Comput Mech (2010) 46:679–701 DOI 10.1007/s00466-010-0509-x
ORIGINAL PAPER
A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates H. Nguyen-Xuan · T. Rabczuk · N. Nguyen-Thanh · T. Nguyen-Thoi · S. Bordas
Received: 5 February 2010 / Accepted: 8 June 2010 / Published online: 23 June 2010 © Springer-Verlag 2010
Abstract In this paper, a node-based smoothed finite element method (NS-FEM) using 3-node triangular elements is formulated for static, free vibration and buckling analyses of Reissner–Mindlin plates. The discrete weak form of the NS-FEM is obtained based on the strain smoothing technique over smoothing domains associated with the nodes of the elements. The discrete shear gap (DSG) method together with a stabilization technique is incorporated into the NS-FEM to eliminate transverse shear locking and to maintain stability of the present formulation. A so-called node-based smoothed stabilized discrete shear gap method (NS-DSG) is then proposed. Several numerical examples are used to illustrate the accuracy and effectiveness of the present method. Keywords Plate bending · Transverse shear locking · Finite element method · Node-based smoothed H. Nguyen-Xuan (B) · T. Nguyen-Thoi Department of Mechanics, Faculty of Mathematics and Computer Science, University of Science, Vietnam National University, HCM, 227 Nguyen Van Cu, District 5, Ho Chi Minh City, Vietnam e-mail:
[email protected] URL: http://www.math.hcmuns.edu.vn/∼nxhung H. Nguyen-Xuan · T. Nguyen-Thoi Division of Computational Mechanics, Faculty of Civil Engineering, Ton Duc Thang University, 98 Ngo Tat To, Binh Thanh District, Ho Chi Minh City, Vietnam T. Rabczuk · N. Nguyen-Thanh Institute of Structural Mechanics (ISM), Bauhaus-University Weimar, Marienstr. 15, 99423 Weimar, Germany S. Bordas School of Engineering, Institute of Theoretical, Applied and Computational Mechanics, Cardiff University, Wales, UK
finite element · Discrete shear gap (DSG) · Stabilization technique
1 Introduction Static, free vibration and buckling analyses of plate structures play an increasing important role in engineering applications. A large amount of research work on plates can be found in the literature reviews [1–5]. The analytical solution approaches are restricted to plates with simple shapes. Effective numerical methods such as finite difference techniques, splinestrip element methods, the finite element method (FEM), and meshfree methods, etc., have been devised to analyze and simulate the behavior of plates. Among these numerical approaches, the FEM is still so far the most popular and reliable tool. During the last three decades, lower-order Mindlin– Reissner plate finite elements have often been preferred due to their simplicity and efficiency. They require only C 0 -continuity for the deflection and the normal rotations. However, these low-order plate elements in the thin plate limit often suffer from shear locking phenomenon due to incorrect transverse forces under bending. Therefore, many formulations have been developed to overcome the shear locking phenomenon and to increase accuracy and stability of numerical methods such as mixed formulation/hybrid elements [6–9], stabilization methods [10,11], the enhanced assumed strain (EAS) methods [12,13], the assumed natural strain (ANS) methods [14–17], etc. Recently, the discrete shear gap (DSG) method [18] which can be considered as an alternative form of the ANS was proposed. The DSG is somewhat similar to the ANS methods in the aspect of modifying the course of certain strains within the element, but is different in the aspect of lack of collocation points. The DSG
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method is therefore independent of the order and form of the element. On an other front of development of finite element technology, Liu et al. have combined the strain smoothing technique [19] used in meshfree methods [20–27] into the finite element method using quadrilateral elements to formulate a cell/element-based smoothed finite element method (SFEM or CS-FEM) [28–30] for 2D solids. It is well known that loworder elements for solid problems certain inherent drawbacks such as overestimation of the stiffness matrix and locking problems. Therefore applying the strain smoothing technique on smoothing domains to these standard FEM models aims to soften the stiffness formulation, and hence can improve significantly the accuracy of solutions in both displacement and stress. In CS-FEM, the smoothing domains are created based on elements, and each element can be subdivided into 1 or several quadrilateral smoothing domains. The theoretical aspects of CS-FEM were fully studied in [29,31,32]. The SFEM has also been extended to general n-sided polygonal elements (nSFEM) [33], dynamic analysis [34,35], plate and shell analysis [36,37], kinematic limit analysis [38] and coupled to partition of unity enrichment [39]. A general framework for this strain smoothing technique in FEM can be found in [40,41]. In the effort to overcome shortcomings of low-order elements, Liu et al. have then extended the concept of smoothing domains to formulate a family of smoothed FEM (S-FEM) models with different applications such as the node-based S-FEM (NS-FEM) [42,43], edge-based S-FEM (ES-FEM) [44–49], face-based S-FEM (FS-FEM) [50,51]. Similar to the standard FEM, these S-FEM models also use a mesh of elements. In these S-FEM models, the discrete weak form is evaluated using smoothed strains over smoothing domains instead of using compatible strains over the elements as in the traditional FEM. The smoothed strains are computed by integrating the weighted (smoothed) compatible strains. The smoothing domains are created based on the features of the element mesh such as nodes [42], or edges [44] or faces [50]. These smoothing domains are linear independent and hence stability and convergence of the S-FEM models are ensured. They cover parts of adjacent elements, and therefore the number of supporting nodes in smoothing domains is larger than that in elements. This leads to bandwidth of the stiffness matrix in S-FEM models increased and the computational cost is hence higher than those in the FEM. However, due to contribution of more supporting nodes in the smoothing domains, S-FEM often produces the solution that is much more accurate than that of the FEM. Therefore in general, when the efficiency of computation (computation time for the same accuracy) in terms of the error estimator versus computational cost is considered, the S-FEM models are more efficient than the counterpart FEM models [43,46,47,51]. It can be argued that these S-FEM models have the features of
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both models: meshfree and FEM. The element mesh is still used but the smoothed strains bring the non-local information from the neighboring elements. A general and rigorous theoretical framework to show properties, accuracy and convergence rates of the S-FEM models was given in [32]. Among these S-FEM models, the NS-FEM [42,43] shows some interesting properties in the elastic solid mechanics such as: 1) it can provide an upper bound to the strain energy; 2) it can avoid volumetric locking without any modification on integration terms; 3) super-accurate and super-convergent properties of stress solutions are gained; and 4) the stress at nodes can be computed directly from the displacement solution without using any post-processing. In this paper, we exploit several interesting properties of the NS-FEM for analyzing plates. The NS-FEM has been already extended to perform adaptive analysis [52], linear elastostatics and vibration 2D solid problems [53]. Also, alpha finite element methods (αFEM) have been recently proposed have been recently proposed as an alternative to the NS-FEM and shown to significantly improve the results obtained by conventional and smoothed FEM techniques, at the cost of the introduction of a problem-dependent parameter α [54–56]. This paper presents a formulation of the node-based smoothed finite element method (NS-FEM) for Reissner– Mindlin plates using only three-node triangular meshes which are easily generated for complicated domains. The evaluation of the discrete weak form is performed by using a strain smoothing technique over smoothing domains associated with nodes of elements. Transverse shear locking can be avoided through the discrete shear gap (DSG) method. The stability and accuracy of NS-FEM formulation is further improved by a stabilization technique to give a so-called node-based smoothed finite element method with a stabilized discrete shear gap method (NS-DSG). Several numerical examples are presented to demonstrate the reliability and effectiveness of the present method. The layout of the paper is as follows. Next section describes the weak form of governing equations and the formulation of 3-node plate element. In Sect. 3, a formulation of NS-FEM with the stabilized discrete shear technique is introduced. Section 4 recalls some techniques relevant to the present approach. Section 5 presents and discusses numerical results. Finally, we close our paper with some concluding remarks.
