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 snapthrough 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