Buckling of thin-walled tubes with an elastic infill

Prepared by Kin Yuen Leung
Supervisor : Prof. Mark Bradford & Dr. Zora Vrcelj

Introduction

In the 21st century, living in the space is a dream for everyone. Some scientists have already embraced planning of building an aerospace station to accomplish this goal. This will require use of structural members that are different from those use conventionally. Currently, composite columns, thin walled steel tubes with concrete infill (Figure 1b), are frequently used in engineering structures. Their growing popularity is due to the substantial economical savings the composite members can provide over their steel counterparts. Thus, this research focuses on the buckling capacity of composite columns when elastic infill, such as polystyrene and some rubbers, is used instead of concrete to achieve light-weight structures, as generally employed in aerospace and other weight-sensitive engineering.

Objective

This project is concerned with generic modeling of the buckling of a square thin-walled rectangular tube with an elastic infill in concentric compression. The aim of the project is to, by means of the energy method, establish a relationship between the variation of the elastic local buckling coefficient and the modulus of the elastic infill. The following are the steps to achieve this goal.

		where: k is the buckling coefficient b is the width of the plate t is thickness of the plate E is the elastic modulus v is poisson’s ratio
	The equation shows that stress is proportional to buckling coefficient.

Project Procedure

Step 1 - Local Buckling Patterns

It has been proven that a concrete filled steel box will sustain higher compressive stresses before buckling locally, compared to a hollow steel box. Since concrete is completely rigid, it will not allow the steel plates to buckle inwards (Figure 1b ). On the other hand, in the hollow steel box section the plates are free to buckle inwards and outwards (Figure 1a ). Therefore, the section with elastic medium (Figure 1c) would have a deformed shape that is somewhere in between that for a hollow section and a concrete filled section because the elastic medium provides some restraint against inward movement, however, it does not completely restrain it.

Step 2 - Displacement function

• A displacement function is required in order to perform the energy method of elastic buckling analysis.
• True buckling shapes in both x and y direction are required for reliable results.
• Edge boundary conditions w(0) = 0 and w(L) = 0 must be achieved with selected displacement function.

The following Fourier series represent the displacement function of elastic infill where n is the number of harmonics.

Step 3 – Programming by using Matlab

Once the displacement functions are defined, the energy method is used to obtain the elastic buckling coefficient of a steel box section filled with an elastic medium. The method involves the principals of virtual work and the Rayleigh-Ritz procedure to convert the buckling conditions into a matrix eigenvalue problem.

By using a programming language Matlab, the eigenvalue problem is easily solved, and the buckling load and the deformed shape, eigenvector, evaluated. The critical buckling loads for a hollow square steel section and a concrete filled section are shown in Figure 2.

Figure 2 (a) Buckling coefficient k vs L/b for hollow section

Figure 2 (b) Buckling coefficient k vs L/b for concrete infill section

It is well known that the buckling coefficients for a hollow steel section and a concrete filled section are k = 4.0 and k = 10.67 respectively (Figure 2 ). Therefore, the buckling coefficient for a section with an elastic infill must lie between 4.0 and 10.67, with the lower limit representing no infill and the upper limit a rigid infill.

Step 4 - Result

The test results obtained were analyzed and illustrated in Figure 3.

Figure 3 (a) Compare the accuracy between 1 and 2 terms

Figure 3 (b) Buckling coefficient vs Restraint parameter

A better accuracy can be achieved if the program adopts a large number of terms (m) in the Fourier series (Figure 3a).

It can be seen from Figure 3b that the local buckling coefficient, kcr increases substantially when the elastic infill is added to a hollow steel tube for which the local buckling coefficient is 4.

Conclusion

Despite the fact that many light materials have low stiffness, when used as an infill in steel square section, the critical local buckling coefficient of such assembly is as high as that of a section with a rigid infill (k=10.67), ie. an infinite stiffness. Hence, elastic infill has a most favorable effect on the elastic buckling response of thin-walled tubes.

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