Stability Analysis of Nonlocal Elastic Columns with Initial Imperfection

Abstract

Investigated herein is the postbuckling behavior of an initially imperfect nonlocal elastic column, which is simply supported at one end and subjected to an axial force at the other movable end. The governing nonlinear differential equation of the axially loaded nonlocal elastic column experiencing large deflection is first established within the framework of Eringen's nonlocal elasticity theory in order to embrace the size effect. Its semianalytical solutions by the virtue of homotopy perturbation method, as well as the successive approximation algorithm, are determined in an explicit form, through which the postbuckling equilibrium loads in terms of the end rotation angle and the deformed configuration of the column at this end rotation are predicted. By comparing the degenerated results with the exact solutions available in the literature, the validity and accuracy of the proposed methods are numerically substantiated. The size effect, as well as the initial imperfection, on the buckled configuration and the postbuckling equilibrium path is also thoroughly discussed through parametric studies.

Discrete and non-local elastica

“…(19) can be efficiently approximated by the Padé's approximant (see, for instance [27][28][29][30][31] for axial wave applications):…”

“…Using the Eringen's model applied at the beam scale, the non-local bending 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 moment -curvature elastic law is written for a non-local EulerBernoulli column as [19,20]<...>…”

“…(29), it is shown that Eringen's non-local elastica problem of the non-local column is exactly governed by the non-linear differential equation Eq. (22) (see also [19] for a similar non-linear differential equation including some initial imperfections). Eq.…”

“…the discrete elastica. It is worth mentioning that the non-local elastica has been already investigated analytically or numerically by Wang et al [17] or Shen [18] for perfectly straight non-local columns, or Xu et al [19] who considered some initial imperfections. Shen [18] also performed some asymptotic expansions in order to obtain some approximated analytical expressions for the post-bifurcated branch.…”

Thermal behavior of elastic columns with second-mode imperfections

“…It should be noted that the mechanical behavior of steel columns with imperfections is different from perfect columns. Often the shape of the imperfection is assumed to be similar to the first buckling mode (symmetric shape) of the perfect columns [13][14][15].However, the imperfection shape of the columns may have the shape of the second buckling mode (anti-symmetric shape). The post-buckling of pinned and cantilevered columns with asymmetric imperfections was studied by Plaut et al [16].…”