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A compound strip method for static and buckling analyses of thin-walled members with transverse stiffeners

https://doi.org/10.1016/j.tws.2021.107829Get rights and content

Highlights

  • Propose a new compound strip method that can analyze members with transverse stiffeners.

  • Validate the accuracy of this new method through static analyses of plates and beams with shell FEM.

  • Validate the accuracy of this new method through buckling analyses of plates, columns and beams with shell FEM.

  • Discuss the influence of transverse stiffeners on buckling behaviors.

Abstract

The objective of this study is to propose and develop a Compound Strip Method (CSM) for static and buckling analyses of thin-walled members with transverse stiffeners. Traditionally thin-walled members are analyzed as prismatic members using computationally efficient methods such as the popular Finite Strip Method (FSM) — spline or semi-analytical. However, with structural systems getting more complicated and also for seeking better performance, transverse components that render the members no longer prismatic are possible, such as spaced transverse stiffeners along the member length. Existing FSM cannot take into account the stiffeners’ effect in the present form. Hence, this study presents a new compound strip method (CSM) based on the spline FSM for static and buckling analyses of thin-walled members with transverse stiffeners. The transverse stiffeners are modeled with shell element formulation by considering in-plane deformation. Transverse stiffeners can be placed flexibly along the member length in the proposed CSM. The developed CSM is then used to analyze plates and thin-walled members with transverse stiffeners. The accuracy of the method is validated with shell finite element solutions through static and buckling analyses of those members. Meanwhile, the impact of transverse stiffeners on the members’ buckling modes is also revealed by the analyses.

Introduction

Thin-walled members have undergone decades of development in seeking material efficiency and have been widely used in aircraft, ships, bridges, and buildings. Consequentially, the variety and complicated profiles of thin-walled sections have been greatly expanded. While the applications of these sections enjoy the boom from new development, the corresponding analysis and design efforts have also become more cumbersome due to the complex instability behaviors of thin-walled members, commonly known as local-plate (or local), distortional, and global buckling. Especially, lots of engineering applications seek to mitigate buckling failures to enhance the buckling strength of the member using stiffeners, either longitudinally or transversely [1], [2]. For instance, a recent study on a trestle structure shown in Fig. 1 explored the application of corrugated sheets (i.e., longitudinal stiffeners) with ribs (i.e., transverse stiffeners) [3]. Experimental studies showed that the buckling failure of the top chord members with corrugated sheets controls the strength. However, the analysis of this buckling behavior can be complicated.

In general, for complex stability problems, the analysis greatly relies on computational methods, such as the Finite Element Method (FEM), and Finite Strip Method (FSM). While FEM can model complex geometry and loading conditions, it demands a lot of modeling efforts. On the other hand, FSM, an old variety of FEM, enjoys quite popular use in the buckling analysis of thin-walled members due to its computational efficiency and much-truncated modeling efforts. FSM was firstly proposed by Wittrick and his colleagues in the late 1960s [4], [5]. It adopts conventional polynomial shape functions in the transverse direction while using specially selected shape functions in the longitudinal direction. Two types of longitudinal functions are commonly used in the FSM formulations: trigonometric functions and spline functions that satisfy specific boundary conditions. Cheung summarized the semi-analytical finite strip method (i.e., based on the trigonometric function) in the well-known text of [6]. The famous signature curve based on the semi-analytical FSM with simply supported boundary conditions is populated by Hancock [7]. Implementations of semi-analytical FSM for other boundary conditions can be found in [8], [9]. Spline FSM utilizes piece-wise polynomial shape functions to represent the longitudinal field, which can adapt to more complex boundary conditions and loading compared to semi-analytical FSM [10], [11]. Many researchers apply splines in plates and shell-type problems exhibiting good ability in analyzing these structures [12], [13], [14], [15]. For instance, Fan [16] implemented a spline Finite strip method (sFSM) for plate structures with different boundaries. Wang [17] used the spline finite strip method to study the buckling of steel jacking pipes embedded in the elastic tensionless foundation.

