美国麻省理工大学：《结构力学》（英文版） Unit 16 Bifurcation Buckling

MT-1620 al.2002 Unit 16 Bifurcation Buckling Readings Rivello 14.1.14.2.144 Paul A Lagace, Ph. D Professor of aeronautics Astronautics and Engineering Systems Paul A Lagace @2001

MIT - 16.20 Fall, 2002 Unit 16 Bifurcation Buckling Readings: Rivello 14.1, 14.2, 14.4 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001

MT-1620 al.2002 V. Stability and Buckling Paul A Lagace @2001 Unit 16-2

MIT - 16.20 Fall, 2002 V. Stability and Buckling Paul A. Lagace © 2001 Unit 16 - 2

MT-1620 al.2002 Now consider the case of compressive loads and the instability they can cause. Consider only static instabilities (static loading as opposed to dynamic loading [ e.g., flutter) From Unified, defined instability via a system becomes unstable when a negative stiffness overcomes the natural stiffness of the structure (Physically, the more you push, it gives more and builds on Itselt Review some of the mathematical concepts. Limit initial discussions to columns Generally, there are two types of buckling/instability Bifurcation buckling Snap-through buckling Paul A Lagace @2001 Unit 16-3

MIT - 16.20 Fall, 2002 Now consider the case of compressive loads and the instability they can cause. Consider only static instabilities (static loading as opposed to dynamic loading [e.g., flutter]) From Unified, defined instability via: “A system becomes unstable when a negative stiffness overcomes the natural stiffness of the structure.” (Physically, the more you push, it gives more and builds on itself) Review some of the mathematical concepts. Limit initial discussions to columns. Generally, there are two types of buckling/instability • Bifurcation buckling • Snap-through buckling Paul A. Lagace © 2001 Unit 16 - 3

MT-1620 Fall 2002 Bifurcation Buckling There are two(or more)equilibrium solutions(thus the solution path bifurcates) from Unified Figure 16.1 Representation of initially straight column under compressive load X Paul A Lagace @2001 Unit 16-4

MIT - 16.20 Fall, 2002 Bifurcation Buckling There are two (or more) equilibrium solutions (thus the solution path “bifurcates”) from Unified… Figure 16.1 Representation of initially straight column under compressive load Paul A. Lagace © 2001 Unit 16 - 4

MT-1620 al.2002 Figure 16.2 Basic load-deflection behavior of initially straight column under compressive load Actual behavior Note: Bifurcation is a mathematical concept. The manifestations in an actual system are altered due to physical realities/imperfections sometimes these differences can be very important (first continue with ideal case Perfect ABC-Equilibrium position, but unstable behavior BD-Equilibrium position There are also other equilibrium positions Imperfections cause the actual behavior to only follow this as asymptotes(will see later) Paul A Lagace @2001 Unit 16-5

MIT - 16.20 Fall, 2002 Figure 16.2 Basic load-deflection behavior of initially straight column under compressive load Actual behavior Note: Bifurcation is a mathematical concept. The manifestations in an actual system are altered due to physical realities/imperfections. Sometimes these differences can be very important. (first continue with ideal case…) Perfect ABC - Equilibrium position, but unstable behavior BD - Equilibrium position There are also other equilibrium positions Imperfections cause the actual behavior to only follow this as asymptotes (will see later) Paul A. Lagace © 2001 Unit 16 - 5

MT-1620 al.2002 Snap-Though Buckling Figure 16.3 Representation of column with curvature(shallow arch) with load applied perpendicular to column P Figure 16.4 Basic load-deflection behavior of shallow arch with transverse load F arch"snaps through" to F when load reaches c Thus there are nonlinear load-deflection curves in this behavior Paul A Lagace @2001 Unit 16-6

MIT - 16.20 Fall, 2002 Snap-Though Buckling Figure 16.3 Representation of column with curvature (shallow arch) with load applied perpendicular to column Figure 16.4 Basic load-deflection behavior of shallow arch with transverse load arch “snaps through” to F when load reaches C Thus, there are nonlinear load-deflection curves in this behavior Paul A. Lagace © 2001 Unit 16 - 6

