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# Shell Element Formulation for Limit Analysis of Thin-Walled Structures ( 박판부재의 붕괴거동해석을 위한 극한해석의 쉘요소 수식화 )

2005.11.29 15:41

저자명 | 김현섭 |
---|---|

년도 | 2000 |

Thin-walled structures are widely used as structural members for the purpose of energy absorption. The most important aspects in the plastic design of these structural members for safety and reliability are their load-carrying capacity under various types of loading and corresponding collapse modes and energy absorption rate. The plastic design method of thin-walled structures can be divided into analytical method and numerical method. The analytical method assumes concentrated deformation at the plastic hinge and obtains collapse mechanism and semi-empirical equation for calculating the collapse load. Although the determination of location and configuration of the plastic hinge becomes very important, it is very difficult to presume the location and configuration of the plastic hinge in complicated structures. As an alternative, numerical method is necessary for collapse analysis of thin-walled structures under various conditions.

The numerical method consists of the incremental elasto-plastic finite element analysis, the incremental rigid-plastic finite element analysis and the finite element limit analysis. Although the elasto-plastic analysis can provide relatively detailed information, the analysis requires large amount of memory for the storage of stress state and much time to check whether yielding occurs. Step increments for the analysis should be small enough to ensure stable convergence and reliable solutions. On the other hand, finite element limit analysis has capability to calculate the collapse load and the collapse mode of complicated structures without any prior conjectures under the assumption that elastic deformation is small enough to be negligible. The analysis with a simple formulation has the advantage of stable convergence, computational efficiency and easy access to work-hardening materials compared to an elasto-plastic analysis. The solution procedure of finite element limit analysis obtains the proportional loading factor by minimizing the dual functional that can also be regarded as the plastic dissipation energy. Since the analysis is waived from tracking the stress-strain curve and obtaining the tangent modulus, it allows relatively large step size without losing the solution reliability. Finite element limit analysis can be efficiently used to predict plastic collapse loads and collapse modes of thin-walled structures with strain-hardening materials.

In this paper, the finite element limit analysis with the degenerated four-node shell element is formulated based on the duality theorem in plasticity. For computational efficiency, the reduced integration technique is employed with the physical stabilization scheme that does not need any user-defined parameter for hourglass control. The analysis considers sequential deformation of structures with strain-hardening. To verify the efficiency of the present analysis method, the collapse analysis of square tube and S-rail was simulated using the finite element limit analysis code.

The axial crushing tests of square tubes were conducted with specified cross-section and various width/height ratios. The cross-section of the test specimens is assigned to correspond to the inextensional and compact mode that shows good energy absorption efficiency. To achieve the quasi-static loading condition, the tests were performed at cross-head speeds of 1mm/min. Finite element limit analysis results are compared with experimental results in order to determine the accuracy in prediction of load-carrying capacity and deformation modes. The comparison demonstrated that finite element limit analysis results are reliable even with relatively large step increments. The large step increment well described continuous folding of square tubes with the appropriate contact treatment.

The collapse analysis of an S-rail was performed using the finite element limit analysis code and its results were compared with elasto-plastic analysis results using the explicit dynamic code PAMCRASH. In view of computational efficiency, the finite element limit analysis was very effective for the structural collapse analysis in comparison of computation time and solution quality with those obtained from an elasto-plastic analysis. The collapse analysis of an S-rail was conducted with variation of the thickness under the specified geometry to estimate the energy absorption capacity with respect to the sheet thickness as a design parameter. The energy absorption ratio with respect to the thickness ratio was found to be well approximated with a quadratic equation. This formula can be used effectively to aid design improvement at the initial stage of the structural design. The collapse analysis result showed that the energy absorption efficiency of the structure was remarkably increased with the stiffened S-rail and effective design improvement could be achieved through adequate stiffening of structures.

The numerical method consists of the incremental elasto-plastic finite element analysis, the incremental rigid-plastic finite element analysis and the finite element limit analysis. Although the elasto-plastic analysis can provide relatively detailed information, the analysis requires large amount of memory for the storage of stress state and much time to check whether yielding occurs. Step increments for the analysis should be small enough to ensure stable convergence and reliable solutions. On the other hand, finite element limit analysis has capability to calculate the collapse load and the collapse mode of complicated structures without any prior conjectures under the assumption that elastic deformation is small enough to be negligible. The analysis with a simple formulation has the advantage of stable convergence, computational efficiency and easy access to work-hardening materials compared to an elasto-plastic analysis. The solution procedure of finite element limit analysis obtains the proportional loading factor by minimizing the dual functional that can also be regarded as the plastic dissipation energy. Since the analysis is waived from tracking the stress-strain curve and obtaining the tangent modulus, it allows relatively large step size without losing the solution reliability. Finite element limit analysis can be efficiently used to predict plastic collapse loads and collapse modes of thin-walled structures with strain-hardening materials.

In this paper, the finite element limit analysis with the degenerated four-node shell element is formulated based on the duality theorem in plasticity. For computational efficiency, the reduced integration technique is employed with the physical stabilization scheme that does not need any user-defined parameter for hourglass control. The analysis considers sequential deformation of structures with strain-hardening. To verify the efficiency of the present analysis method, the collapse analysis of square tube and S-rail was simulated using the finite element limit analysis code.

The axial crushing tests of square tubes were conducted with specified cross-section and various width/height ratios. The cross-section of the test specimens is assigned to correspond to the inextensional and compact mode that shows good energy absorption efficiency. To achieve the quasi-static loading condition, the tests were performed at cross-head speeds of 1mm/min. Finite element limit analysis results are compared with experimental results in order to determine the accuracy in prediction of load-carrying capacity and deformation modes. The comparison demonstrated that finite element limit analysis results are reliable even with relatively large step increments. The large step increment well described continuous folding of square tubes with the appropriate contact treatment.

The collapse analysis of an S-rail was performed using the finite element limit analysis code and its results were compared with elasto-plastic analysis results using the explicit dynamic code PAMCRASH. In view of computational efficiency, the finite element limit analysis was very effective for the structural collapse analysis in comparison of computation time and solution quality with those obtained from an elasto-plastic analysis. The collapse analysis of an S-rail was conducted with variation of the thickness under the specified geometry to estimate the energy absorption capacity with respect to the sheet thickness as a design parameter. The energy absorption ratio with respect to the thickness ratio was found to be well approximated with a quadratic equation. This formula can be used effectively to aid design improvement at the initial stage of the structural design. The collapse analysis result showed that the energy absorption efficiency of the structure was remarkably increased with the stiffened S-rail and effective design improvement could be achieved through adequate stiffening of structures.