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 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 MIT - 16.20 Fall, 2002 P σ = A If σ = σ compressive ultimate before P = P cr , then failure occurs by material failure in compression “squashing ” Using the “slenderness ratio ” previously defined: P π 2 E cr 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 σσσ 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 σ cu What happens in the transition region? Paul A. Lagace ? 2001 Unit 18 - 4 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 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 MIT - 16.20 Fall, 2002 Figure 18.6 Representation of change in location of centroid of cross- section due to local yielding lower modulus, E T < 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 E I . 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 18.7 Representation of beam-column with tapered cross-section E I = E I (x) Thus, the governing equation is: 2 2 d 2 ? EI dw ? + P dw = 0 dx 2 ? ? dx 2 ?? dx 2 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 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 E I ). 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 MIT - 16.20 Fall, 2002 The Poisso n ’s ratio enters into play here. For an isotropic plate get: π 2 EI P = cr l 2 ( 1 ? ν 2 ) where: l = a I = 1/12 bh 3 2 ? σ = P cr = π 2 E ? h ? cr bh 12 1 ? ν 2 ? a ? whereas the column buckling load is P = π 2 EI = π 2 EAh 2 ? σ = π 2 E ? h ? 2 cr l 2 l 2 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 - 1 0 MIT - 16.20 Fall, 2002 π π w = w sin mx sin ny mn a b --> can have contributions of many modes. Will depend on boundary conditions on all four sides. --> See Rivello , Ch. 15 Even more complicated for orthotropic plates as the modulus varies in the two directions. (must also look at buckling due to shear loads) --> See Jones, Ch. 5 Note : for some “weird ” anisotropic plates with shear couplings, can get buckling under tension. ( Key question : Is there an induced compressive stress in some direction?) think back to basic definition of instability … Paul A. Lagace ? 2001 Unit 18 - 1 1 MIT - 16.20 Fall, 2002 (e) Cylinders (“ thin-walled things ”, like shells) Have dealt with “ globa l ” instabilities. However, buckling can also be a “ local ” instability. Figure 18.10 Representation of crippling in thin cylinder under axial compressive load local “crippling ” total axial load: P = σ (2 π ) R h for an isotropic cylinder: σ = 0 . 606 E h cr (linear) R Paul A. Lagace ? 2001 Unit 18 - 1 2 MIT - 16.20 Fall, 2002 The actual load where the local instability sets in is less than that predicted from linear theory due to imperfections in both geometry and loading: . σ cr ( actual ) ≈ ( 01 5 to 0 . 9 ) σ cr ( linear ) --> See Rivello , Ch. 15 (f) Reinforced Plates A common design in aerospace structures (and many other structures) is to reinforce a plate with stiffeners: Figure 18.11 Representation of plate with stiffeners Paul A. Lagace ? 2001 Unit 18 - 1 3 MIT - 16.20 Fall, 2002 The buckling can take place at several levels ? buckling of panels between stiffness ? buckling of “parts” of stiffeners (e.g., flange, web) ? global instability This can occur on a progressive basis. Analysis often uses only stiffness as carrying the load for buckling (axial load) or talk about “effective width ” of skin (this was previously discussed in talking about general shell beams and holds true for buckling) (g) Post-buckling When talked about buckling, talked about bifurcation. In that case w --> ∞ as P --> P cr (with imperfections). In reality, a structure can carry load after buckling ( “ post- buckling ” behavior). Paul A. Lagace ? 