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Research Papers

# Fast and Robust Full-Quadrature Triangular Elements for Thin Plates/Shells With Large Deformations and Large Rotations

[+] Author and Article Information
Hui Ren

MSC Software Corporation,
201 Depot Street,
Suite 100,
Ann Arbor, MI 48104
e-mail: hui.ren@mscsoftware.com

Manuscript received April 30, 2014; final manuscript received March 24, 2015; published online June 10, 2015. Assoc. Editor: Javier Garcia de Jalon.

J. Comput. Nonlinear Dynam 10(5), 051018 (Sep 01, 2015) (13 pages) Paper No: CND-14-1112; doi: 10.1115/1.4030212 History: Received April 30, 2014; Revised March 24, 2015; Online June 10, 2015

## Abstract

A new formulation for plates/shells with large deformations and large rotations is derived from the principles of continuum mechanics and calculated using the absolute nodal coordinate formulation (ANCF) techniques. A class of triangular elements is proposed to discretize the plate/shell formulation, which does not suffer from shear locking or membrane locking issue, and full quadrature can be performed to evaluate the integrals of each element. The adaptability of triangular elements enables the current approach to be applied to plates and shells with complicated shapes and variable thicknesses. The discretized mass matrix is constant, and the elastic force and stiffness matrix are polynomials of the generalized coordinates with constant coefficients. All the coefficients can be evaluated accurately beforehand, and numerical quadrature is not required in each time step of the simulation, which makes the current approach superior in numerical efficiency to most other approaches. The accuracy, robustness, and adaptability of the current approach are validated using both finite element benchmarks and multibody system standard tests.

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## Figures

Fig. 1

Three types of triangular elements discussed in this work: (a) linear strain element; (b) Lagrange quadratic strain element; and (c) Hermite quadratic strain element

Fig. 2

The elements in the membrane and bending plate patch tests

Fig. 7

Tessellation of the circular plate using (a) 600 linear strain elements and (b) 216 quadratic strain elements

Fig. 8

The mode shapes of the first six modes of a free circular plate

Fig. 10

Equilibria of the clamped circular plate under (a) a uniform pressure and (b) a concentrated force at the center

Fig. 9

Deflections of a clamped circular plate under (a) a uniform pressure and (b) a concentrated force, calculated from the linear strain elements (solid), and the linear plate theory (dashed)

Fig. 11

(a) Schematic of a hemisphere under a concentrated load and (b) tessellation of the hemisphere

Fig. 12

(a) The deflection-load response curve of the rubber hemisphere and (b) the collapsed configuration of the hemisphere

Fig. 13

The (a) x, (b) y, and (c) z tip motions of the plate pendulum, calculated using 5 × 5 × 2 linear strain triangular elements (dashed), 3 × 3 × 2 Hermite quadratic strain triangular elements (dashed-dotted), and 4 × 4 rectangular thin plate elements in Ref. [7] (dotted), and the results are compared with those calculated using 16 × 16 rectangular thin plate elements (solid)

Fig. 14

The (a) x, (b) y, and (c) z tip motions of the plate pendulum, calculated by 16 × 16 × 2 linear strain triangular elements (dashed), 8 × 8 × 2 Hermite quadratic strain triangular elements (dashed-dotted), and 8 × 8 rectangular thin plate elements in Ref. [7] (dotted), and the results are compared with those calculated using 16 × 16 rectangular thin plate elements (solid)

Fig. 15

The configurations of the flexible plate pendulum at time = 0, 0.2, 0.3, 0.4, 0.5 s

Fig. 3

The (a) schematic and (b) tessellation of the twisted beam

Fig. 4

Time histories of the displacements at the tip under (a) the out of plane force and (b) the in-plane force, calculated using the linear strain element approach (dotted), the Hermite quadratic element approach (dashed), and the rectangular thin shell elements in Ref. [7] (dashed-dotted); and the static benchmark results (solid) are also presented for comparison

Fig. 5

(a) Configuration and (b) tessellation of the Scordelis–Lo roof

Fig. 6

(a) Nonlinear displacement distribution and (b) vertical displacement field of the Scordelis–Lo roof

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