美国麻省理工大学：《结构力学》（英文版） Unit 18 Other Issues In Buckling/Structural Instability

MT-1620 al.2002 Unit 18 Other lssues In Buckling/structural Instability Readings. Rivello 143,14.5,146,14.7( (read these at least, others at your leisure") Ch.15.Ch.16 Timoshenko Theory of Elastic Stability Jones Mechanics of Composite Materials. Ch. 5 Paul A Lagace, Ph. D Professor of aeronautics Astronautics and Engineering Systems Paul A Lagace @2001

MIT - 16.20 Fall, 2002 Unit 18 Other Issues In Buckling/Structural Instability Readings: Rivello Timoshenko Jones 14.3, 14.5, 14.6, 14.7 (read these at least, others at your “leisure”) Ch. 15, Ch. 16 Theory of Elastic Stability Mechanics of Composite Materials, Ch. 5 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001

MT-1620 al.2002 Have dealt, thus far, with perfect columns, loading eccentricities, and beam-columns. There are, however, many more issues in buckling/(static) structural instability, most of which will try to touch on a) Buckling versus Fracture Have looked at columns that are long enough such that they buckle However, it is possible that the material compressive ultimate stress may be reached before the static instability occurs Consider short/'squat column(saw in Unified) Figure 18.1 Representation of short column under compressive load Paul A Lagace @2001 Unit 18-2

MIT - 16.20 Fall, 2002 Have dealt, thus far, with perfect columns, loading eccentricities, and beam-columns. There are, however, many more issues in buckling/(static) structural instability, most of which will try to touch on. (a) Buckling versus Fracture Have looked at columns that are long enough such that they buckle. However, it is possible that the material compressive ultimate stress may be reached before the static instability occurs. Consider short/”squat” column (saw in Unified) Figure 18.1 Representation of short column under compressive load Paul A. Lagace © 2001 Unit 18 - 2

MT-1620 al.2002 丁 If o=compressive ultimate before P=Par, then failure occurs by material failure in compression squashing Using the slenderness ratio" previously defined E Where. define a column by its slenderness ratio and can plot the behavior and failure mode of various columns Paul A Lagace @2001 Unit 18-3

MIT - 16.20 Fall, 2002 P σ = A If σ = σcompressive ultimate before P = Pcr, then failure occurs by material failure in compression “squashing” Using the “slenderness ratio” previously defined: P π2 cr E σ cr = = ′ ρ 2 A (l ) where: l l′ = c “define” a column by its slenderness ratio and can plot the behavior and “failure mode” of various columns… Paul A. Lagace © 2001 Unit 18 - 3

MT-1620 al.2002 Figure 18.2 Summary plot showing general behavior of columns based on stress level versus slenderness ratio Euler curve ED-compressive yield actual I Transition behavior Regions of values depend on E and o What happens in the transition region? Paul A Lagace @2001 Unit 18-4

σ MIT - 16.20 Fall, 2002 Figure 18.2 Summary plot showing general behavior of columns based on stress level versus slenderness ratio actual behavior Euler curve compressive yield Regions of values depend on E and σcu What happens in the transition region? Paul A. Lagace © 2001 Unit 18 - 4

MT-1620 al.2002 (b Progressive Yielding Figure 108.3 Typical stress-strain plot for a ductile metal (in compression e As the column is loaded there is some deflection due to slight mperfections this means the highest load is at the outer part of the beam-column Paul A Lagace @2001 Unit 18-5

MIT - 16.20 Fall, 2002 (b) Progressive Yielding Figure 18.3 Typical stress-strain plot for a ductile metal (in compression) As the column is loaded, there is some deflection due to slight imperfections. This means the highest load is at the outer part of the beam-column. Paul A. Lagace © 2001 Unit 18 - 5

