Asymptotically accurate and locking-free finite element implementation of first order shear deformation theory for plates

TL;DR

The paper develops an inherently shear-locking-free, asymptotically exact first-order shear deformation theory (FSDT) for linear-elastic plates by reexpressing the problem in rescaled coordinates and rotation variables. By introducing $ar{x}_eta=x_eta/h$ and $ar{oldsymbol{ar{C1}}}=holdsymbol{C3}$ and redefining the kinematics with $ar{oldsymbol{ar{C1}}}$, the bending and shear stiffnesses are rendered $O(1)$, enabling an FE formulation that is both locking-free and asymptotically accurate when discretized with $C^1$-continuous isogeometric elements. The weak form is derived with a clear separation into bending, shear, and external work, and the discretization leads to a block-structured stiffness matrix solved within a Kratos-based isogeometric framework using NURBS patches and Bézier extraction. Numerical results for circular and rectangular plates show excellent agreement with analytical FSDT solutions and full 3D elasticity, validating the locking-free behavior and the asymptotic accuracy of the approach; a square concrete plate example demonstrates applicability to real materials. The work lays a foundation for extensions to FSDT-based shells, functionally graded plates, dynamics, and nonlinear buckling, promising computational efficiency without sacrificing theoretical rigor in thickness-dominated regimes.

Abstract

A formulation of the asymptotically exact first-order shear deformation theory for linear-elastic homogeneous plates in the rescaled coordinates and rotation angles is considered. This allows the development of its asymptotically accurate and shear-locking-free finite element implementation. As applications, numerical simulations are performed for circular and rectangular plates, showing complete agreement between the analytical solution and the numerical solutions based on two-dimensional theory and three-dimensional elasticity theory.

Figures (11)

• Figure 1: Cross section of a plate
• Figure 2: Structure of the computational code.
• Figure 3: Numerical example 4.1(a) (Circular plate with clamped edge): Multipatch geometry (left) and contour plot of scaled deflection $\left( \dfrac{u}{h \varepsilon \bar{R}^4} \right)$ at $\bar{R}=10$ (right).
• Figure 4: Numerical example 4.1(a) (Circular plate with clamped edge): Convergence rate of displacement ($L_2$ error) with $\bar{R}=10$ (left), $\bar{R}=100$ (middle) and $\bar{R}=1000$ (right).
• Figure 5: Numerical example 4.1(a) (Circular plate with clamped edge): Normalized deflection and rotation angle along the radial direction for $\bar{R}=10$ (left) and $\bar{R}=1000$ (right).