2001 Unit 18 - 1 4 MIT - 16.20 Fall, 2002 Figure 18.12 Representation of post-buckling behavior via load- deflection plot actual behavior w/post-buckling capability perfect linear behavior behavior with imperfection The critical assumption which breaks down is: SMALL DEFORMATIONS Must now account for geometrical nonlinear effects. ( Note : material nonlinear effects will also enter in as approach σ cu ) Paul A. Lagace ? 2001 Unit 18 - 1 5 MIT - 16.20 Fall, 2002 Consider: Post-Buckling of a beam-column (the issues are the same for a plate) Figure 18.13 Representation of post-buckling of a beam-column For large deflections, the moment-curvature equation is: Ms () = E I d θ ds where: d θ = curvature ds Look at a beam section: Paul A. Lagace ? 2001 Unit 18 - 1 6 MIT - 16.20 Fall, 2002 dw = sin θ ds differentiating: 2 dw d θ = cos θ ds 2 ds or: 2 d θ 1 dw = ds cos θ ds 2 with: dx cos θ = = ? ds dw 2 2 = ds ds ? ? ? ? ? ? ? dw ds 2 1 or: cos θ = sin ? 1 2 θ Paul A. Lagace ? 2001 Unit 18 - 1 7 MIT - 16.20 Fall, 2002 So: curvature = ? ? ? ? ? ? ? d ds dw ds dw ds θ 1 1 2 2 2 = For “moderate ” angles θ : (via expansion) 2 2 d θ = ? ? 1 + 1 ? dw ? + H.O.T. ?? dw ds ? 2 ? ds ? ? ds 2 nonlinear term In the absence of any primary moment, the moment at any point s is due to the deflection w at that point: Figure 18.14 Representation of resultants along the beam-column + ∑ M = 0 ? M + Pw = 0 ? M = -Pw Paul A. Lagace ? 2001 Unit 18 - 1 8 MIT - 16.20 Fall, 2002 So the basic Post-Buckling equation becomes: 2 2 ? 1 ? dw ? ? dw P ? 1 + 2 ? ds ? + H.O.T. ?? ds 2 + EI w = 0 ? This can be solved via ? Numerical Techniques ? Energy Method --> Effect appears to “stiffen ” the behavior Consider one (numerical) technique known as the … Galerkin Method 1. Assume a mode that satisfies the boundary conditions w = q 1 sin π s/ l unknown assumed mode shape satisfies all coefficient boundary conditions Paul A. Lagace ? 2001 Unit 18 - 1 9 MIT - 16.20 Fall, 2002 2. Integrate a weighted average of the solution and the Ordinary Differential Equation (Are minimizing the “residuals ” ) l s ? Differential ? ∫ 0 q 1 sin π ? ? ds = 0 l ? equation ? assumed mode shape 2 2 ? ? dw P + w Here: 1 1 ? ? dw ?? + 2 ? ds ? ? ? ds 2 EI This gives: π 2 3 π 4 Pl ? q 1 ? q 1 1 6 l 3 + q 1 = 0 2 l 2 EI Solving: ? π 4 2 π 2 Pl ? q 1 ? ? 16 l 3 q 1 + 2 l ? 2 EI ? ? = 0 Paul A. Lagace ? 2001 Unit 18 - 2 0 MIT - 16.20 Get: q 1 = ? 2 ? q 1 = q 1 = Fall, 2002 0 (trivial solution: w = 0) 2 8 l 2 ? P l ? π 2 ?? π 2 EI ? 1 ?? this latter gives: 0 . 903 l Note : q 1 only for P/ P cr > 1 P P cr 1 ? Plotting P/ P cr vs . w c (= q 1 ) gives: Figure 18.15 Representation of load versus center deflection for post- buckled beam-column based on Galerkin Method Paul A. Lagace ? 2001 Unit 18 - 2 1 MIT - 16.20 Fall, 2002 For w c / l > 0.3, include more terms: 2 4 1 ? dw ? 3 ? dw ? 1 + ? ? + ? ? + K 2 ? ds ? 8 ? ds ? See : Rivello , Timoshenko & Gere --> Postscript: Buckling and Failure When is a structure that buckles considered to have failed? ? Recall discussion of “failure ” at beginning of term. ? There is not (physically …only mathematically) “ a ” point of buckling. What happens is that deflection increases with load very rapidly. ? If failure is deflection-based, look at deflection; if stress/strain-base, look at that … Paul A. Lagace ? 2001 Unit 18 - 2 2 MIT - 16.20 Fall, 2002 Must consider : ? pre -buckling behavior: imperfections cause deflections and stresses. These may cause failure before “ buckling ” ? post -buckling behavior: “ extra” stiffening at large deflections may result in ability to carry deflections and stresses such that failure is after “buckling ” Paul A. Lagace ? 2001 Unit 18 - 2 3