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

Józef Szybiński; Piotr Ruta

doi:10.2478/sgem-2025-0021

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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

Volume 47 (2025): Issue 4 (December 2025)

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 $x = 0$ .

Table 1

Critical load values when the load acts on the centre of the upper flange.

$Ψ_{f}$	$Ψ_{w}$
$Ψ_{f}$	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⁾

Source: Author’s contribution.

¹⁾ $S_{w} = \frac{q_{cr} (ψ_{f} = 0,1 * i, ψ_{w} = 0,8)}{q_{cr} (ψ_{f} = 0,1 * i, ψ_{w} = 0)}; i = 0, 1, \dots, 8$ ²⁾ $S_{f} = \frac{q_{cr} (ψ_{f} = 0,8, ψ_{w} = 0,1 * j)}{q_{cr} (ψ_{f} = 0, ψ_{w} = 0,1 * j)}; j = 0, 1, \dots, 8$ ³⁾ $S_{fw} = \frac{q_{cr} (ψ_{f} = 0,8, ψ_{w} = 0,8)}{q_{cr} (ψ_{f} = 0, ψ_{w} = 0)} .$

Table 2

Critical load values when the load acts on the centre of the web.

$Ψ_{f}$	$Ψ_{w}$
$Ψ_{f}$	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

Source: Author’s contribution.

Table 3

Critical load values when the load acts on the centre of the lower flange.

$Ψ_{f}$	$Ψ_{w}$
$Ψ_{f}$	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

Source: Author’s contribution.

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 W_ind–F_ind 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.

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DOI: https://doi.org/10.2478/sgem-2025-0021 | Journal eISSN: 2083-831X (formerly 0137-124X) | Journal ISSN: 0137-6365

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Language: English

Page range: 1 - 14

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

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Related subjects:

Physics,

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.

Volume 47 (2025): Issue 4 (December 2025)

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

Figures & Tables

Figure 1

Figure 2

Figure 3

Table 1

Table 2

Table 3

Figure 4

Figure 5

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