MT-1620 Fall 2002 For"deeper arches antisymetric behavior is possible Figure 16.5 Representation of antsy metric buckling of deeper arch under transverse load 2 /(flops over) before snapping throug h Figure 16.6 Load-deflection behavior of deeper arch under transverse load ABCDEF-symmetric snap through ABF-antisymmetric behavior E Paul A Lagace @2001 Unit 16-7

MIT - 16.20 Fall, 2002 For “deeper” arches, antisymetric behavior is possible Figure 16.5 Representation of antisymetric buckling of deeper arch under transverse load (flops over) before snapping through Figure 16.6 Load-deflection behavior of deeper arch under transverse load ABCDEF - symmetric snap￾through ABF - antisymmetric behavior A D E • • • Paul A. Lagace © 2001 Unit 16 - 7

MT-1620 al.2002 Will deal mainly with Bifurcation Buckling First consider the " perfect case: uniform column under end load First look at the simply-supported case. column is initially straight Load is applied along axis of beam Perfect column only axial shortening occurs(before instability), i.e., no bending Figure 16.7 Simply-supported column under end compressive load EI= constant Paul A Lagace @2001 Unit 16-8

MIT - 16.20 Fall, 2002 Will deal mainly with… Bifurcation Buckling First consider the “perfect” case: uniform column under end load. First look at the simply-supported case…column is initially straight • Load is applied along axis of beam • “Perfect” column ⇒ only axial shortening occurs (before instability), i.e., no bending Figure 16.7 Simply-supported column under end compressive load EI = constant Paul A. Lagace © 2001 Unit 16 - 8

MT-16.20 al.2002 Recall the governing equation W el- t p dx dx Notice that p does not enter into the equation on the right hand side(making the differential equation homogenous), but enters as a coefficient of a linear differential term This is an eigenvalue problem. Let W=已 this gives EI i0.0 E epeated roots= need to look for more solutions End up with the following general homogenous solution W= Asin-x+ Bcos -x+C+ Dx El Paul A Lagace @2001 Unit 16-9

MIT - 16.20 Fall, 2002 Recall the governing equation: 4 2 EI dw + P dw = 0 dx 4 dx2 --> Notice that P does not enter into the equation on the right hand side (making the differential equation homogenous), but enters as a coefficient of a linear differential term This is an eigenvalue problem. Let: λ x w = e this gives: λ4 + P λ2 = 0 EI ⇒ λ = ± P EI i 0, 0 repeated roots ⇒ need to look for more solutions End up with the following general homogenous solution: w = Asin P EI x + B cos P EI x + C + Dx Paul A. Lagace © 2001 Unit 16 - 9

MT-1620 al.2002 where the constants A, B, C, d are determined by using the Boundary Conditions For the simply-supported case, boundary conditions are x=0W=0 M=EⅠ @x=e w=0 M=0 From: W(X=0)=0→B+C=0 M(X=0)=0→-EIB=0 C=0 El WX=0)=0→Asn,,l+Dl=0 El D=0 (x=O) 0→-EⅠAsin1=0 El El Paul A Lagace @2001 Unit 16-10

MIT - 16.20 Fall, 2002 where the constants A, B, C, D are determined by using the Boundary Conditions For the simply-supported case, boundary conditions are: @ x = 0 w = 0 2 M = E I dw = 0 dx2 @ x = l w = 0 M = 0 From: w(x = 0) = 0 ⇒ B + C = 0 B = 0 ⇒ M(x = 0) = 0 ⇒ − EI P B = 0 C = 0 EI w(x = l) = 0 ⇒ Asin P EI l + Dl = 0 ⇒ D = 0 M(x = l) = 0 ⇒ − EI P Asin P EI l = 0 EI Paul A. Lagace © 2001 Unit 16 - 10