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Research Papers

# Lateral-Torsional Stability Boundaries for Polygonally Depth-Tapered Strip Cantilevers Under Multi-Parameter Point Load Systems—An Analytical Approach

[+] Author and Article Information

Department of Civil Engineering,INESC Coimbra,  University of Coimbra FCTUC-Pólo II,Rua Luís Reis Santos, 3030-788 Coimbra, Portugalanisio@dec.uc.pt

Noël Challamel

LIMATB – Laboratoire d’Ingénierie des MATériaux de Bretagne,  Université Européenne de Bretagne – Université de Bretagne Sud Rue de Saint Maudé, BP 92116, 56321 Lorient Cedex, Francenoel.challamel@univ-ubs.fr

Paulo Providência

Department of Civil Engineering,INESC Coimbra,  University of Coimbra FCTUC-Pólo II,Rua Luís Reis Santos, 3030-788 Coimbra, Portugalprovid@dec.uc.pt

Dinar Camotim

Department of Civil Engineering and Architecture, ICIST/IST,  Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisbon, Portugaldcamotim@civil.ist.utl.pt

It is generally believed that analytical solutions (in the above sense) exist for only relatively few, very simple buckling problems involving columns, beams and plates. The monographs by Elishakoff [3] and Wang et al.  [4] prove that there are, in fact, not just a few such solutions. However, as far as the LTB of tapered strip beams is concerned, the general belief is entirely warranted.

The subscripts “−” and “+” have therefore slightly different meanings, but the same underlying spirit, when applied to functions defined on different intervals. This is unlikely to cause confusion.

These complex analytic functions (M(a,b,z) is an entire function of z and U(a,b,z) is an analytic function of z on the slit complex plane $C$\]−∞,0]) belong to the class of confluent hypergeometric functions—see Refs. [7,22-26] for previous applications in structural mechanics. The theory of confluent hypergeometric functions is discussed at great length and detail in the monographs by Buchholz [19], Slater [21] and Tricomi [18]; a useful summary of facts and relations is given in Ref. [20].

1

Corresponding author.

J. Appl. Mech 79(6), 061015 (Sep 21, 2012) (11 pages) doi:10.1115/1.4006459 History: Received June 10, 2011; Revised March 03, 2012; Posted March 29, 2012; Published September 21, 2012; Online September 21, 2012

## Abstract

This paper reports an analytical study on the elastic lateral-torsional buckling behavior of strip cantilevers (i) whose depth is given by a monotonically decreasing polygonal function of the distance to the support and (ii) which are subjected to an arbitrary number of independent conservative point loads, all acting in the same “downward” direction. The study is conducted on the basis of a one-dimensional (beam) mathematical model. A specialized model problem, consisting of a two-segment cantilever acted by two loads, applied at the free end and at the junction between segments, is first considered in detail for it “contains all the germs of generality”. It is shown that the governing differential equations can be integrated in terms of confluent hypergeometric functions or Bessel functions (themselves special cases of confluent hypergeometric functions). This allows us to establish exactly the characteristic equation for this structural system, which implicitly defines its stability boundary. Moreover, it is shown that the methods used to solve the model problem also apply to the general problem. A couple of parametric illustrative examples are discussed. Some analytical solutions are compared with the results of shell finite element analyses—a good agreement is found.

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## Figures

Figure 1

Two-segment strip cantilever acted by independent loads applied at the free end and at the junction between segments

Figure 2

Two segment strip cantilever with 0  <  α  =  β  <  1

Figure 3

Two-segment strip cantilever with 0  <  α  =  β  <  1

Figure 4

Illustrative example 1

Figure 5

Illustrative example 1 – stability boundaries (×– shell FE analyses)

Figure 6

Illustrative example 2

Figure 7

Illustrative example 2 – nondimensional (a) critical loads (× – shell FE analyses) and (b) critical load-to-volume ratios

Figure 8

Illustrative example 2 – LTB modes obtained with Cast3M (α  =  0.2, ρ  =  0.5)

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