2 The formulation of 3-node plate element 2.1 Discrete governing equations We consider a domain Ω ⊂ R2 occupied by the reference middle surface of a plate. Let w and β = (βx , β y )T be the transverse displacement and the rotations about the y and x
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Fig. 1 a 3-Node triangular element; b Local coordinates
axes, see Fig. 1, respectively. We assume that the material is homogeneous and isotropic with Young’s modulus E and Poisson’s ratio ν, the governing differential equations of the static and dynamic Mindlin–Reissner plates can be expressed in the following form [57]
For the free vibration analysis of a Mindlin–Reissner plate model, a weak form may be derived from the dynamic form of the principle of virtual work under the assumptions of first order shear-deformation plate theory [58]: find ω ∈ R+ and 0 = (w, β) ∈ V such that
ρt 3 2 ω β = 0 in Ω 12 kt∇ · γ + ρtω2 w = p in Ω
(2)
∀(v, η) ∈ V0 , a(β, η) + kt (∇w − β, ∇v − η)
1 3 2 = ω ρt (w, v) + ρt (β, η) 12
w = w0 , β = β 0 on Γ = ∂Ω
(3)
∇ · Db κ(β) + ktγ +
(1)
where t is the plate thickness, ρ is the mass density, ω is the natural frequency, p = p(x, y) is the transverse loading per unit area, k = μE/2(1 + ν), μ = 5/6 is the shear correction factor and Db is the tensor of bending modulus given by ⎤ ⎡ 1 ν 0 Et 3 b ⎣ν 1 0 ⎦ D = (4) 12 1 − ν 2 0 0 1−ν 2 1 κ = [∇β + (∇β)T ], γ = ∇w − β 2
(5)
Ω
κ(β) : D : κ(η) dΩ
a(β, η) =
{(∇βx )T σˆ 0 ∇ηx + (∇β y )T σˆ 0 ∇η y } dΩ Ω
(6)
with B denotes a set of the essential boundary conditions and the L 2 inner products are given as (w, v) = wv dΩ, (β, η) = β · η dΩ,
(∇w)T σˆ 0 ∇v dΩ,
b1 (w, v) =
(7)
Ω
b
where σˆ 0 =
σ 0x
σ 0x y
σ 0x y σ 0y
(11)
Let us assume that the bounded domain Ω is discretized into
Ne Ne finite elements such that Ω ≈ e=1 Ω e and Ω i ∩ Ω j = ∅ , i = j. The finite element solution of the static problem of a low-order1 element model for the Mindlin–Reissner plate is to find (w h , β h ) ∈ V h such that ∀(v, η) ∈ V0h , a(β h , η)+kt (∇w h −β h , ∇v−η) = ( p, v)
Ω
The weak form of the static equilibrium equations is: find (w, β) ∈ V such that ∀(v, η)∈V0 , a(β, η)+kt (∇w − β, ∇v − η)=( p, v)
(10)
b2 (β, η) =
V0 = {(v, η) : v ∈ H 1 (Ω), η ∈ H 1 (Ω)2 : v = 0, η = 0 on ∂Ω}
∀(v, η) ∈ V0 , a(β, η) + kt (∇w − β, ∇v − η)
1 = λcr tb1 (w, v) + t 3 b2 (β, η) 12
Ω
where ∇ = (∂/∂ x, ∂/∂ y)T is the gradient vector. Let V and V0 be defined as V = {(w, β) : w ∈ H 1 (Ω), β ∈ H 1 (Ω)2 } ∩ B
In the case of in-plane buckling analyses and assuming prebuckling stresses σˆ 0 , nonlinear strains appear and the weak form can be reformulated as [58]: find ω ∈ R+ and 0 = (w, β) ∈ V s.t.
where
The bending κ and shear strains γ are defined as
(9)
(8)
(12) 1
In our case a three-node triangular linear finite element.
