Strain gradient differential quadrature beam finite elements

Zhang, Bo; Li, Heng; Kong, Liulin; Wang, Jizhen; Shen, Huoming

doi:10.1016/j.compstruc.2019.01.008

Cited by 27 publications

(16 citation statements)

References 77 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In a standard interval (ie,

\overset{true‾}{X} \in false[- 1, 1 false]

), the integral points and weighting coefficients for the fourth‐order GLQ rule are as follows 51 :

{\overset{X}{true}}_{1} = - 1, {\overset{X}{true}}_{2} = - 1 / \sqrt{5}, {\overset{X}{true}}_{3} = 1 / \sqrt{5}, {\overset{X}{true}}_{4} = 1,

C_{1} = C_{4} = 1 / 6, C_{2} = C_{3} = 5 / 6 .

…”

Section: Strain Gradient Differential Quadrature Finite Elementmentioning

confidence: 99%

“…The underlying cause is that the present element uses the bi-cubic Hermitian interpolation technique to express the trial functions of kinematic variables, as seen from Equation (17). For its one-dimensional reduced model (ie, Timoshenko beam model), Zhang et al 28,51 have validated such shear locking-free performance explicitly. Table 13 further demonstrates the shear locking-free behavior of our element for solving the free vibration problem of a simply supported plate with L X /h ranging from 10 to 50 000, where a 24 × 24 mesh is chosen and the Navier solution 17 based on the Kirchhoff plate model is provided for the sake of comparison.…”

Section: Convergence Comparison Of the Present And Available Elementsmentioning

confidence: 99%

“…Afterward, these authors 54 proposed an incomplete C 2 ‐continuous DQFE for the MSGT‐based Kirchhoff plate model. Using the DQFEs proposed in References 51, 52, Zhang et al 55 analyzed the coupling effects of surface energy, strain gradient, and inertia gradient on the behavior of freely vibrating small‐scale beams. Ishaquddin and Gopalakrishnan 56 constructed a novel weak‐form QEM for a one‐parameter gradient elastic Euler‐Bernoulli beam model.…”

Section: Introductionmentioning

confidence: 99%

“…Next, we use the Gauss-Lobatto quadrature (GLQ) and DQ rules to discretize Equation (14). In a standard interval (ie, X ∈ [−1, 1]), the integral points and weighting coefficients for the fourth-order GLQ rule are as follows 51 :…”

mentioning

confidence: 99%

See 3 more Smart Citations

Strain gradient differential quadrature finite element for moderately thick micro‐plates

Zhang

Kong

et al. 2020

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

Summary In this study, we integrate the advantages of differential quadrature method (DQM) and finite element method (FEM) to construct a C1‐type four‐node quadrilateral element with 48 degrees of freedom (DOF) for strain gradient Mindlin micro‐plates. This element is free of shape functions and shear locking. The C1‐continuity requirements of deflection and rotation functions are accomplished by a fourth‐order differential quadrature (DQ)‐based geometric mapping scheme, which facilitates the conversion of the displacement parameters at Gauss‐Lobatto quadrature (GLQ) points into those at element nodes. The appropriate application of DQ rule to non‐rectangular domains is proceeded by the natural‐to‐Cartesian geometric mapping technique. Using GLQ and DQ rules, we discretize the total potential energy functional of a generic micro‐plate element into a function of nodal displacement parameters. Then, we adopt the principle of minimum potential energy to determine element stiffness matrix, mass matrix, and load vector. The efficacy of the present element is validated through several examples associated with the static bending and free vibration problems of rectangular, annular sectorial, and elliptical micro‐plates. Finally, the developed element is applied to study the behavior of freely vibrating moderately thick micro‐plates with irregular shapes. It is shown that our element has better convergence and adaptability than that of Bogner‐Fox‐Schmit (BFS) one, and strain gradient effects can cause a significant increase in vibration frequencies and a certain change in vibration mode shapes.

show abstract

“…In a standard interval (ie,

\overset{true‾}{X} \in false[- 1, 1 false]

), the integral points and weighting coefficients for the fourth‐order GLQ rule are as follows 51 :

{\overset{X}{true}}_{1} = - 1, {\overset{X}{true}}_{2} = - 1 / \sqrt{5}, {\overset{X}{true}}_{3} = 1 / \sqrt{5}, {\overset{X}{true}}_{4} = 1,

C_{1} = C_{4} = 1 / 6, C_{2} = C_{3} = 5 / 6 .

…”

Section: Strain Gradient Differential Quadrature Finite Elementmentioning

confidence: 99%

Section: Convergence Comparison Of the Present And Available Elementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Strain gradient differential quadrature finite element for moderately thick micro‐plates

Zhang

Kong

et al. 2020

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…随后, 借助分片检验的理论分析, 多个满足分片检验条件的三角形和四边形偶应力/应变梯度理论单元被陆续提出 [12,[14][15][16][17][18] . 近年来, 国内外学者将不同的有限元方法用于构造求解偶应力/应变梯度理论问题的高精度单元 [19][20][21][22][23] .…”

unclassified

A polygonal element for couple stress/strain gradient elasticity based on SBFEM and spline interpolation

Chen¹,

Li²

2021

Sci. Sin.-Phys. Mech. Astron.

View full text Add to dashboard Cite

Variational formulation and differential quadrature finite element for freely vibrating strain gradient Kirchhoff plates

Zhang

Kong

et al. 2020

Z Angew Math Mech

Self Cite

View full text Add to dashboard Cite

In this paper, we apply the energy variational principle to arrive at the differential equation of motion and all appropriate boundary conditions for strain gradient Kirchhoff micro‐plates. The resulting sixth‐order boundary value problem of free vibration is solved by a thirty‐six‐DOF four‐node differential quadrature plate finite element. The C2‐continuity condition of the deflection is guaranteed by devising a sixth‐order differential quadrature‐ based geometric mapping scheme that can transform the displacement parameters at Gauss‐Lobatto quadrature points into those at element nodes. The total potential energy of a generic micro‐plate element is firstly discretized in terms of nodal parameters. It is then minimized to obtain the formulation of element stiffness and mass matrices. For comparison reasons, a Hermite interpolation‐based strain gradient finite element is provided. With the help of the symbolic computation system Maple, the explicit algebraic relationship between the stiffness (or mass) matrices of two types of elements is derived. Convergence and comparison studies are conducted to show the efficacy of our element in the free vibration analysis of macro/micro‐ plates. Finally, we apply the developed method to study the size‐dependent vibration behavior of micro‐plates with uniform or stepped thickness. Numerical examples reveal that strain gradient effects can change the vibration mode shapes, not the vibration frequencies alone.

show abstract

Strain gradient differential quadrature beam finite elements

Cited by 27 publications

References 77 publications

Strain gradient differential quadrature finite element for moderately thick micro‐plates

Strain gradient differential quadrature finite element for moderately thick micro‐plates

A polygonal element for couple stress/strain gradient elasticity based on SBFEM and spline interpolation

Variational formulation and differential quadrature finite element for freely vibrating strain gradient Kirchhoff plates

Contact Info

Product

Resources

About