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Research Papers

# Analysis of Thin-Walled Structures With Longitudinal and Transversal Stiffeners

[+] Author and Article Information
E. Carrera

Professor of Aerospace Structures
and Aeroelasticity
e-mail: erasmo.carrera@polito.it

E. Zappino

Ph.D. student
e-mail: enrico.zappino@polito.it

M. Petrolo

Research Assistant
e-mail: marco.petrolo@polito.it
Department of Mechanical
and Aerospace Engineering,
Politecnico di Torino,
Corso Duca degli Abruzzi 24,
10129 Torino, Italy

The difference between the EBBT and the higher order solution has been defined as $ΔEB%=(wEBBT-w)/|wEBBT|×100$.

In the classical models based on displacements degrees of freedom, $Θ$ may be defined as $tan(Θ)=∂ux/∂z=∂uz/∂x$, considering Eq. (7) the twisting angle in the CUF formulation can be written as $tan(Θx)=∂ux/∂z=ux2+x ux5+2z ux6+…,$$tan(Θz)=∂uz/∂x=uz3+2x uz4+z uz5+….$ This equation shows that the twisting angle is not a property of the whole section but a property of the point, also the equivalence $∂ux/∂z=∂uz/∂x$ is not automatically satisfied. In order to compare the results with those from literature in this work only the constant term is considered in the evaluation of the twisting angle; therefore $θ$ has been defined as $Θ=(arctan(ux2)+arctan(uz3))/2$.

1Corresponding author.

Manuscript received June 6, 2011; final manuscript received May 7, 2012; accepted manuscript posted June 6, 2012; published online October 29, 2012. Assoc. Editor: Krishna Garikipati.

J. Appl. Mech 80(1), 011006 (Oct 29, 2012) (12 pages) Paper No: JAM-11-1175; doi: 10.1115/1.4006939 History: Received June 06, 2011; Revised May 07, 2012; Accepted June 06, 2012

## Abstract

This paper proposes the use of a one-dimensional (1D) structural theory to analyze thin walled structures with longitudinal and transverse stiffeners. The 1D theory has hierarchical features and it is based on the unified formulation (UF) which has recently been introduced by Carrera. UF permits us to introduce any order of expansion (N) for the unknown displacements over the cross section by preserving the compact form of the related governing equations. In this paper the latter are written in terms of finite element matrices. The same 1D structural theory is used to model a given thin-walled structure composed of stiffened (longitudinal and transverse) and unstiffened parts. It is shown that an appropriate choice of N permits us to accurately describe strain/stress fields of both transverse and longitudinal stiffeners. Comparisons with available results from open literature highlight the efficiency of the proposed model. Moreover, a set of sample problems are proposed and compared with plate/shell formulations from a commercial finite element (FE) software.

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## Figures

Fig. 1

Example of reinforced structures

Fig. 2

Reference frame of the model

Fig. 3

Multicomponent capabilities of CUF 1D models

Fig. 4

“Very low” aspect ratio beam model

Fig. 5

Cross-section displacements (m) at y = L/2. Sixth order theory.

Fig. 6

Stress (Pa) distributions at y = L/2, sixth order theory

Fig. 7

Models considered in the bending analysis

Fig. 8

Circular cross-section reference system

Fig. 9

Comparisons between model 1 and model 3, 3D view

Fig. 10

Comparisons between model 1 and model 3 of the cross-section deformations at different longitudinal axis positions

Fig. 11

Effects of ribs on the warping of the section at y = L/4 from a lateral point of view

Fig. 12

Effects of the rib on the percentage gap between higher order theories and EBBT

Fig. 13

Reference system and loads of the C-shaped cross section

Fig. 14

Effects of the ribs on the torsion of a beam with a C-shaped cross section

Fig. 15

Reference system and geometry of the different models. Case 1: only skin; case 2: skin and ribs; case 3: skin and stringers; case 4: skin, stringers, and ribs.

Fig. 16

Displacements in the z direction of thin-walled cylinder with an internal pressure. Comparisons between the with-ribs model (case 2) and without-ribs model (case 1). Displacements amplified of a ×100 factor.

Fig. 17

Displacements in the z direction of thin-walled cylinder with ribs under an internal pressure load. Comparisons between the with-stringers model (case 4) and without-stringers model (case 2). Displacements amplified of a ×100 factor.

Fig. 18

Section displacements of thin-walled cylinder with ribs under an internal pressure load. Section analyzed at y = 7.5. Comparisons between the with-stringers model (case 4) and without-stringers model (case 2). Displacements amplified of a ×100 factor.

Fig. 19

3D view of the displacements of the different models, undeformed in white, amplification factor ×300. (a) Case 1, (b) case 2, (c) case 3, (d) case 4.

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