Perfect Reduced Equilibrium Equations and Cross-sectional Stress Resultants for Small Elastic Orientation in Nonlinear Spatial Euler-Bernoulli Beam

Subject Areas : Solid Mechanics

S. Mahdi Khorsandijou ^{1 *}

1 - Islamic Azad University- South Tehran Branch

Received: 2025-09-15 Accepted : 2025-09-17 Published : 2025-09-12

Keywords: Perfect Reduced Weak Form of Strain Energy Variation, Perfect Reduced Equilibrium Equations, Perfect Reduced Cross-sectional Stress Resultants, Nonlinear Spatial Euler-Bernoulli Beam, Elastic Orientation,

Abstract :

Using variational indication and minimizing strain energy of a geometrically-nonlinear linearly-elastic isotropic homogeneous spatial Euler-Bernoulli beam, the equilibrium governing formulations are found. Therefore, the weak form of the strain energy variation was derived for large elastic orientation and on the basis of which the perfect weak form of the strain energy variation was extracted for small elastic orientation. The obtained perfect equilibrium equations include the balance of axial forces, torsional moment and two perpendicular flexural moments. The obtained cross-sectional stress resultants include axial force, two perpendicular shearing forces, torsional moment and two perpendicular flexural moments. The assumption of small elastic orientation is imposed to the perfect weak form of strain energy variation in the present manuscript, but it is imposed to imperfect strain, and then to imperfect nonweak form of strain energy variation in the classical nonlinear 3-D Euler-Bernoulli beam theory. Imperfect strain causes strain variation and strain energy variation to miss a few terms. Moreover, imperfect nonweak form of strain energy variation causes the weak form of the strain energy variation to miss many terms. As a result, in this theory equilibrium equations and cross-sectional stress resultants suffer from the lack of missed terms. The lost terms are revealed in this manuscript.

References:

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