An efficient and high-precision algorithm for solving multiple deformation modes of elastic beams

doi:10.1007/s10483-025-3292-7

Abstract

Abstract:

The elliptic integral method (EIM) is an efficient analytical approach for analyzing large deformations of elastic beams. However, it faces the following challenges. First, the existing EIM can only handle cases with known deformation modes. Second, the existing EIM is only applicable to Euler beams, and there is no EIM available for higher-precision Timoshenko and Reissner beams in cases where both force and moment are applied at the end. This paper proposes a general EIM for Reissner beams under arbitrary boundary conditions. On this basis, an analytical equation for determining the sign of the elliptic integral is provided. Based on the equation, we discover a class of elliptic integral piecewise points that are distinct from inflection points. More importantly, we propose an algorithm that automatically calculates the number of inflection points and other piecewise points during the nonlinear solution process, which is crucial for beams with unknown or changing deformation modes.

Key words: elastic beam, elliptic integral, deformation mode transition, equilibrium path, high-precision algorithm

2010 MSC Number:

O343

Yunzhou WANG, Binbin ZHENG, Lingling HU, Nan SUN, Minghui FU. An efficient and high-precision algorithm for solving multiple deformation modes of elastic beams. Applied Mathematics and Mechanics (English Edition), 2025, 46(9): 1753-1770.

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Figures/Tables 11

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Table 1

The relationship between different beam lengths and arch heights"

Method	$l$ /mm	25	26	27	28	29
FEM	$h$ /mm	6.61	6.80	6.95	7.13	7.30
EIM	$h$ /mm	6.58	6.75	6.92	7.09	7.25
Error	\	0.5%	0.7%	0.4%	0.6%	0.7%

Table 1

Fig. 9

Table 2

The post-buckling deformation process of the cantilever beam under varying load"

Deformation step	Applied load and calculated results					Deformation shape of the beam
Deformation step	$P$ /N	$M B$ /(N·mm)	$β / (∘)$	$u x B$ /mm	$u y B$ /mm	Deformation shape of the beam
1	1	$− 2.2$	43.5	54	8.3
2	1	15.2	89.4	34.8	14.2
3	1	17	226.9	$− 10$	$− 40.6$
4	2.9	$− 16.4$	226.9	$− 6$	$− 52$
5	11.4	$− 64.5$	129.5	$− 6$	$− 33$
6	19.8	75	123.8	4	$− 30.5$
7	28.6	200	188.6	9	$− 27$
8	36.4	69.7	255.7	9	$− 68.1$

Table 2

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