Research Papers

Design of Axially Graded Columns Under a Central Force

[+] Author and Article Information
Roberta Santoro

Dipartimento di Ingegneria Civile,
Informatica, Edile, Ambientale
e Matematica Applicata,
University of Messina,
C.da Di Dio,
98166 Messina, Italy
e-mail: roberta.santoro@unime.it

Isaac Elishakoff

Department of Ocean and Mechanical
Florida Atlantic University,
Boca Raton, FL 33431-0991
e-mail: elishako@fau.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 24, 2010; final manuscript received April 18, 2013; accepted manuscript posted May 6, 2013; published online September 16, 2013. Assoc. Editor: Glaucio H. Paulino.

J. Appl. Mech 81(2), 021001 (Sep 16, 2013) (6 pages) Paper No: JAM-10-1295; doi: 10.1115/1.4024400 History: Received August 24, 2010; Revised April 18, 2013; Accepted May 06, 2013

Structures made of functionally graded materials have attracted much interest recently. The idea is to create a material that fulfills a specified function in accordance to the identified purpose of structure's utilization. In this study, a semi-inverse problem is posed of determining the needed variation of the axial grading of an inhomogeneous column subjected to a central force, generalizing the ungraded column case. Remarkably, it turns out that for a specific combination of parameters, there could exist three different axially graded columns, that possess the same buckling load. Whereas this fact is not immediately apparent, the proposed formulation leads to the design of the axially graded column in such a manner that the buckling load is not less than a prespecified load. Purpose-oriented design demands columns have at least a prespecified buckling load. To solve the problem, a combined analytical-numerical procedure is developed in this study.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Koizumi, M., 1993, Ceramic Transactions, Functionally Gradient Materials, Vol. 34, J. B.Holt, M.Koizumi, T.Hirai, and Z.Munir, eds., The American Ceramic Society, Westerville, OH, pp. 3–10.
Suresh, S., and Mortensen, A., 1998, Fundamentals of Functionally Graded Materials, IOM Communications Ltd., London, UK.
Ng, T. Y., Lam, K. Y., Liew, K. M., and Reddy, J. N., 2001, “Dynamic Stability Analysis of Functionally Graded Cylindrical Shells Under Periodic Axial Loading,” Int. J. Solids Struct., 38(8), pp. 1295–1309. [CrossRef]
Javaheri, R., and Eslami, M. R., 2002, “Buckling of Functionally Graded Plates Under In-Plane Compressive Loading,” ZAMM, 82(4), pp. 277–283. [CrossRef]
Papargyri-Beskou, S., Tsepoura, K. G., Polyzos, D., and Beskos, D. E., 2003, “Bending and Stability of Gradient Elastic Beams,” Int. J. Solids Struct., 40(2), pp. 385–400. [CrossRef]
Najafizadeh, M. M., and Eslami, M. R., 2002, “First-Order-Theory-Based Thermoplastic Stability of Functionally Graded Material Circular Plates,” AIAA J., 40(7), pp. 1444–1450. [CrossRef]
Najafizadeh, M. M., and Eslami, M. R., 2002, “Buckling Analysis of Circular Plates of Functionally Graded Materials Under Uniform Radial Compression,” Int. J. Mech. Sci., 44(12), pp. 2479–2493. [CrossRef]
Elishakoff, I., 2005, Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions, CRC Press, Boca Raton, FL.
Elishahoff, I., 2000, “Resurrection of the Method of Successive Approximations to Yield Closed-Form Solutions for Vibrating Inhomogeneous Beams,” J. Sound Vib., 234(2), pp. 349–362. [CrossRef]
Elishakoff, I., 2001, “Inverse Buckling Problems for Inhomogeneous Columns,” Int. J. Solids Struct., 38(3), pp. 457–464. [CrossRef]
Elishakoff, I., and Rollot, O., 1999, “New Closed-Form Solutions for Buckling of a Variable Stiffness Column by Mathematics,” J. Sound Vib., 224(1), pp. 172–182. [CrossRef]
Elishakoff, I., and Guedé, Z., 2001, “A Remarkable Nature of the Effect of Boundary Conditions an Closed-Form Solutions for Vibrating Inhomogeneous Bernoulli-Euler Beams,” Chaos, Solitons Fractals, 12(4), pp. 659–704. [CrossRef]
Caliò, I., and Elishakoff, I., 2004, “Can a Trigonometric Function Serve Both as the Vibration and the Buckling Mode of an Axially Graded Structure?,” Mech. Based Des. Struct. Mach., 32(4), pp. 401–421. [CrossRef]
Elishakoff, I., and Endres, J., 2005, “Extension of Euler's Problem to Axially Graded Columns: 260 Years Later,” J. Intelligent Mater. Syst. Struct., 16(1), pp. 77–83. [CrossRef]
Elishakoff, I., Gentilini, C., and Santoro, R., 2006, “Some Conventional and Unconventional Educational Column Stability Problems,” Int. J. Struct. Stability and Dyn., 6(1), pp. 139–151. [CrossRef]
Timoshenko, S. P., and Gere, J. M., 1961, Theory of Elastic Stability, McGraw-Hill, New York, pp. 55–57.
Gajewski, A., and Zyczkowski, M., 1969, “Optimal Shaping of an Elastic Homogeneous Bar Compressed by a Polar Force,” Bull. Acad. Pol. Sci., Series Sci. Techniques, 17(10), p. 479; Rozpr. Inżyn.17/2, 299 (in Polish).
Mladenov, K. A., and Sugiyama, Y., 1983, “Buckling of Elastic Cantilevers Subjected to a Polar Force: Exact Solution,” Trans. Jpn. Soc. Aero. Space Sci., 26(72), pp. 80–90.
Sugiyama, Y., Mladenov, K. A., and Fusayasu, K., 1983, “Stability and Vibration of Elastic Systems Subjected to a Central Force,” Reports of the Faculty of Engineering Tottori University Japan, 14(1), pp. 1–13.


Grahic Jump Location
Fig. 1

Elastic cantilever subjected to a central force

Grahic Jump Location
Fig. 5

Variation of D(ξ)/|b2| for a simply-supported column at both ends (solution 1)

Grahic Jump Location
Fig. 4

Variation of D(ξ)/|b2| for a column loaded by a central force for values αL=-0.2 and αL=-0.1

Grahic Jump Location
Fig. 3

Variation of D(ξ)/|b2| for a column loaded by a central force for different values of αL (-0.9 < αL < -0.3)

Grahic Jump Location
Fig. 6

Variation D(ξ)/|b2| for a simply-supported column at both ends (a) solution 2, solid line; (b) solution 3, dashed line

Grahic Jump Location
Fig. 7

Variation of D(ξ)/|b2| for a column loaded by a central force for different values of αL (-0.1 < αL < 1) (first possible axial grading)

Grahic Jump Location
Fig. 8

Variation of D(ξ)/|b2| for a column loaded by a central force for different values of αL (-0.1 < αL < 1) (second possible axial grading)

Grahic Jump Location
Fig. 2

Variation of D(ξ)/|b2| for a clamped-pinned column

Grahic Jump Location
Fig. 9

Variation of D(ξ)/|b2| for a column loaded by a central force for different values of αL (-0.1 < αL < 1) (third possible axial grading)




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In