Ritz Variational Method for Solving the Elastic Buckling Problems of Thin-Walled Beams with Bisymmetric Cross-Sections
Bibliographic record
Abstract
The Ritz variational method was used in this study to solve the lateral torsional buckling problem of simply supported thin-walled beam with doublysymmetric cross-section.Two considered cases of loading were uniform bending moments applied at the two ends, and a point load applied vertically at the midspan.The problem was presented in variational form as the problem of minimizing the total potential energy functional, , with respect to the unknown parameters of the generalized displacement modal functions.The total potential energy functional was found to be a function of two unknown displacement buckling functions v(x) and (x) and their derivatives with respect to the longitudinal coordinate axis.Suitable displacement buckling functions that satisfy the Dirichlet boundary conditions at the ends were used as trial functions to obtain the Ritz variational problem as the minimization of with respect to the generalized buckling modal displacement amplitudes c1n and c2n.The Ritz variational equations were obtained as the minimum conditions for with respect to c1n and c2n.The equations were solved for the two cases considered and the buckling moments found for the nth buckling mode from solving the resulting system of homogeneous algebraic equations.It was found that the expressions obtained for the buckling moments in each considered case were the exact expressions obtained by other researchers in literature who solved using classical mathematical methods.It was further found that for each considered case the critical buckling moment occurred at the first buckling mode, and the critical buckling moment expressions for each case agreed with exact solutions from the literature.The effectiveness of the Ritz variational method was thus illustrated for stability problems of thin-walled beams with Dirichlet boundary conditions.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".