This paper is concerned with buckling analysis of a nonuniform column with classical∕nonclassical boundary conditions and subjected to a concentrated axial force and distributed variable axial loading, namely, the generalized Euler’s problem. Exact solutions are derived for the buckling problem of nonuniform columns with variable flexural stiffness and under distributed variable axial loading expressed in terms of polynomial functions. Then, more complicated buckling problems are considered such as that the distribution of flexural stiffness of a nonuniform column is an arbitrary function, and the distribution of axial loading acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for such problems is reduced to Bessel equations and other solvable equations for seven cases by means of functional transformations. A class of exact solutions for the generalized Euler’s problem involved a nonuniform column subjected to an axial concentrated force and axially distributed variable loading is obtained herein for the first time in literature.

1.
Euler
,
L.
, 1757, “
Sur La Forces Des Colunnes
,”
Memoires de l’Academie Royale des Sciences et Belles Letters, Berlin
,
13
, pp.
252
282
.
2.
Greenhill
,
A. G.
, 1881, “
Determination of the Greatest Height Consistent With Stability That a Vertical Pole or Mast Can Be Made, and of the Greatest Height to Which a Tree of Given Proportions Can Grow
,”
Proc. Cambridge Philos. Soc.
0068-6735,
4
, pp.
765
774
.
3.
Morley
,
A.
, 1917, “
Critical Loads for Long Tapering Struts
,”
Engineering (London)
0013-7782,
104
, p.
295
.
4.
Dinnik
,
A. N.
, 1912, “
Buckling Under Own Weight
,”
Proceedings of the Don Polytechnic Institute
,
1
, p.
19
.
5.
Dinnik
,
A. N.
, 1955, “
Selected Papers, Application of Bessel Function to the Theory of Elasticity Problems
,”
Ukrainian Academy of Science
,
2
, pp.
73
87
.
6.
Karman
,
T. V.
, and
Biot
,
M. A.
, 1940,
Mathematical Methods in Engineering
,
McGraw-Hill
,
New York
.
7.
Timoshenk
,
S. P.
, and
Gere
,
G. M.
, 1961,
Theory of Elastic Stability
,
McGraw-Hill
,
New York
.
8.
Gere
,
J. M.
, and
Carter
,
W. O.
, 1962, “
Critical Buckling Loads for Tapered Columns
,”
ASCE J. Struct. Div.
0044-8001,
88
(
1
), pp.
1346
1354
.
9.
Ermopoulos
,
J. C.
, 1986, “
Buckling of Tapered Bars Under Stepped Axial Loads
,”
J. Struct. Eng.
0733-9445,
112
(
6
), pp.
1346
1354
.
10.
Williams
,
F. W.
, and
Aston
,
G.
, 1989, “
Exact or Lower Bound Tapered Column Buckling Loads
,”
J. Struct. Eng.
0733-9445,
115
(
5
), pp.
1088
1100
.
11.
Arbabi
,
F.
, and
Li
,
F.
, 1991, “
Buckling of Variable Cross-Section Columns—Integral Equation Approach
,”
J. Struct. Eng.
0733-9445,
117
(
8
), pp.
2426
2441
.
12.
Siginer
,
A.
, 1992, “
Buckling of Columns of Variable Flexural Rigidity
,”
J. Eng. Mech.
0733-9399,
118
(
3
), pp.
640
643
.
13.
Li
,
Q. S.
,
Cao
,
H.
, and
Li
,
G.
, 1994, “
Stability Analysis of Bars With Multi-Segments of Varying Cross-Section
,”
Comput. Struct.
0045-7949,
53
(
5
), pp.
1085
1089
.
14.
Li
,
Q. S.
,
Cao
,
H.
, and
Li
,
G.
, 1995, “
Stability Analysis of Bars With Varying Cross-Section
,”
Int. J. Solids Struct.
0020-7683,
32
(
21
), pp.
3217
3228
.
15.
Elishakoff
,
I.
, 2000, “
A Closed-Form Solution for the Generalized Euler Problem
,”
Proc. R. Soc. London, Ser. A
1364-5021,
456
, pp.
2409
2417
.
16.
Elishakoff
,
I.
, 2001, “
Inverse Buckling Problem for Inhomogeneous Columns
,”
Int. J. Solids Struct.
0020-7683,
38
, pp.
457
464
17.
Elishakoff
,
I.
, 2001, “
Euler’s Problem Revisited: 222 Years Later
,”
Meccanica
0025-6455,
36
, pp.
265
272
.
18.
Elishakoff
,
I.
, and
Guédé
,
Z.
, 2001, “
Novel Closed-Form Solutions in Buckling of Inhomogeneous Columns Under Distributed Variable Loading
,”
Chaos, Solitons Fractals
0960-0779,
12
, pp.
1075
1089
.
19.
Elishakoff
,
I.
, 2005,
Eigenvalues of Inhomogeneous Structures
,
CRC
,
Boca Raton, FL
.
20.
Kamke
,
E.
, 1972,
Differentialgleichungen, Lö sungsmethoden und Lö sungen
,
Chelsea
,
New York
.
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