MT-1620 al.2002 Unit 3 Review of) Language of Stress/Strain Analysis Readings B.M,PA2.A.3,A.6 Rivello 2.1.2.2 t&G Ch. 1(especially 1.7) Paul A Lagace, Ph. D Professor of aeronautics Astronautics and Engineering Systems Paul A Lagace @2001
MIT - 16.20 Fall, 2002 Unit 3 (Review of) Language of Stress/Strain Analysis Readings: B, M, P A.2, A.3, A.6 Rivello 2.1, 2.2 T & G Ch. 1 (especially 1.7) Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001
MT-1620 al.2002 Recall the definition of stress o= stress= intensity of internal force at a point Figure 3.1 Representation of cross-section of a general body →Fn △F Stress=im\△A △A→0 There are two types of stress on(fn1. Normal (or extensional): act normal to the plane of the element os( Fs) 2. Shear: act in-plane of element Sometimes delineated ast Paul A Lagace @2001 Unit 3-p. 2
MIT - 16.20 Fall, 2002 Recall the definition of stress: σ = stress = “intensity of internal force at a point” Figure 3.1 Representation of cross-section of a general body Fn Fs ∆F Stress = lim ∆A ∆A → 0 There are two types of stress: • σn (Fn) 1. Normal (or extensional): act normal to the plane of the element • σs (Fs) 2. Shear: act in-plane of element Sometimes delineated as τ Paul A. Lagace © 2001 Unit 3 - p. 2
MT-1620 al.2002 And recall the definition of strain e= strain ="percentage deformation of an infinitesimal element Figure 3.2 Representation of 1-Dimensional Extension of a body △L △L 8 =limL Again, there are two types of strain 1. Normal (or extensional): elongation of element E. 2. Shear: angular change of element Sometimes delineated as y Figure 3.3 stration of shear Deformation shear deformation/ Paul A Lagace @2001 Unit 3-p. 3
MIT - 16.20 Fall, 2002 And recall the definition of strain: ε = strain = “percentage deformation of an infinitesimal element” Figure 3.2 Representation of 1-Dimensional Extension of a body ∆L ε = lim L L → 0 Again, there are two types of strain: εn 1. Normal (or extensional): elongation of element εs 2. Shear: angular change of element Sometimes delineated as γ Figure 3.3 Illustration of Shear Deformation shear deformation! Paul A. Lagace © 2001 Unit 3 - p. 3
MT-1620 al.2002 Since stress and strain have components in several directions we need a notation to represent these(as you learnt initially in Unified Several possible Tensor(indicial) notation · Contracted notation will review here Engineering notation and g/ve examples Matrix notation in recitation IMPORTANT: Regardless of the notation the equations and concepts have the same meaning =earn be comfortable with be able to use all notations Tensor(or Summation) notation Easy"'to write complicated formulae Easy' to mathematically manipulate Elegant, rigorous Use for derivations or to succinctly express a set of equations or a long equation Paul A Lagace @2001 Unit 3-p. 4
MIT - 16.20 Fall, 2002 Since stress and strain have components in several directions, we need a notation to represent these (as you learnt initially in Unified) Several possible • Tensor (indicial) notation • Contracted notation will review here • Engineering notation and give examples • Matrix notation in recitation IMPORTANT: Regardless of the notation, the equations and concepts have the same meaning ⇒ learn, be comfortable with, be able to use all notations Tensor (or Summation) Notation • “Easy” to write complicated formulae • “Easy” to mathematically manipulate • “Elegant”, rigorous • Use for derivations or to succinctly express a set of equations or a long equation Paul A. Lagace © 2001 Unit 3 - p. 4
MT-1620 al.2002 EXample:X=号y Rules for subscripts NOTE: index= subscript Latin subscripts(m, n, p, q, . . take on the values 1, 2, 3 (3-D) Greek subscripts(a,B, y.take on the values 1, 2(2-D) When subscripts are repeated on one side of the equation within one term, they are called dummy indices and are to be summed on Thus: 手y1=∑手y1 Bty1+9… do not sum on i! Subscripts which appear only once on the left side of the equation within one term are called free indices and represent a separate equation Paul A Lagace @2001 Unit 3-p. 5
