Optimization of the dimensions of a tapered thin-walled cantilever, aimed at reaching a maximum critical buckling load Cover

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Optimization of the dimensions of a tapered thin-walled cantilever, aimed at reaching a maximum critical buckling load

Studia Geotechnica et Mechanica

By: Józef Szybiński and Piotr Ruta

Open Access

|Nov 2025

Figures & Tables

Coordinate system and local basis vectors for a nonprismatic thin-walled beam.

Description of displacements for the cross-section.

Cross-section at beam’s midspan

x
=
0

x=0

. — Cross-section at beam’s midspan x = 0 x=0 .

Relationship between the critical load value and taper coefficients for “admissible” systems of type W(m), F(n), W-F(k) when load acts on the (a) upper flange, (b) web centre, and (c) lower flange.

Relationship between the critical load value and taper coefficients for “admissible” systems Wind–Find when the load acts on the (a) upper flange, (b) web centre, and (c) lower flange. Diagrams for a set of discrete values and diagrams for discrete values continuous approximation are shown in top and bottom figures, respectively.

Critical load values when the load acts on the centre of the upper flange_

Ψ f {\Psi }_{f}	Ψ w {\Psi }_{w}
Ψ f {\Psi }_{f}	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	S _w ¹⁾
0.0	14.77	15.72	16.76	17.88	19.10	20.43	21.90	23.58	25.50	1.73
0.0	14.77	15.72	16.71	17.73	18.79	19.90	21.05	22.24	23.47	1.73
0.1	15.75	16.72
0.1	15.74	16.79	17.79	18.94	20.21	21.60	23.15	24.91	26.92	1.71
0.2	16.67		18.73
0.2	16.67	17.66	18.97	19.91	21.20	22.63	24.23	26.04	28.07	1.68
0.3	17.55			20.78
0.3	17.54	18.53	19.60	21.31	22.08	23.53	25.15	26.96	28.97	1.65
0.4	18.38				22.86
0.4	18.34	19.34	20.39	21.56	23.78	24.29	25.89	27.66	29.60	1.61
0.5	19.18					24.93
0.5	19.05	20.12	21.15	22.29	23.54	26.31	26.46	28.14	29.99	1.56
0.6	20.03						26.91
0.6	19.67	20.93	21.92	23.00	24.19	25.49	28.79	28.47	30.21	1.51
0.7	20.99							28.94
0.7	20.14	21.84	22.75	23.75	24.85	26.04	27.34	31.04	30.43	1.45
0.8	22.04								30.72
0.8	20.43	22.81	23.65	24.56	25.57	26.67	27.89	29.26	32.79	1.39
S _f ²⁾	1.49	1.45	1.41	1.37	1.34	1.31	1.27	1.24	1.20	2.08 ³⁾

Critical load values when the load acts on the centre of the web_

Ψ f {\Psi }_{f}	Ψ w {\Psi }_{w}
Ψ f {\Psi }_{f}	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	S _w
0.0	24.35	25.40	26.46	27.53	28.60	29.69	30.82	32.06	33.57	1.38
0.0	24.36	25.39	26.40	27.38	28.35	29.30	30.23	31.15	32.06	1.38
0.1	26.59	27.66	28.74	29.82	30.92	32.03	33.19	34.47	36.04	1.36
0.1	26.57	27.69	28.74	29.82	30.92	32.03	33.19	34.47	36.04	1.36
0.2	28.66	29.73	30.80	31.88	32.98	34.10	35.27	36.56	38.12	1.33
0.2	28.64	29.73	31.00	31.88	32.98	34.10	35.27	36.56	38.12	1.33
0.3	30.54	31.59	32.64	33.70	34.78	35.88	37.02	38.28	39.81	1.30
0.3	30.52	31.59	32.64	34.22	34.78	35.88	37.02	38.28	39.81	1.30
0.4	32.21	33.21	34.21	35.22	36.25	37.29	38.39	39.59	41.04	1.27
0.4	32.18	33.21	34.21	35.22	37.25	37.29	38.39	39.59	41.04	1.27
0.5	33.64	34.56	35.49	36.43	37.37	38.34	39.35	40.47	41.85	1.24
0.5	33.56	34.56	35.49	36.43	37.37	39.97	39.35	40.47	41.85	1.24
0.6	34.87	35.71	36.55	37.39	38.23	39.11	40.02	41.07	42.36	1.21
0.6	34.58	35.71	36.55	37.39	38.23	39.11	42.21	41.07	42.36	1.21
0.7	36.10	36.84	37.57	38.31	39.06	39.84	40.67	41.59	42.83	1.19
0.7	35.16	36.84	37.57	38.31	39.06	39.84	40.67	43.73	42.83	1.19
0.8	37.50	38.15	38.79	39.43	40.09	40.78	41.53	42.41	43.30	1.15
0.8	35.16	38.15	38.79	39.43	40.09	40.78	41.53	42.41	44.18	1.15
S _f	1.54	1.50	1.47	1.43	1.40	1.37	1.35	1.32	1.29	1.78

Critical load values when the load acts on the centre of the lower flange_

Ψ f {\Psi }_{f}	Ψ w {\Psi }_{w}
Ψ f {\Psi }_{f}	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	S _w
0.0	33.07	34.29	35.46	36.58	37.66	38.68	39.69	40.74	42.09
0.0	33.18	34.36	35.48	36.54	37.54	38.47	39.35	40.16	40.91	1.27
0.1	36.87	38.11
0.1	36.85	38.11	39.28	40.39	41.45	42.45	43.42	44.47	45.87	1.24
0.2	40.42		42.78
0.2	40.37	41.64	42.92	43.85	44.87	45.82	46.76	47.79	49.17	1.22
0.3	43.69			46.99
0.3	43.67	44.87	45.96	47.44	47.94	48.84	49.71	50.68	52.01	1.19
0.4	46.66				50.57
0.4	46.65	47.76	48.78	49.71	51.49	51.36	52.12	53.00	54.24	1.16
0.5	49.24					53.30
0.5	49.22	50.23	51.13	51.93	52.65	54.85	53.93	54.70	55.84	1.13
0.6	51.42						55.28
0.6	51.25	52.28	53.03	53.68	54.25	54.76	57.26	55.98	57.03	1.11
0.7	53.34							56.93
0.7	52.58	54.05	54.65	55.17	55.61	56.02	56.47	58.36	58.09	1.09
0.8	55.21								58.84
0.8	52.96	55.77	56.24	56.63	56.96	57.28	57.66	58.23	57.69	1.07
S _f	1.67	1.63	1.59	1.55	1.51	1.48	1.45	1.43	1.40	1.78

DOI: https://doi.org/10.2478/sgem-2025-0021 | Journal eISSN: 2083-831X | Journal ISSN: 0137-6365

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Language: English

Submitted on: May 12, 2025

Accepted on: Sep 3, 2025

Published on: Nov 17, 2025

Published by: Wroclaw University of Science and Technology

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

thin-walled beam,

Related subjects:

Geosciences, other,

Materials sciences,

Porous materials,

Mechanics and fluid dynamics

© 2025 Józef Szybiński, Piotr Ruta, published by Wroclaw University of Science and Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.