 4
MIT - 16.20 Fall, 2002 Define the “Airy” Stress Function = φ(x, y) English a scalar mathematician ∂2φ σxx = 2 + V (8 -1) ∂y ∂2φ σyy = ∂x2 + V (8 - 2) 2φ σxy = − ∂ (8 - 3) ∂ ∂x y where: V = potential function for body forces fx and fy fx = − ∂V fy = − ∂V ∂x ∂y V exists if ∇ x f = 0 (curl) that is: ∂fx = ∂fy ∂y ∂x Paul A. Lagace © 2001 Unit 8 - p. 4
MT-1620 al.2002 Recall that curl f=0→“ conservative" field gravity forces spring forces etc What does that compare to in fluids? rrotational flow Look at how p has been defined and what happens if we place these equations(8-1-8-3)into the plane stress equilibrium equations do 0 f=0 (E1) yy (E2) aX we then get Paul A Lagace @2001 Unit 8-p. 5
MIT - 16.20 Fall, 2002 Recall that curl f = 0 ⇒ “conservative” field - gravity forces - spring forces - etc. What does that compare to in fluids? Irrotational flow Look at how φ has been defined and what happens if we place these equations (8-1 - 8-3) into the plane stress equilibrium equations: ∂σxx + ∂σxy + fx = 0 (E1) ∂x ∂y ∂σxy ∂σyy ∂x + ∂y + fy = 0 (E2) we then get: Paul A. Lagace © 2001 Unit 8 - p. 5