1,183 research outputs found
A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam based on the weak form quadrature element method
The purpose of this paper is to provide a high-order finite element method (FEM) formulation of nonlocal nonlinear nonlocal graded Timoshenko based on the weak form quadrature element method (WQEM). This formulation offers the advantages and flexibility of the FEM without its limiting low-order accuracy. The nanobeam theory accounts for the von Kármán geometric nonlinearity in addition to Eringen’s nonlocal constitutive models. For the sake of generality, a nonlinear foundation is included in the formulation. The proposed formulation generates high-order derivative terms that cannot be accounted for using regular first- or second-order interpolation functions. Hamilton’s principle is used to derive the variational statement which is discretized using WQEM. The results of a WQEM free vibration study are assessed using data obtained from a similar problem solved by the differential quadrature method (DQM). The study shows that WQEM can offer the same accuracy as DQM with a reduced computational cost. Currently the literature describes a small number of high-order numerical forced vibration problems, the majority of which are limited to DQM. To obtain forced vibration solutions using WQEM, the authors propose two different methods to obtain frequency response curves. The obtained results indicate that the frequency response curves generated by either method closely match their DQM counterparts obtained from the literature, and this is despite the low mesh density used for the WQEM systems
Isogeometric analysis for functionally graded microplates based on modified couple stress theory
Analysis of static bending, free vibration and buckling behaviours of
functionally graded microplates is investigated in this study. The main idea is
to use the isogeometric analysis in associated with novel four-variable refined
plate theory and quasi-3D theory. More importantly, the modified couple stress
theory with only one material length scale parameter is employed to effectively
capture the size-dependent effects within the microplates. Meanwhile, the
quasi-3D theory which is constructed from a novel seventh-order shear
deformation refined plate theory with four unknowns is able to consider both
shear deformations and thickness stretching effect without requiring shear
correction factors. The NURBS-based isogeometric analysis is integrated to
exactly describe the geometry and approximately calculate the unknown fields
with higher-order derivative and continuity requirements. The convergence and
verification show the validity and efficiency of this proposed computational
approach in comparison with those existing in the literature. It is further
applied to study the static bending, free vibration and buckling responses of
rectangular and circular functionally graded microplates with various types of
boundary conditions. A number of investigations are also conducted to
illustrate the effects of the material length scale, material index, and
length-to-thickness ratios on the responses of the microplates.Comment: 57 pages, 14 figures, 18 table
Meshless methods for shear-deformable beams and plates based on mixed weak forms
Thin structural theories such as the shear-deformable Timoshenko beam and Reissner-Mindlin
plate theories have seen wide use throughout engineering practice to simulate the response of
structures with planar dimensions far larger than their thickness dimension. Meshless methods
have been applied to construct numerical methods to solve the shear deformable theories.
Similarly to the finite element method, meshless methods must be carefully designed to overcome
the well-known shear-locking problem. Many successful treatments of shear-locking in
the finite element literature are constructed through the application of a mixed weak form. In
the mixed weak form the shear stresses are treated as an independent variational quantity in
addition to the usual displacement variables.
We introduce a novel hybrid meshless-finite element formulation for the Timoshenko beam
problem that converges to the stable first-order/zero-order finite element method in the local
limit when using maximum entropy meshless basis functions. The resulting formulation is free
from the effects shear-locking.
We then consider the Reissner-Mindlin plate problem. The shear stresses can be identified as
a vector field belonging to the Sobelov space with square integrable rotation, suggesting the use
of rotated Raviart-Thomas-Nedelec elements of lowest-order for discretising the shear stress field. This novel formulation is again free from the effects of shear-locking.
Finally we consider the construction of a generalised displacement method where the shear
stresses are eliminated prior to the solution of the final linear system of equations. We implement
an existing technique in the literature for the Stokes problem called the nodal volume
averaging technique. To ensure stability we split the shear energy between a part calculated
using the displacement variables and the mixed variables resulting in a stabilised weak form. The method then satisfies the stability conditions resulting in a formulation that is free from
the effects of shear-locking.Open Acces
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