美国麻省理工大学：《结构力学》英文版 Unit 7 Transformations and Other Coordinate Systems

MT-1620 al.2002 Unit 7 Transformations and other Coordinate Systems Readings R 2-4,2-5,2-7.29 BMP 5.6.57,5.14,64,6.8,69,6.11 T&g 13,Ch.7(74-83) On "other coordinate systems T&G 27,54,55,60,61 Paul A Lagace Ph D Professor of aeronautics Astronautics and engineering systems Paul A Lagace @2001

MIT - 16.20 Fall, 2002 Unit 7 Transformations and Other Coordinate Systems Readings: R 2-4, 2-5, 2-7, 2-9 BMP 5.6, 5.7, 5.14, 6.4, 6.8, 6.9, 6.11 T & G 13, Ch. 7 (74 - 83) On “other” coordinate systems: T & G 27, 54, 55, 60, 61 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001

MT-1620 al.2002 As we've previously noted, we may often want to describe a structure in various axis systems this involves Transformations Axis, Deflection, Stress, Strain, Elasticity Tensors) e.g., loading axes <-- material principal axis Figure 7.1 Unidirectional Composite with Fibers at an Angle fibers Know stresses along loading axes, but want to know stresses (or whatever)in axis system referenced to the fiber Paul A Lagace @2001 nit7-p. 2

MIT - 16.20 Fall, 2002 As we’ve previously noted, we may often want to describe a structure in various axis systems. This involves… Transformations (Axis, Deflection, Stress, Strain, Elasticity Tensors) e.g., loading axes material principal axis Figure 7.1 Unidirectional Composite with Fibers at an Angle fibers Know stresses along loading axes, but want to know stresses (or whatever) in axis system referenced to the fiber. Paul A. Lagace © 2001 Unit 7 - p. 2

MT-1620 Fall 2002 Problem: get expressions for(whatever) in one axis system in terms of (whatever in another axis system (Review from Unified) Recall: nothing is inherently changing, we just describe a body from a different reference Use (tilde) to indicate transformed axis system Figure 7.2 General rotation of 3-d rectangular axis system ar cartesian) g Paul A Lagace @2001 Unit 7-p 3

MIT - 16.20 Fall, 2002 Problem: get expressions for (whatever) in one axis system in terms of (whatever) in another axis system (Review from Unified) Recall: nothing is inherently changing, we just describe a body from a different reference. Use ~ (tilde) to indicate transformed axis system. Figure 7.2 General rotation of 3-D rectangular axis system (still rectangular cartesian) Paul A. Lagace © 2001 Unit 7 - p. 3

MT-1620 Fall 2002 Define this transformation via direction cosines lmn= cosine of angle from ym to y Notes: by convention, angle is measured positive counterclockwise(+ CCw) (not needed for cosine) mn since cos is an even function CoS(0)=CoS(-8 (reverse direction) But ≠ angle differs by 20 The order of a tensor governs the transformation needed an nth order tensor requires an nth order transformation (can prove by showing link of order of tensor to axis system via governing equations) Paul A Lagace @2001 Unit 7-p 4

MIT - 16.20 Fall, 2002 “Define” this transformation via direction cosines ~ l~mn = cosine of angle from ym to yn Notes: by convention, angle is measured positive counterclockwise (+ CCW) (not needed for cosine) l~ mn = lnm ~ since cos is an even function cos (θ) = cos (-θ) (reverse direction) But l~ ~ mn ≠ lmn angle differs by 2θ! The order of a tensor governs the transformation needed. An nth order tensor requires an nth order transformation (can prove by showing link of order of tensor to axis system via governing equations). Paul A. Lagace © 2001 Unit 7 - p. 4

MT-1620 al.2002 Quantity. Transformation Equation Physical Basis Stress mp ng pq equilibrium Strain mp ng pq geometr Axis Wm=lmip xp geometry Displacement geometr Elasticity tensor E Hookes law mnp pt qustus In many cases, we deal with 2-D cases (replace the latin subscripts by greek subscripts) e. g, OaB=a0 Br Bt (These are written out for 2-D in the handout Paul A Lagace @2001 Unit 7-p 5

MIT - 16.20 Thus: Quantity Transformation Equation ˜ Stress σmn = l mp l ˜ nq ˜ σpq Strain ε˜ = l ˜ l ˜ mn ε mp nq pq Axis x˜m = l mp x ˜ p Displacement u˜m = l mp u ˜ p Fall, 2002 Physical Basis equilibrium geometry geometry geometry Elasticity Tensor E˜ mnpq = l lns l pt l mr˜ ˜ ˜ qu ˜ Erstu Hooke’s law In many cases, we deal with 2-D cases (replace the latin subscripts by greek subscripts) e.g., σ˜ αβ = ll αθ˜ βτ˜ σθτ (These are written out for 2-D in the handout). Paul A. Lagace © 2001 Unit 7 - p. 5