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where the finite element spaces, V h and V0h , are defined by V h = {(w h , β h ) ∈ H 1 (Ω) × H 1 (Ω)2 , w h |Ω e ∈ P1 (Ω e ), β h |Ω e ∈ P1 (Ω e )2 } ∩ B V0h
(13)
= {(v , η ) ∈ H (Ω) × H (Ω) : v = 0, h
h
1
1
2
h
ηh = 0 on ∂Ω}
(14)
where P1 (Ω e ) stands for the set of polynomials of degree 1 for each variable. The finite element problem of the free vibration modes is to find the natural frequency ωh ∈ R+ and 0 = (w h , β h ) ∈ V h such that ∀(v, η) ∈ V0h , a(β h , η) + kt (∇w h − β h , ∇v − η)
1 (15) = (ωh )2 ρt (w h , v) + ρt 3 (β h , η) 12 and for the buckling analysis is to find the critical buckling h ∈ R+ and 0 = (w h , β h ) ∈ V h load λcr ∀(v, η) ∈ V0h , a(β h , η) + kt (∇w h − β h , ∇v − η)
1 3 h h h = λcr tb1 (w , v) + t b2 (β , η) 12
(16)
Since only linear triangular elements are used to obtain stiffness matrices, the finite Reissner–Mindlin plate-bending element approximation is simply interpolated using the linear basis functions for both deflection and rotations without any additional variables (C 0 -continuity for the transverse displacement and the normal rotations). Hence, the bending and geometric strains are constant and unchanged from the standard finite elements while the transverse shear strains contain linear interpolated functions. It is known that these low-order plate elements in the thin plate limit often suffer from shear locking. In order to avoid shear locking, the discrete shear gaps (DSG) [18] which were proposed to reform the shear strains are adopted. As a result of using the threenode triangular elements, the shear strains γ DSG become then constant. Table 1 Summary elements
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Fig. 2 Triangular elements and smoothing domains associated with nodes
2.2 Brief on the DSG3 element In the linear triangular DSG3 element [18], the finite element approximation (w h , β h ) is simply interpolated using the linear basis functions for both deflection and rotations without any additional variables. The bending strains used in the standard finite elements are unchanged while the transverse shear strains are reformulated by the interpolated shear gaps. The shear strains can be expressed as a reduced operator Rh : H 1 (Ω e ) → Γ h (Ω e ), where Γ h (Ω e ) is defined as: Γ h (Ω e ) = γ h |Ω e = (J−1 )T γˆ h , γˆ h = [γ1h γ2h ]T (17) The shear strain can be rewritten in the incorporation of reduction operator as γ DSG (w h , β h ) = ∇w h − Rh β h = (J−1 )T γˆ h
MITC4
Four node mixed interpolation of tensorial component [14]
MIN3
Three node Mindlin [15]
DSG3
Discrete shear gap triangle element [18]
(18)
ES-DSG3
Edge-based smoothed discrete shear gap triangular element [46]
Q4BL
Quadrilateral bubble linked [69]
DKMQ
Discrete Kirchhoff Mindlin quadrilateral [70]
ANS4
Four node assumed natural strain [71]
ANS9
Nine node Assumed natural strain [72]
RPIM
Radial point interpolation method [73]
Pb-2 Ritz
Two-dimensional polynomial function Rayleigh–Ritz method [74]
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Δwγ11 = Δwγ31 = Δwγ12 = Δwγ22 = 0 1 1 Δwγ21 = w2 −w1 + a(βx1 +βx2 )+ b(β y1 +β y2 ) (20) 2 2 1 1 Δwγ32 = w3 − w1 + d(βx1 + βx3 ) + c(β y1 + β y3 ) 2 2
Fig. 3 Patch test of the element (E = 100,000; ν = 0.25; t = 0.01)
Table 2 Patch test Methods
w5
θx5
θ y5
m x5
m x5
m x y5
MIN3
0.6422 1.1300 −0.6400 −0.0111 −0.0111 −0.0033
DSG3
0.6422 1.1300 −0.6400 −0.0111 −0.0111 −0.0033
ES-DSG3 0.6422 1.1300 −0.6400 −0.0111 −0.0111 −0.0033
with a = x2 − x1 , b = y2 − y1 , c = y3 − y1 , d = x3 − x1 and (wi , βxi , β yi ), i = 1, 2, 3 are the degree of freedoms at node i of the element. In order to further improve the accuracy of approximate solutions and to stabilize shear force oscillations appearing in the triangular element, the stabilization technique [59] can be used here. The idea for the stabilization of the original DSG3 element was also introduced in [60]. With this remedy, the DSG3 element problem to the static problem is to find (w h , β h ) ∈ V h such that ∀(v, η) ∈ V0h , a(β h , ηh ) + kt (γ DSG (w h , β h ), γ DSG (v, η)) = ( p, v) (21)
NS-DSG3 0.6422 1.1300 −0.6400 −0.0111 −0.0111 −0.0033
x,ξ y,ξ is the Jacobian matrix of the bilinear where J = x,η y,η mapping from the local triangular element Ωˆ into the physical triangular element Ω e (see Fig. 1b) and γˆ h contains the derivatives of the shape functions (N1 = 1 − ξ − η, N2 = ξ, N3 = η) that are only constant (cf. Bletzinger al. [18] for more detail)
where (γ DSG (w h , β h ), γ DSG (v, η)) = (∇w h −Rh β h , ∇v−Rh η) =
Ne e=1
=
Ne e=1
⎡
γ1h
⎤
γˆ h = ⎣ γ h ⎦ = 2
N2,ξ Δwγ21 + N3,η Δwγ31 N2,ξ Δwγ22 + N3,η Δwγ32
(19)
where Δwγi ς (i = 1, 2, 3) are the discrete shear gaps at the triangular element nodes related to the ς -local coordinate axis (ς = 1, 2) that are reported as
t2
t2 (∇w h − Rh β h , ∇v − Rh η) L 2 (Ω e ) + αh 2e
t2 (∇w h − Rh β h ).(∇v − Rh η)Ae t 2 + αh 2e
(22)
where h e is the longest length of the edges of the element, α is a positive constant and Ae is the area of the triangular element. It is evident that the original DSG3 element is recovered when α = 0. For free vibration and buckling problems, the second terms ((∇w h − β h , ∇v − η)) in the left hand side of Eq. (15) and Eq. (16) are replaced by the terms in Eq. (22). In what follows, we utilize these constant strains to establish a formulation of a node-based smoothed triangular
Fig. 4 Square plate model: a simply supported plate; b full clamped plate
(a)
(b)
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Normalized central moment
Normalized deflection w
1.05 1 0.95 0.9 0.85 Exact solu. MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3
0.8 0.75 0.7
5
10
15
20
25
1.05 1 0.95 0.9 Exact solu. MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3
0.85 0.8 0.75
30
5
10
Number of elements per edge
15
25
30
(a)
1.5
1.5
0.5 0 −0.5 −1
MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3
−1.5 −2
0.6
0.8
1
1.2
1.4
1.6
10
1
log (Relative error of central moment)[%])
10
log (Relative error of central deflection)[%])
(a)
−2.5 0.4
20
Number of elements per edge
1
0.5
0
−0.5
−1
MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3
−1.5
−2 0.4
1.8
0.6
0.8
log10(h)
1
1.2
1.4
1.6
1.8
log10(h)
(b)
(b)
Fig. 5 Simply supported plate (t/L = 0.01): a normalized central deflection; b relative error under log–log scale
Fig. 6 Simply supported plate (t/L = 0.01): a normalized central moment; b relative error under log–log scale
element with the stabilized discrete shear gap technique (NS-DSG3) for Reissner–Mindlin plates.
