The Impact of Corner Radii

In existing models with GBT the comers are always modeled as sharp, no comer radii is included. Comparison of FSM models with and without corner radii modeled, as shown in Figure 8a support the notion that the basic behavior is little influenced by small corner radii. A slight change in distortional buckling is observed, but this is still less than a few percent difference between the models.

However, from the standpoint of modal decomposition and the constrained FSM methods developed herein, the addition of corner radii has an important effect. Figure 8b illustrates the prediction of the local and distortional modes with three different models: an all mode, or conventional FSM analysis, and constrained models including only the D modes, and L modes. In addition, the figure also provides the predicted trasnverse displacements for the first minima (local buckling mode) predicted in the half-wavelength vs. buckling load plot. Even though the only change to the model is the addition of rounded corners, now the D modes do a better job of capturing the first minima than the L modes. Comparing this to the model with sharp corners of Figure 5a, the difference is striking. In fact, the predicted buckled shape via the L mode analysis is far too stiff when compared to a conventional (all mode) FSM analysis, and does not capture the appropriate deformations.

Comparing the transverse displacements in local buckling (Figure 8b); the L modes allow rotation only for the main nodes, and due to the angle changes around the corner all of the corner nodes are main nodes, thus the L modes effectively prevent the rotation of the corner as a global (rigid-body-like rotation). On the other hand, D modes allow transverse displacements, even though it is a special case of transverse flexure only, but since there are a many nodes in the vicinity of the corners, these transverse displacements result in a reasonable approximation of the real (all-mode) displacements. The addition of the corners also have a pronounced influence on the relationship between warping displacements and in-plane deformations. With the corners modeled explicitly, a gradient in the warping displacements through the corners allows a rotation to be engaged that closely mimics the corner rotations in the L modes of the sharp corner model. This may be best seen in the observed warping displacements, as given in Figure 9.

(a) FSM model results with sharp and round corners (b) decomposed model and predicted local modes

Figure 8. Impact of modeling cross-section with round corners on (a) FSM and (b) decomposition

The warping (longitudinal) displacements for the local buckling minima are provided in Figure 9. Comparison of Figure 9a and b, demonstrates that inclusion of corner radii alters the expected warping displacements. In a sharp corner model, the warping displacements only exist near the flange/lip juncture and have little impact on the actual result. With corner radii included in the model non-negligible warping also exists at the web/flange juncture; and further the warping distribution across the flange is approximately uniform. For constrained FSM models (Figure 9c and d) the D modes are unsuccessful in reproducing the actual (all mode) warping displacements, but the rounded corners allow warping at the web/flange juncture to engage bending in the web – and thus reasonably approximate the all mode local minima. Of course, the analysis consisting of L modes has, by definition, no warping, and in this case is a poor predictor.

sharp corner round corner model

(a) all modes (b) all modes, (c) D modes (d) L modes

Figure 9. FSM predicted warping displacements for the local buckling minima

The constrained FSM method, based on GBT’s mechanical assumptions, provides a unique means of decomposing a solution; however, it can not yet provide a general tool for modal classification and/or identification. The primary reason for this, is that reliance on heuristic definitions that describe the in­plane deformations; or simply point to minima in an FSM plot; are not themselves based on strict mechanical assumptions. Agreed upon definitions are needed in order to advance the field; however even simple issues such as models with corner radius included, demonstrate the challenges inherent in employing strict mechanical definitions. The corner radius models also suggest that further exploration of the relationship between criterion 3 regarding transverse flexure, and local plate bending are needed.

Conclusions

This paper provides an introduction to a new numerical method whereby general purpose finite element or finite strip methods can be constrained to deformation fields consistent with a particular class of buckling modes. The mechanical assumptions employed for this decomposition are based on Generalized Beam Theory, and implemented in a finite strip analysis. The decomposition (or constraining) of the conventional cross-section stability solution provides increased numerical efficiency and the potential to investigate cross-section stability behavior in a more detailed fashion. Some challenges exist when the approach is adopted; primarily related to the assumptions inherent in conventional solutions. For example, with global buckling modes it is shown that the constrained finite strip solutions are stiffer than analytic solutions due to the underlying constitutive relations. Models with and without corner radii present additional challenges for the mechanical definitions used to decompose the modes. Mechanics-based definitions of the cross-section stability modes do not agree well with current heuristic definitions when corner radii are present. Reasons for this difference are discussed in the paper. Modal decomposition brings the basic tools of Generalized Beam Theory for use in the more general finite strip method context, in doing so, both the advantages and limitations of such an approach are highlighted.

References

Adany, S., Schafer, B. W. (2004). “Buckling mode classification of members with open thin-walled cross­sections.” Fourth Int’l Conf. on Coupled Instabilities in Metal Structures, Rome, Italy, 27-29 Sept., 2004 Adany, S., Schafer, B. W. (2005a). “Buckling mode decomposition of single-branched open cross-section members via finite strip method: derivation.” Elsevier, Thin-walled Structures, (Submitted)

Adany, S., Schafer, B. W. (2005b). “Buckling mode decomposition of single-branched open cross-section members via finite strip method: application and examples.” Elsevier, Thin-walled Structures, (Submitted) Cheung, Y. K., Tham, L. G. (1998). The Finite Strip Method. CRC Press.

Schafer, B. W. (1997). Cold-Formed Steel Behavior and Design: Analytical and Numerical Modeling of Elements and Members with Longitudinal Stiffeners. Ph. D. Disseration. Cornell University, Ithaca, NY. Schafer, B. W., Adany, S. (2005). “Understanding and classifying local, distortional and global buckling in open thin-walled members.” Tech. Session and Mtg., Structural Stability Research Council. Montreal, Canada. Silvestre, N., Camotim, D. (2002a). “First-order generalised beam theory for arbitrary orthotropic materials.” Thin-Walled Structures, Elsevier, 40 (9) 755-789.

Silvestre, N., Camotim, D. (2002b). “Second-order generalised beam theory for arbitrary orthotropic materials.” Thin-Walled Structures, Elsevier, 40 (9) 791-820.

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