An important way to illustrate transformation of stress and strain in 2-0 3002 MT-1620 Fall via Mohr's circle(recall from Unified ) This was actually used B.C. (before calculators). It is a geometrical representation of the transformation (See handout (you will get to work with this in a problem set) Also recal‖l (Three) Important Aspects Associated with Stress/Strain Transformations 1. Principal Stresses/Strains(AXes): there is a set of axes into which any state of stress /strain can be resolved such that there are no shear stresses/strains >O depend on applied loads Ei depend on applied loads and material response Thus note For general materials axes for principal strain axes for principal stress Generally Chave nothing to do with) material principal axes* principal axes of stress/strain Paul A Lagace @2001 Unit 7-p 6

MIT - 16.20 Fall, 2002 An important way to illustrate transformation of stress and strain in 2-D is via Mohr’s circle (recall from Unified). This was actually used B.C. (before calculators). It is a geometrical representation of the transformation. (See handout). (you will get to work with this in a problem set). Also recall… (Three) Important Aspects Associated with Stress/Strain Transformations 1. Principal Stresses / Strains (Axes): there is a set of axes into which any state of stress / strain can be resolved such that there are no shear stresses / strains --> σij depend on applied loads --> εij depend on applied loads and material response Thus, note: For general materials… axes for principal strain ≠ axes for principal stress Generally: (have nothing to do with) material principal axes ≠ principal axes of stress / strain Paul A. Lagace © 2001 Unit 7 - p. 6

MT-1620 al.2002 Find via roots of equation 12 13 12 0 eigenvalues:o1,σ1 I9 II (same for strain) 2. Invariants: certain combinations of stresses strains are invariant with respect to the axis system Most important: E(extensional stresses /strains)=Invariant very useful in back-of-envelope /quick check" calculations 3. EXtreme shear stresses/strains:(in 3-D)there are three planes along which the shear stresses /strains are maximized These values are often used in failure analysis(recall Tresca condition from unified These planes are oriented at 45 to the planes defined by the principal axes of stress strain(use rotation to find these) Paul A Lagace @2001 Unit 7-p. 7

MIT - 16.20 Fall, 2002 Find via roots of equation: σ11 − τ σ12 σ13 σ12 σ22 − τ σ23 = 0 σ13 σ23 σ33 − τ eigenvalues: σI, σII, σIII (same for strain) 2. Invariants: certain combinations of stresses / strains are invariant with respect to the axis system. Most important: Σ (extensional stresses / strains) = Invariant very useful in back-of-envelope / “quick check” calculations 3. Extreme shear stresses / strains: (in 3-D) there are three planes along which the shear stresses / strains are maximized. These values are often used in failure analysis (recall Tresca condition from Unified). These planes are oriented at 45° to the planes defined by the principal axes of stress / strain (use rotation to find these) Paul A. Lagace © 2001 Unit 7 - p. 7

Not only do we sometimes want to change the orientation of MT-1620 al.20 the axes we use to describe a body but we find it more convenient to describe a body in a coordinate system other than rectangular cartesian thus consider Other Coordinate Systems The“ easiest" case is Cylindrical (or Polar in 2-D)coordinates Figure 7.3 Loaded disk Figure 7. 4 Stress around a hole Paul A Lagace @2001 Unit 7-p8

MIT - 16.20 Fall, 2002 Not only do we sometimes want to change the orientation of the axes we use to describe a body, but we find it more convenient to describe a body in a coordinate system other than rectangular cartesian. Thus, consider… Other Coordinate Systems The “easiest” case is Cylindrical (or Polar in 2-D) coordinates Figure 7.3 Loaded disk Figure 7.4 Stress around a hole Paul A. Lagace © 2001 Unit 7 - p. 8

MT-1620 al.2002 Figure 7.5 Shaft Define the point p by a different set of coordinates other than y1, y2, y3 Figure 7.6 Polar coordinate representation ,2 Volume s rde 2 2 Paul A Lagace @2001 Unit 7-p 9

MIT - 16.20 Fall, 2002 Figure 7.5 Shaft Define the point p by a different set of coordinates other than y1, y2, y3 Figure 7.6 Polar coordinate representation Volume = rdθdrdz Paul A. Lagace © 2001 Unit 7 - p. 9

MT-1620 al.2002 Useθ,r, z Where 1= r cos 0 y- = rSIn are the"mapping"functions Now the way we describe stresses, etc. change >Differentia/ element is now different Rectangular cartesian Figure 7. 7 Differential element in rectangular cartesian system 3 1 dy Volume = dy1 Paul A Lagace @2001

MIT - 16.20 Fall, 2002 Use θ, r, z where: y1 = r cos θ y2 = r sinθ y3 = z are the “mapping” functions Now the way we describe stresses, etc. change… --> Differential element is now different Rectangular cartesian Figure 7.7 Differential element in rectangular cartesian system Volume = dy1 dy2 dy3 Paul A. Lagace © 2001 Unit 7 - p. 10