ated by connecting sequentially the mid-edge-points to the centroids of the surrounding triangular elements of node k as shown in Fig. 2 ¯ and the average shear strains The average curvatures κ, γ¯ over the cell Ω (k) are defined by
3 A formulation of NS-FEM with stabilized discrete shear technique In the NS-FEM [42,43], the domain discretization is the same as that of the standard FEM using Ne triangular elements, but the integration required in the weak form of the FEM is now performed based on the nodes, and strain smoothing technique [19] is utilized. In such a nodal integration process, the problem domain Ω is again divided into a set Nn Ω (k) and of smoothing domains Ω (k) such as Ω ≈ k=1 j) (i) ( Ω ∩ Ω = ∅, i = j in which Nn is the total number of nodes of the problem domain. For triangular elements, the smoothing domain Ω (k) associated with the node k is cre-
123
k
Ne 1 e 1 κ dΩ = (k) A κi 3 i A i=1 Ω (k) 1 γ¯ k (w h , β h ) = (k) γ DSG (w h , β h ) dΩ A
1 κ¯ k = (k) A
(23)
Ω (k)
Nek
=
1 1 e DSG h h A γ (w , β ) 3 i i A(k) i=1
(24)
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1.04
1.1
1.02
Normalized central deflection w
C
1
Normalized strain energy
1 0.98 0.96 0.94
Exact sol. MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3
0.92 0.9 0.88
0
5
10
15
20
25
30
Exact sol. MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3
0.9
0.8
0.7
0.6
0.5 0
35
5
10
15
log10(Relative error of central deflection)[%])
MITC4 (r = 1.12) MIN3 (r = 1.42) DSG3 (r = 1.3) ES−DSG3 (r = 1.6) NS−DSG3 (r=0.99)
1.6 1.4
log10(Error in strain energy)
30
2
1.8
1.2 1 0.8 0.6 0.4 0.2 0 −0.2 0.4
1.5 1 0.5 0 −0.5 −1 MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3
−1.5 −2 −2.5 0.4
0.6
0.8
1
1.2
0.6
0.8
1.4
10
Fig. 7 Simply supported plate (t/L = 0.01): a strain energy; b convergence rate
¯ ∇β ¯ x, The average gradients related to geometric strains ∇w, (k) ¯ ∇β y over the smoothing domain Ω are given by
1.6
1.8
A
(k)
=
Nk
e 1 dΩ = Aie 3
(26)
i=1
k
Ne Ne 1 e 1 e h 1 1 ¯ A ∇wi , ∇βxk = (k) A β , 3 i 3 i xi A(k) A i=1
k
Ne 1 e h 1 A ∇β yi = (k) 3 i A
1.4
Fig. 8 Convergence in the deflection of clamped plate (t/L = 0.001): a normalized central deflection; b relative error under log–log scale
Ω (k)
i=1
1.2
(b)
(b)
k
1
log10(h)
log (h)
¯ yk ∇β
25
(a)
(a)
¯ k= ∇w
20
Number of elements per edge
Number of elements per edge
(25)
i=1
where A(k) is the area of the smoothing domain Ω (k) and is computed by
in which Aie is the area of the ith element attached to node k and Nek is the number of elements associated with node k illustrated in Fig. 2. With the above definitions, a solution of the NS-FEM with the stabilized discrete shear gap technique using three—node triangular elements (NS-DSG3) is now established. The NSDSG3 solution to the static problem is to find (w h , β h ) ∈ V h such that
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Normalized central moment
1.2 1.1 1 0.9 0.8
Normalized strain energy
1.05 Exact sol. MITC4 DSG3 MIN3 ES−DSG3 NS−DSG3
1
0.95
0.9 Exact solu. MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3
0.7
0.85 0.6 0.5 0.4
0.8 0
5
10
15
20
25
5
10
15
(a)
1.5 1
log10(Error in energy norm)
log10(Relative error of central moment)[%])
1.8
0.5 0 −0.5 −1
MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3
−1.5 −2
0.8
1
1.2
1.4
1.6
1.6
MITC4 (r = 1.0) MIN3 (r = 1.1) DSG3 (r = 0.9) ES−DSG3 (r = 1.03) NS−DSG3 (r = 0.97)
1.4
1.2
1
0.8 1.8
log (h)
0.4
10
0.6
(b)
1
1.2
1.4
10
∀(v, η) ∈ V0h , a(β ¯ h , ηh )+kt (γ¯ (w h , β h ), γ¯ (v, η)) = ( p, v) (27) The NS-DSG3 solution of the free vibration modes is to find the natural frequency ωh ∈ R+ and 0 = (w h , β h ) ∈ V h such that ¯ h , η) + kt (γ¯ (w h , β h ), γ¯ (v, η)) ∀(v, η) ∈ V0h , a(β
1 3 h h 2 h = (ω ) ρt (w , v) + ρt (β , η) 12
0.8
log (h)
Fig. 9 Clamped plate (t/L = 0.001): a Normalized central moment; b relative error under log–log scale
(b) Fig. 10 Clamped plate model (t/L = 0.001): a strain energy; b convergence rate
∀(v, η) ∈ V0h , a(β ¯ h , η) + kt (γ¯ (w h , β h ), γ¯ (v, η))
1 3¯ h h h ¯ = λcr t b1 (w , v) + t b2 (β , η) 12
a(β ¯ , η) = (28)
(29)
where a(·, ¯ ·) is a smoothed bilinear form given by
h
and for the buckling analysis is to find the critical buckling h ∈ R+ and 0 = (w h , β h ) ∈ V h such that load λcr
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30
2
2
0.6
25
(a)
Number of elements per edge
−2.5 0.4
20
Number of elements per edge
30
Nn
κ¯ k : Db : η¯ k A(k)
(30)
k=1
¯ ∇β ¯ x, and the geometric terms related to the gradients (∇w, (k) ¯ ∇β y ) over the smoothing domain Ω are given by
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c
Central deflection (100w D/pL4)
0.425
0.42
0.415
0.41
0.405 10
100
1000
10000
100000
1000000
Ratio L/t
(a)
4 Central displacements wc/(pL /1000D)
0.8 Exact DSG3 MIN3 NS−DSG3
Morley sol. Q4BL DKMQ MITC4 DSG3 ES−DSG3 NS−DSG3
0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3
5
Central deflection (100wcD/pL4)
0.155 Exact DSG3 MIN3 NS−DSG3
0.15
10
15
20
25
30
Number of element per edge
Fig. 13 Convergence of central deflection for skew Morley’s plate
0.145
(γ¯ (w , β ), γ¯ (v, η)) = h
0.14
h
Nn k=1
× γ¯ k (w h , β h ) · γ¯ k (v, η)A(k) (32)
0.135
0.13
0.125 10
100
1000
10000
100000
1000000
Ratio L/t
(b) Fig. 11 Performance of NS-DSG3 with varying L/t ratios: a Simply supported plate; b Clamped plate
Fig. 12 A simply supported skew Morley’s model
b¯1 (w h , v) =
t2 t 2 + αh 2k
Nn
¯ k )T σˆ 0 ∇v ¯ k A(k) , (∇w
k=1
(31) Nn h T h T ¯ xk ) σˆ 0 ∇η ¯ yk ) σˆ 0 ∇η ¯ xk+(∇β ¯ yk A(k) b2 (β , η) = (∇β h
k=1
and modified shear terms are now obtained by performing the smoothing operation via the smoothing domain Ω (k) :
√ in which h k = A(k) is considered as the character length of the smoothing domain Ω (k) . The necessity of stabilization for lower order plate elements in bending was shown in [59,60]. Stabilization significantly improves the accuracy in the case of very thin plates and distorted meshes and to reduce oscillations of transverse shear forces. It is found from numerical experiments that the stabilization parameter α fixed at 0.1 can produce the reasonable accuracy for all cases tested. The stiffness matrix of NS-DSG3 becomes too flexible, if α is chosen too large. On the contrary, the accuracy of the solution will reduce due to the oscillation of shear forces, if α is chosen too small. So far, how to obtain an optimal value of parameter α is an open question.