Although widely used in investigating buckling behaviors of thin-walled members for its computational efficiency and low modeling effort, significant limitations of FSM exist due to its intrinsic simplification of the model, particularly, requiring the section to be prismatic and selections of longitudinal shape functions. When transverse stiffeners are present, traditional FSM has difficulty capturing the influence of stiffeners. To overcome this, in the early literature, the compound strip method (CSM) was presented by Puckett and Gutkowski [18] by incorporating the stiffeners as beam and column elements and adding the elements’ stiffnesses directly to the strip matrix. This method used trigonometric functions as longitudinal shape functions and was limited for the plates with transverse beams and internal supports. Chen [19] proposed a compound finite strip method to study buckling of plates with stiffeners at any location adopting a cubic B-spline function that has C2 continuity as a longitudinal shape function replacing continuous Fourier series. Displacement fields of stiffener elements were constrained to relative strip displacement fields. In-plane deformation of stiffeners cannot be considered because stiffener elements were simplified to beam elements. More recently, Abbasi et al. [20] proposed a compound strip method based on the semi-analytical FSM to consider the fasteners’ presence along the member length. The presented method allows for modeling arbitrarily-located discrete fasteners by simplifying the fastener as a beam element with adjustable stiffness properties and the associated stiffness of the beam elements are incorporated in the global stiffness matrix. The method is intended particularly in modeling built-up sections.

Given the fact that current FSM can only address the plate sections with transverse stiffeners or members with fasteners, a new compound strip method (CSM) is proposed in this paper to appropriately capture transverse stiffeners’ influence on buckling behaviors of thin-walled members. The selection of stiffener elements and incorporation of strip and stiffener elements are keys for CSM. Given that intermediate nodes can be imaginarily extracted from spline functions, the new CSM is developed based on the B-spline finite strip method. The stiffeners are modeled using the shell element. The stiffness of the in-plane displacement is incorporated into the strip matrix. Transverse stiffeners can be located arbitrarily along the member length in the new CSM. The validity of the proposed method is demonstrated through comparisons of numerical examples with the existing analytical and numerical solutions of the finite element method (FEM). The paper firstly validates the results of static analysis to ensure the feasibility of the new compound strip method and then continues numerical studies for buckling analyses. This new CSM can potentially be used to analyze more complicated sections and optimize the member design with transverse stiffeners.

Section snippets

Compound Strip Method formulation

As mentioned before, FSM is a variant of more commonly used FEM. In this new Compound Strip Method (CSM), cubic B-spline functions are adopted for the longitudinal shape functions and the classic polynomial beam shape functions are used in the transverse direction with the cross-section. For stiffeners, shell element formulation with in-plane deformation is derived and incorporated into the strip stiffness. The geometric stiffness of stiffeners is also formulated and incorporated for buckling

Static analysis application and validation

The application of the new Compound Strip Method (CSM) in static analysis is investigated first. Plates with stiffeners and a channel section with transverse stiffeners are studied in this section.

Application and validation of linear elastic buckling analysis of thin-walled members

The application of the new CSM for buckling analysis is enabled through the integration of geometric stiffness in Section 2.3.2 into the eigenvalue problem in Eq. (17). The study herein will utilize three thin-walled members to highlight the application of the new CSM including the plate, channel, and lipped channel sections. The impact of the number and location of stiffeners on members is discussed here along with the validation against the shell FEM solutions. For all the examples studied in

Discussions

Shell FEM is primarily used for the validation of the developed CSM. It is crucial to ensure the equivalent boundary in the two methods. An interesting finding is that FEM models used to verify the classical FSM are slightly different from the new CSM or sFSM. The Longitudinal displacements of the whole section at the mid-span of members are set to zero as a result of using sinusoidal series in classical FSM. Thus, all nodes of the mid-span would be longitudinally constrained in the FEM model

Conclusions

A new Compound Strip Method based on the spline Finite Strip Method is proposed and developed to perform linear elastic static and buckling analysis of thin-walled members with transverse stiffeners. The effect of the stiffeners is modeled with shell elements by taking into account the in-place degrees of freedom. The approach ensures that the potential interaction between stiffeners and the flange/web of members could be appropriately considered and transverse stiffeners can be placed flexibly

CRediT authorship contribution statement

Yanguo Hou: Conceptualization, Methodology, Formal analysis, Investigation, Validation, Writing - original draft, Writing - review & editing. Zhanjie Li: Conceptualization, Methodology, Supervision, Investigation, Writing - original draft, Writing - review & editing. Jinghai Gong: Conceptualization, Methodology, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (27)

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