MIT - 16.20 Fall, 2002 Example: xi = fij yj • Rules for subscripts NOTE: index ≡ subscript • Latin subscripts (m, n, p, q, …) take on the values 1, 2, 3 (3-D) • Greek subscripts (α, β, γ …) take on the values 1, 2 (2-D) • When subscripts are repeated on one side of the equation within one term, they are called dummy indices and are to be summed on Thus: 3 fij yj = ∑ fij yj j=1 But fij yj + gi ... do not sum on i ! • Subscripts which appear only once on the left side of the equation within one term are called free indices and represent a separate equation Paul A. Lagace © 2001 Unit 3 - p. 5
MT-1620 Fall 2002 Thus X Key concept: The letters used for indices have no inherent meaning in and of themselves Thus:xi= y is the same as: X,=fs ys or x =fi yi Now apply these concepts for stress/strain analysis 1. Coordinate System Generally deal with right-handed rectangular Cartesian: y Paul A Lagace @2001 Unit 3-p. 6
MIT - 16.20 Fall, 2002 Thus: xi = ….. ⇒ x1 = ….. x2 = ….. x3 = ….. Key Concept: The letters used for indices have no inherent meaning in and of themselves Thus: xi = fij yj is the same as: xr = frs ys or xj = f y ji i Now apply these concepts for stress/strain analysis: 1. Coordinate System Generally deal with right-handed rectangular Cartesian: ym Paul A. Lagace © 2001 Unit 3 - p. 6
MT-1620 al.2002 Figure 3.4 Right-handed rectangular Cartesian coordinate system 3 Z 25 Compare notations y1,X Tensor Engineering X Z Note: Normally this is so, but always check definitions in any article, book, report, etc. Key issue is self-consistency, not consistency with a worldwide standard (an official one does not exIs Paul A Lagace @2001 Unit 3-p. 7
MIT - 16.20 Fall, 2002 Figure 3.4 Right-handed rectangular Cartesian coordinate system y3 , z y2, y Compare notations y1 , x y z 3 y y 2 y x 1 Tensor Engineering Note: Normally this is so, but always check definitions in any article, book, report, etc. Key issue is self-consistency, not consistency with a worldwide standard (an official one does not exist!) Paul A. Lagace © 2001 Unit 3 - p. 7
MT-1620 al.2002 2. Deformations/Displacements (3) Figure 3.5 ●p(y1,y2y), smalp (deformed position) PYY2 Y3) Capita/P (original position) Um= plym)-P(ym) Compare notations Direction in ensor Engineering Engineering u uVw xyz Paul A Lagace @2001 Unit 3-p. 8
MIT - 16.20 Fall, 2002 2. Deformations/Displacements (3) Figure 3.5 p(y1, y2, y3), small p (deformed position) P(Y1, Y2, Y3) Capital P (original position) um = p(ym) - P(ym) --> Compare notations Tensor Engineering Direction in Engineering u1 u x u2 v y u3 w z Paul A. Lagace © 2001 Unit 3 - p. 8
MT-1620 al.2002 3. Components of Stress (6) o“ Stress tensor' 2 subscripts= 2nd order tensor 6 independent components Extensional Shear O Note: stress tensor is symmetric due to equilibrium(moment)considerations Meaning of subscripts stress acts in n-direction stress acts on face with normal vector in the m-direction Paul A Lagace @2001 Unit 3-p. 9
⇒ σ MIT - 16.20 Fall, 2002 3. Components of Stress (6) σmn “Stress Tensor” 2 subscripts ⇒ 2nd order tensor 6 independent components Extensional σ11 σ22 σ33 Note: stress tensor is symmetric σmn = σnm Shear σ12 = σ21 σ23 = σ32 σ13 = σ31 due to equilibrium (moment) considerations Meaning of subscripts: σmn stress acts in n-direction stress acts on face with normal vector in the m-direction Paul A. Lagace © 2001 Unit 3 - p. 9
MT-1620 al.2002 Figure 3.6 Differential element in rectangular system 4 NOTE: If face has a negative normal positive stress is in negative direction -- Compare notations Tensor Engineering 23 sometimes 0 used for shear stresses Paul A Lagace @2001 Unit 3-p. 10
MIT - 16.20 Fall, 2002 Figure 3.6 Differential element in rectangular system NOTE: If face has a “negative normal”, positive stress is in negative direction --> Compare notations = τyz sometimes = τxz used for = τxy shear stresses Tensor Engineering σ11 σx σ22 σy σ33 σz σ23 σyz σ13 σxz σ12 σxy Paul A. Lagace © 2001 Unit 3 - p. 10