4 Relationship to similar techniques The present method can be considered as an alternative form of nodally integrated techniques in finite element formulations [61–68]. The crucial idea of these methods is to formulate a nodal deformation gradient via a weighted average of the surrounding element values. The major contribution to the nodal-integral method has been pointed out by Bonet and Burton [61]. In their approach, the node-based formulation is applied to the volumetric component of the strain energy in order to eliminate volumetric locking of the
123
688
Comput Mech (2010) 46:679–701 10
Morley sol. Q4BL DKMQ MITC4 DSG3 ES−DSG3 NS−DSG3
2.3
2
Central max principal moment (Mmax/qL /100)
2.4
2.2 2.1
4.4
x 10
Reference DKMQ MITC4 Q4BL DSG3 ES−DSG3 NS−DSG3
4.2
4
Strain energy
2 1.9 1.8 1.7
3.8
3.6
3.4
1.6
3.2 1.5
5
10
15
20
25
30
3
Number of elements per edge
(a)
10
20
30
40
50
60
Number of elements per edge
2
Central min principal moment (Mmin/qL /100)
Fig. 15 Strain energy of a simply supported skew Morley’s plate Morley sol. Q4BL DKMQ MITC4 DSG3 ES−DSG3 NS−DSG3
1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7
5
10
15
20
25
30
Number of elements per edge
(b) Fig. 14 Skew Morley’s plate: a central max principle moment; b central min principle moment
standard tetrahedral element. Subsequently, Dohrmann et al. [62] proposed a nodally averaged formulation for entire components of the strain energy (i.e. including deviatoric component). This approach is simple while the method possesses very interesting properties such as 1) the upper bound property in strain energy; 2) free of volumetric locking; 3) superaccurate and super-convergent properties of stress solutions; 4) the stress at nodes computed directly from the displacement solution without using any post-processing. Further improvements on the stability condition of the nodally averaged formulation have also been devised in Bonet et al.
123
[63], Puso and Solberg [65] and Gee et al. [66]. Recently, a weighted-residual method that is used to weakly impose both the equilibrium equation and the kinematic equation was also introduced to create the average nodal strain formulation [67,68] for a variety of solid and plate elements. The NS-FEM approach originates from the computation of smoothed strains via the smoothing domains associated with nodes of elements. In the NS-FEM, the way to create smoothing domains is similar to nodal backgrounds in Dohrmann et al. [62]. Furthermore, the NS-FEM works for arbitrary n-sided polygonal elements. When only linear triangular or tetrahedral elements are used, the NS-FEM produces the same results as the method proposed by Dohrmann et al. [62]. The extension of the nodally integrated techniques to Reissner–Mindlin plate elements has been proposed very recently in [68]. Numerical results show some advantages of the nodally integrated formulation compared to the standard FEM. However, it was proved in [68] that these formulations can not fulfill sufficiently, in general, constant bending strain patch test with an arbitrary mesh. While only static plate problems were studied in [68], we dealt with static, free vibration and buckling solutions of Reissner–Mindlin plates using the NS-FEM. Moreover, we show numerically the fulfilment of the constant bending strain patch test. Although in several cases the numerical results using the NS-FEM are slightly less accurate than those using the ES-FEM,2 its performance is much better than several other existing 2
This method for analysis of plates has already been investigated in [46].
Comput Mech (2010) 46:679–701
689
Fig. 16 Supported and clamped plate
Table 3 A non-dimensional frequency parameter of a SSSS plate (a/b = 1)
t/a
Method DSG3 (%)
0.005
0.1
Table 4 A non-dimensional frequency parameter of a CCCC plate (a/b = 1)
Mode
t/a
0.1
NS-DSG3 (%)
Exact [76] 4.443
1
4.5131 (1.58)
4.4641 (0.48)
4.4509 (0.18)
2
7.1502 (1.78)
7.0870 (0.88)
7.0441 (0.27)
7.025
3
7.3169 (4.15)
7.1193 (1.34)
7.0572 (0.46)
7.025
4
9.3628 (5.36)
9.0582 (1.94)
8.9519 (0.74)
8.886
5
10.3772 (4.45)
10.1444 (2.11)
9.9944 (0.60)
9.935
6
10.3772 (4.45)
10.1489 (2.15)
9.9954 (0.61)
9.935
1
4.3943 (0.56)
4.3846 (0.33)
4.3725 (0.06)
4.37
2
6.8227 (1.23)
6.7922 (0.77)
6.7516 (0.17)
6.74
3
6.8587 (1.76)
6.8196 (1.18)
6.7627 (0.34)
6.74
4
8.5447 (2.33)
8.4744 (1.49)
8.3801 (0.36)
8.35
5
9.4557 (2.56)
9.3666 (1.60)
9.2278 (0.08)
9.22
6
9.4616 (2.62)
9.3698 (1.62)
9.2285 (0.09)
9.22
Mode
Method DSG3 (%)
0.005
ES-DSG3 (%)
ES-DSG3 (%)
NS-DSG3 (%)
Exact [76]
1
6.1786 (3.00)
6.0355 (0.61)
5.9693 (−0.50)
5.999
2
8.8759 (3.60)
8.6535 (1.00)
8.5085 (−0.70)
8.568
3
9.0680 (5.83)
8.7081 (1.64)
8.5269 (−0.48)
8.568
4
11.2452 (8.05)
10.6584 (2.42)
10.3880 (−0.18)
10.407
5
12.2182 (6.50)
11.7430 (2.36)
11.3913 (−0.70)
11.472
6
12.2992 (6.97)
11.7720 (2.38)
11.4215 (−0.67)
11.498
1
5.7616 (0.90)
5.7250 (0.26)
5.6746 (−0.62)
5.71
2
7.9935 (1.44)
7.9211 (0.52)
7.8158 (−0.81)
7.88
3
8.0525 (2.19)
7.9627 (1.05)
7.8313 (−0.62)
7.88
4
9.5772 (2.65)
9.4499 (1.29)
9.2686 (−0.66)
9.33
5
10.4153 (2.82)
10.2631 (1.31)
10.0222 (−1.06)
10.13
6
10.4697 (2.85)
10.3126 (1.30)
10.0683 (−1.09)
10.18
methods. In addition, nodally integrated techniques for Reissner–Mindlin plate formulations found in the literature are very limited. This may be due to the various difficulties generated from plate models. Therefore, the motivation of this work is to complement the nascent body of litera-
ture on nodally integrated formulations for static, free vibration and buckling analyses of Reissner–Mindlin plates. Further developments of the present technique for plates with complicated behaviors or shell problems will be investigated in forthcoming papers.