MT-1620 Fall 2002 Figure 18.4 Representation of region of highest stress in cross-section of beam-column ighest compressive stress Thus, this outer part is the first part to yield As the material yields, the modulus decreases Figure 18.5 Representation of tangent modulus angent modulus AE This changes the location of the centroid Paul A Lagace @2001 Unit 18-6

MIT - 16.20 Fall, 2002 Figure 18.4 Representation of region of highest stress in cross-section of beam-column highest compressive stress Thus, this outer part is the first part to yield. As the material yields, the modulus decreases. Figure 18.5 Representation of tangent modulus tangent modulus This changes the location of the centroid… Paul A. Lagace © 2001 Unit 18 - 6

MT-1620 Fall 2002 Figure 18.6 Representation of change in location of centroid of cross section due to local yielding lower modulus, E<E E This continues and it may eventually squash" or buckle(or a combination See rivello 14.6 c)Nonuniform Beam-Columns Have looked only at beams with uniform cross-sectional property El. Now let this vary with x(most likely I, not E) EXample: Tapered section Paul A Lagace @2001 Unit 18-7

MIT - 16.20 Fall, 2002 Figure 18.6 Representation of change in location of centroid of cross￾section due to local yielding lower modulus, ET See Rivello 14.6 (c) Nonuniform Beam-Columns Have looked only at beams with uniform cross-sectional property EI. Now let this vary with x (most likely I, not E). Example: Tapered section Paul A. Lagace © 2001 Unit 18 - 7

MT-16.20 al.2002 Figure 18.7 Representation of beam-column with tapered cross-section EIE EI I(x) Thus, the governing equation is El P dx(a dx dx must keep this inside the derivative Solve this via numerical techniques Energy Methods Galerkin Finite Element method · Finite Difference Rayleigh-Ritz -->See rive/lo 14.3 Paul A Lagace @2001 Unit 18-8

MIT - 16.20 Fall, 2002 Figure 18.7 Representation of beam-column with tapered cross-section EI = EI(x) Thus, the governing equation is: 2 2 d 2 EI dw  + P dw = 0 dx2  dx2  dx2 must keep this “inside” the derivative Solve this via numerical techniques: • Energy Methods • Galerkin • Finite Element Method • Finite Difference • Rayleigh-Ritz --> See Rivello 14.3 Paul A. Lagace © 2001 Unit 18 - 8

MT-1620 Fall 2002 (a)Buckling of Plates Thus far have considered aone-dimensional problem(structural property of main importance is l, besides EI). Now have a two-dimensional structure(a"plate" Figure 18.8 Representation of plate under compressive load Pin-sliding Q Free Paul A Lagace @2001 Unit 18-9

MIT - 16.20 Fall, 2002 (d) Buckling of Plates Thus far have considered a “one-dimensional” problem (structural property of main importance is l, besides EI). Now have a two-dimensional structure (a “plate”): Figure 18.8 Representation of plate under compressive load Pin-sliding Free Paul A. Lagace © 2001 Unit 18 - 9

MT-1620 al.2002 The poisson s ratio enters into play here. For an isotropic plate ge 兀2EⅠ Where a I =1/12 bh e/h bh 121-v2a whereas the column buckling load is 兀2 E TEAh 丌2E/h C212 The buckled shape will have components in both directions Figure 18.9 Representation of deflection of buckled square plate with all sides simply-suppo Paul A. Lagace @2001 Unit 18-10

MIT - 16.20 Fall, 2002 The Poisson’s ratio enters into play here. For an isotropic plate get: π2 EI P cr = l2 (1 − ν2 ) where: l = a I = 1/12 bh3 2 ⇒ σ = Pcr = π2 E  h cr bh 12 1 − ν2  a whereas the column buckling load is P = π2EI = π2EAh2 ⇒ σ = π2E  h 2 cr l2 l2 12 cr 12  l The buckled shape will have components in both directions: Figure 18.9 Representation of deflection of buckled square plate with all sides simply-supported Paul A. Lagace © 2001 Unit 18 - 10