[Paper Review] Analytical solution for free vibrations of simply supported transversally inextensible homogeneous rectangular plate

Milan Batista|arXiv (Cornell University)|Jul 15, 2010

Composite Structure Analysis and Optimization4 references3 citations

TL;DR

This paper presents an exact analytical solution for free vibrations of a simply supported, transversally inextensible, homogeneous rectangular plate using three-dimensional elasticity theory. By solving the equilibrium equations under sinusoidal loading and inextensibility constraints, the authors derive closed-form expressions for displacement and stress components, with asymptotic expansions for thin plates. The key contribution is a benchmark solution for validating higher-order plate theories.

ABSTRACT

In article, the exact solution of sinusoidal loaded simply supported elastic transversally inextensible rectangular plate is given. The expressions for displacement and stress components are derived and asymptotic expansion with respect to plate thickness are present. The frequency factors for plate thickness to width ratio 0.01, 0.1, 0.2 and 0.4 and various ratios of plate length to width are given.

Motivation & Objective

• To provide an exact three-dimensional analytical solution for free vibrations of a simply supported, transversally inextensible rectangular plate.
• To fill the gap in the literature for an exact benchmark solution under transverse inextensibility, a key assumption in Mindlin and higher-order plate theories.
• To enable direct comparison of plate theories against an exact solution, improving assessment of their accuracy.
• To derive frequency factors for various thickness-to-width ratios and aspect ratios, supporting validation of numerical and approximate methods.

Proposed method

• Formulating the problem using three-dimensional elasticity equations with transverse inextensibility (w_z = 0).
• Applying double Fourier series expansions for displacement and stress components, assuming harmonic time dependence.
• Deriving a system of ordinary differential equations for the axial and transverse displacement functions U(z) and V(z).
• Solving the system using hyperbolic functions and introducing normalized coordinate ζ = z/h ∈ [-1,1].
• Applying boundary conditions at the simply supported edges and plate faces to derive a homogeneous system for unknown coefficients.
• Obtaining the characteristic equation and non-trivial solution via determinant conditions, leading to frequency factors and stress/displacement expressions.

Experimental results

Research questions

• RQ1 What is the exact three-dimensional solution for free vibrations of a simply supported, transversally inextensible rectangular plate under sinusoidal loading?
• RQ2 How do the displacement and stress components vary through the thickness and in the plane of the plate?
• RQ3 What are the frequency factors for different plate aspect ratios and thickness-to-width ratios?
• RQ4 How does the asymptotic expansion for thin plates compare to classical Reissner's approximation?
• RQ5 To what extent can this exact solution serve as a benchmark for evaluating the accuracy of plate theories?

Key findings

• The exact analytical solution is derived for displacement and stress components using hyperbolic functions and a characteristic equation based on boundary conditions.
• The frequency factors are computed for thickness-to-width ratios of 0.01, 0.1, 0.2, and 0.4, and various length-to-width ratios, providing a comprehensive benchmark.
• For a square plate with h/a = 0.4 and ν = 0.3, the first mode frequency factor is 1.3994, with relative errors of 0.17% for FOPT and 0.01% for HOPT.
• The asymptotic expansion for thin plates recovers Reissner’s classical approximation for stress components, validating its use in higher-order theories.
• The solution confirms that higher-order plate theories (HOPT) significantly outperform first-order theories (FOPT) in accuracy, especially for higher modes.
• The derived expressions for stress resultants and transverse shear forces are consistent with classical plate theory formulations, enabling direct comparison.