123
690
Comput Mech (2010) 46:679–701 1.9
1.7
Normalized frequencies
5 Numerical examples
Exact DSG3 (mode1) DSG3 (mode2) DSG3 (mode3) DSG3 (mode4) DSG3 (mode5) NS−DSG3 (mode1) NS−DSG3 (mode2) NS−DSG3 (mode3) NS−DSG3 (mode4) NS−DSG3 (mode5)
1.8
1.6 1.5 1.4
In what follows, the present element (NS-DSG3) is compared to several published elements from the literature, summarized in Table 1. Static, free vibration and buckling and analyses of square, rectangular, circular and triangular plates are considered.
1.3
5.1 Static analysis 1.2
5.1.1 Patch test
1.1 1 0.9
4
6
8
10
12
14
16
Number of elements per edge
(a) 1.6
Normalized frequencies
√ Table 5 A non-dimensional frequency parameter = ωa 2 ρt/D of a square plate (t/a = 0.005) with various boundary conditions
Exact DSG3 (mode1) DSG3 (mode2) DSG3 (mode3) DSG3 (mode4) DSG3 (mode5) NS−DSG3 (mode1) NS−DSG3 (mode2) NS−DSG3 (mode3) NS−DSG3 (mode4) NS−DSG3 (mode5)
1.5
1.4
1.3
The patch test is introduced to examine the convergence of finite elements. It is checked if the element is able to reproduce a constant distribution of all quantities for arbitrary meshes. A rectangular plate is modeled by several
Plate type
SSSF
1.2
1.1
SFSF 1
0.9
4
6
8
10
12
14
16
Number of elements per edge
CCCF
(b) Fig. 17 Convergence of normalized frequency a/b = 1; t/a = 0.005: a SSSS plate; b CCCC plate
h /
exact
with CFCF
Fig. 18 The circular plates and a typical mesh
123
Mode
Methods DSG3
ES-DSG3
NS-DSG3
Exact [76]
1
11.7720
11.6831
11.6481
11.685
2
28.3759
27.8382
27.7495
27.756
3
41.9628
41.4312
41.0324
41.197
4
61.5092
59.6720
59.0864
59.066
1
9.6673
9.6425
9.6224
9.631
2
16.3522
16.1239
16.0636
16.135
3
37.6792
36.9054
36.8072
36.726
4
39.5026
39.2167
38.9116
38.945
1
24.2848
23.8947
23.6016
24.020
2
41.7698
40.1998
39.9046
40.039
3
65.0068
63.5127
61.8530
63.493
4
80.9461
77.8776
77.1318
76.761
1
22.3437
22.1715
21.8768
22.272
2
27.1814
26.4259
26.1366
26.529
3
45.8829
43.9273
43.6459
43.664
4
62.5225
62.9466
62.9090
64.466
Comput Mech (2010) 46:679–701 Table 6 The parameterized natural frequencies √ = ωa 2 ρt/D of a clamped circular plate with t/(2R) = 0.01
Table 7 The parameterized natural frequencies √ = ωa 2 ρt/D of a clamped circular plate with t/(2R) = 0.1
691
Mode
Methods DSG3
ES-DSG3
NS-DSG3
ANS4 [71]
ANS9 [72]
Exact [77]
1
10.2941
10.2402
10.2580
10.2572
10.2129
10.2158
2
21.6504
21.3966
21.4620
21.4981
21.2311
21.2600
3
21.6599
21.4096
21.4800
21.4981
21.2311
21.2600
4
35.9885
35.3012
35.4611
35.3941
34.7816
34.8800
5
35.9981
35.3277
35.5009
35.5173
34.7915
34.8800
6
41.1864
40.3671
40.6001
40.8975
39.6766
39.7710
7
53.4374
52.0138
52.3402
52.2054
50.8348
51.0400
8
53.5173
52.1013
52.4428
52.2054
50.8348
51.0400
9
64.2317
62.3053
62.8261
63.2397
60.6761
60.8200
10
64.4073
62.4665
62.9265
63.2397
60.6761
60.8200
11
74.2254
71.6554
72.2458
71.7426
69.3028
69.6659
12
74.3270
71.7269
72.3162
72.0375
69.3379
69.6659
13
91.4366
87.7019
88.5316
88.1498
84.2999
84.5800
14
91.5328
87.7861
88.6825
89.3007
84.3835
84.5800
Mode
Methods DSG3
ES-DSG3
NS-DSG3
ANS4 [71]
Exact [77]
1
9.3012
9.2527
9.2789
9.2605
9.240
2
18.0038
17.8372
17.9195
17.9469
17.834
3
18.0098
17.8428
17.9366
17.9469
17.834
4
27.6010
27.2344
27.4301
27.0345
27.214
5
27.6082
27.2391
27.4531
27.6566
27.214
6
30.9865
30.5173
30.7906
30.3221
30.211
7
37.9464
37.2817
37.6719
37.2579
37.109
8
37.9817
37.3128
37.7152
37.2579
37.109
9
43.9528
43.0626
43.6325
43.2702
42.409
10
44.0324
43.1328
43.6664
43.2702
42.409
11
48.9624
47.8823
48.5592
47.7074
47.340
12
48.9793
47.8976
48.5887
47.8028
47.340
13
57.2487
55.7747
56.7283
56.0625
54.557
14
57.2776
55.8052
56.7876
57.1311
54.557
triangular elements as shown in Fig. 3. The boundary deflection is assumed to be w(x, y) = (1+x +2y+x 2 +x y+y 2 )/2. It is seen from Table 2 that the NS-DSG3 element passes the constant bending patch test within machine precision. 5.1.2 Square plates Consider the model of a square plate (length L, thickness t) with simply supported and clamped boundary conditions,
respectively, subjected to a uniform load p = 1 as shown in Fig. 4. The material parameters are: Young’s modulus E = 1,092,000 and Poisson’s ratio ν = 0.3. Due to symmetry, only the below left quadrant of the plate is modeled and uniform meshes N × N with N = 2, 4, 8, 16, 32 are employed. For a simply supported plate, Fig. 5 illustrates the convergence of the normalized deflection and the relative error on a log–log plot. The central moments are depicted in Fig. 6. The
123
692
Comput Mech (2010) 46:679–701
Fig. 19 A triangular cantilever plates and mesh of it
strain energy together with its convergence rate for a relation t/L = 0.01 are shown in Fig. 7. It is observed that the NSDSG3 element reveals higher accuracy the original DSG3. It is seen that all the results of the NS-DSG3 converge to the exact value from above. More details concerning upper bound solutions provided by NS-FEMs are given in [42]. For the convergence of the deflection, the MITC4 element is the most effective. For the convergence of moment and energy, the ES-DSG3 element is superior. Based on the above results, it can be concluded that the NS-DSG3 gives relatively good results compared with the MIN3, MITC4 and ES-DSG3 elements. For a clamped plate, numerical results are displayed in Figs. 8, 9 and 10. It is seen that the NS-DSG3 produces upper bound solutions to the exact value and shows high reliability compared to the other elements. For the deflection, the NS-DS3 convergence is slower than the other elements with fine meshes. For the central moment, the NS-DSG3 element produces very reasonable results compared with the MITC4 and ES-DSG3 models. Figure 10 plots the convergence in strain energy and its energy error for a relation t/L = 0.001. It is again seen that the present element can produce an upper bound in strain energy and its result shows good agreement with those from MITC4 and ES-DSG3 for this case.
123
Now we illustrate the performance of the NS-DSG3 when the plate becomes very thin. Theoretically, it is well known that shear effect will reduce when the ratio of length– thickness (L/t) increases. Hence, solutions of Reissner– Mindlin theory will approach solutions of Kirchhoff theory. Figure 11 plots the central deflection with respect to the L/t ratio. It is found that the NS-DSG3 is locking-free in the thin plate limit. In addition, the results of NS-DSG3 are more accurate than those of the original DSG3 element. 5.1.3 Skew plate subjected to a uniform load Let us consider a rhombic plate subjected to a uniform load p = 1 as shown in Fig. 12. This plate was originally studied by Morley [75]. Geometry and material parameters are length L = 100, thickness t = 0.1, Young’s modulus E = 10.92 and Poisson’s ratio ν = 0.3. The values of the deflection, principle moments and strain energy derived from the NS-DSG3 in comparison with those of other methods are shown in Figs. 13, 14 and 15, respectively. It is observed that the NS-DSG3 can produce upper bound solutions to the exact value. Although the NS-DSG3 is less accurate than the ES-DSG3, it shows very good performance compared to the other elements in the literature.
Comput Mech (2010) 46:679–701 Table 8 The parameterized natural frequencies = ωa 2 (ρt/D)1/2 /π of triangular platess with t/b = 0.001
693 ϕo
Mode
Methods DSG3
0◦
15◦
30◦
45◦
60◦
ES-DSG3
NS-DSG3
ANS4 [71]
Rayleigh–Ritz [79]
1
0.6252
0.6242
0.6235
0.624
2
2.3890
2.3789
2.3737
2.379
0.625 2.377
3
3.3404
3.3159
3.3040
3.317
3.310
4
5.7589
5.7124
5.6828
5.724
5.689
5
7.8723
7.7919
7.7312
7.794
7.743
6
10.3026
10.1547
10.0674
10.200
1
0.5855
0.5840
0.5835
0.583
0.586
2
2.1926
2.1833
2.1783
2.181
2.182
3
3.4528
3.4163
3.4024
3.413
3.412
4
5.3481
5.3020
5.2777
5.303
5.279
5
7.3996
7.3112
7.2517
7.289
7.263
6
10.2498
10.0779
9.9925
10.095
1
0.5798
0.5766
0.5745
0.575
2
2.1880
2.1778
2.1720
2.174
2.178
3
3.7157
3.6539
3.6287
3.638
3.657
4
5.5983
5.5361
5.5095
5.534
5.518 7.109
–
– 0.578
5
7.2814
7.1628
7.0954
7.139
6
10.7753
10.5108
10.3948
10.477
1
0.6006
0.5923
0.5855
0.588
0.593
2
2.3564
2.3359
2.3243
2.324
2.335
3
4.2795
4.1699
4.1289
4.126
4.222
4
6.5930
6.4424
6.3662
6.381
6.487
5
7.8615
7.6658
7.5973
7.614
7.609
6
11.7850
11.3496
11.1757
11.224
1
0.6497
0.6261
0.6298
0.613
2
2.7022
2.6101
2.5709
2.564
2.618
3
5.6491
5.4283
5.3683
5.353
5.521
–
– 0.636
4
8.3505
7.7333
7.5127
7.460
8.254
5
10.7757
10.3756
10.2769
10.306
10.395
6
14.6003
13.3296
12.9519
12.942
–
5.2 Free vibration of plates In this section, we examine the accuracy and efficiency of the NS-DSG3 element for analyzing natural frequencies of plates. The plate may have free (F), simply (S) supported or clamped (C) edges. A non-dimensional frequency parameter is often used for the presentation of the results for regular meshes.
5.2.1 Square plates Let’s consider square plates of length a, width b and thickness t as shown in Fig. 16. The material parameters are Young’s
modulus E = 2.0 × 1011 , Poisson’s ratio ν = 0.3 and the density mass ρ = 8,000. For comparison, the plate is modeled with uniform meshes of 16 elements per side. A nondimensional frequency parameter = (ω2 ρa 4 t/D)1/4 is used, where D = Et 3 /(12(1 − ν 2 )) is the flexural rigidity of the plate. Thin and thick SSSS plates corresponding to length-to-width ratios, a/b = 1 and thickness-to-length t/a = 0.005 (and t/a = 0.1) are considered in this problem. The geometry of the plate and the typical mesh are shown in Fig. 16. The convergence of the first six modes corresponding to meshes using 16 × 16 rectangular elements is presented in Tables 3 and 4. The relative error percentages compared with the exact results are given in parentheses. The NS-DSG3
123
694
Comput Mech (2010) 46:679–701 0.65 DSG3 ES−DSG3 NS−DSG3 Rayleigh−Ritz Pb2−Rayleigh−Ritz ANS4
0.64 0.63
DSG3 ES−DSG3 NS−DSG3 Rayleigh−Ritz Pb2−Rayleigh−Ritz ANS4
5.5
5
Mode 3
Mode 1
0.62 0.61 0.6
4.5
4
0.59 3.5
0.58
0
10
20
30
40
50
3
60
0
10
20
angle
30
40
50
60
50
60
angle 8.5 DSG3 ES−DSG3 NS−DSG3 Rayleigh−Ritz Pb2−Rayleigh−Ritz ANS4
2.7
8
7.5
Mode 4
2.6
Mode 2
2.5
2.4
7
6.5
2.3
6
2.2
5.5
2.1
DSG3 ES−DSG3 NS−DSG3 Rayleigh−Ritz Pb2−Rayleigh−Ritz ANS4
5
0
10
20
30
40
50
60
0
10
20
30
40
angle
angle
Fig. 20 Convergence of the frequencies of triangular plates with t/a = 0.001
shows good agreement with the exact results [76]. It is seen that computed frequencies of the NS-DSG3 element are more accurate than those of the DSG3 element, and slightly more accurate than those of ES-DSG3 element. The convergence of computed frequencies of SSSS and CCCC plates is also displayed in Fig. 17. It is clear that the NS-DSG3 element outperforms the DSG3 element. In addition, the reliability of the NS-DSG3 element is also shown for the four sets of various boundary conditions in this example: SSSF, SFSF, CCCF, and CFCF. The first four lowest frequencies are listed in Table 5.
5.2.2 Circular plates In this example, a circular plate with a clamped boundary is studied as shown in Fig. 18. The problem parameters are
123
Young’s modulus E = 2.0 × 1011 , Poisson’s ratio ν = 0.3, radius R = 5 and mass density ρ = 8000. The plate is discretized with 848 triangular elements with 460 nodes. Two thickness-span ratios t/(2R) = 0.01 and 0.1 are considered. Table 6 summarizes the frequencies of circular plate with the thickness-span ratio t/(2R) = 0.01 derived from the NS-DSG3 in comparison with other elements. The results of the NS-DSG3 element are closer to the analytical solutions [77,78] compared to those of the DSG3 element. It also is a good competitor to the ES-DSG3 element and quadrilateral plate elements such as the Assumed Natural Strain solutions (ANS4) with 432 quadrilateral elements [71] and the higher order Assumed Natural Strain solutions (ANS9) [72]. In case of the thickness-span ratio t/(2R) = 0.1, as shown in Table 7, reasonable results are obtained for the NS-DSG3 element.
Comput Mech (2010) 46:679–701 Table 9 The parameterized natural frequencies = ωa 2 (ρt/D)1/2 /π of a triangular plates with t/b = 0.2
695 ϕ◦
0◦
15◦
30◦
45◦
60◦
Mode
Methods DSG3
ES-DSG3
NS-DSG3
ANS4 [71]
Rayleigh–Ritz [79]
1
0.5830
0.5823
0.5816
0.582
0.582
2
1.9101
1.9040
1.8980
1.915
1.900
3
2.4176
2.4083
2.3989
2.428
2.408
4
3.9772
3.9559
3.9337
3.984
3.936
5
5.0265
4.9954
4.9599
5.018
–
6
5.9521
5.8994
5.8454
5.944
–
1
0.5449
0.5441
0.5433
0.545
0.544
2
1.7803
1.7749
1.7693
1.764
1.771
3
2.3959
2.3854
2.3752
2.420
2.386
4
3.6668
3.6467
3.6266
3.608
3.628
5
4.8504
4.8208
4.7868
4.820
–
6
5.6057
5.5385
5.4760
5.431
–
1
0.5339
0.5328
0.5316
0.532
0.533
2
1.7815
1.7754
1.7693
1.773
1.772
3
2.4356
2.4206
2.4077
2.437
2.419
4
3.6085
3.5842
3.5631
3.591
3.565
5
4.7829
4.7444
4.7043
4.765
–
6
5.4532
5.3377
5.2481
5.323
–
1
0.5412
0.5391
0.5371
0.541
0.540
2
1.8977
1.8882
1.8800
1.884
1.885
3
2.5304
2.5004
2.4820
2.518
2.489
4
3.7518
3.7035
3.6730
3.748
3.674
5
4.8188
4.6800
4.5794
4.740
–
6
5.4304
5.2256
5.1299
5.292
–
1
0.5634
0.5588
0.5556
0.559
0.559
2
2.0837
2.0623
2.0496
2.095
2.059
3
2.5355
2.4356
2.4114
2.483
2.396
4
4.0862
3.8009
3.7229
3.910
3.590
5
4.6612
4.3393
4.2779
4.517
–
6
5.9782
5.5835
5.4814
5.763
–
5.2.3 Triangular plates Let us consider cantilever (CFF) triangular plates with various shape geometries, see Fig. 19. The material parameters are Young’s modulus E = 2.0 × 1011 , Poisson’s ratio ν = 0.3 and mass density ρ = 8,000. A non-dimensional frequency parameter = ωa 2 (ρt/D)1/2 /π of triangular square plates with aspect ratio t/a = 0.001 and 0.2 is calculated. The mesh of 744 triangular elements with 423 nodes is used to analyze the convergence for various skew angles such as ϕ ◦ = 0◦ , 15◦ , 30◦ , 45◦ , 60◦ . The first four modes of the thin triangular plate (t/a = 0.001) are shown in Table 8 and Fig. 20. The NS-DSG3 element is also compared to the ANS4 and ES-DSG3 elements
and two other well-known numerical methods such as the Rayleigh–Ritz method [79] and the pb-2 Ritz method [74]. The frequencies of the NS-DSG3 are often bounded by these reference models. Note that our method only uses three primitive DOFs at each vertex node without adding any additional DOFs. For a thick plate, the results are shown in Table 9. The first eight mode shapes of cantilever triangular square plates are illustrated in Fig. 21. It is clear that the NS-DSG3 is stable. 5.3 Buckling of plates In the following examples, the buckling load factor is defined as K = λcr b2 /(π 2 D) where b is the edge width of the plate,
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Comput Mech (2010) 46:679–701
The results of the NS-DSG3 are more accurate than those of the DSG3. The NS-DSG3 produces the best result for the SSSS plate case, but it is slightly less accurate than the ES-DSG3 and some other methods for the CCCC plate, see Table 11. Therefore the accuracy of the NS-DSG3 seems to be a problem-dependent. Next, we consider the buckling load factors of SSSS, CCCC, CFCF plates with thickness-to-width ratios t/b = 0.05; 0.1. The results given in Table 12 using NS-DSG3 compare well with several other methods. We also consider simply supported plates with various thickness-to-width ratios, t/b = 0.05; 0.1; 0.2 and lengthto-width ratios, a/b = 0.5; 1.0; 1.5; 2.0; 2.5. The buckling factors for a 16 × 16 mesh are described in Fig. 24 and Table 13. The axial buckling modes of simply-supported rectangular plates with thickness-to-width ratios t/b = 0.01 and various length-to-width ratios, a/b = 1.0; 1.5; 2.0; 2.5 are shown in Fig. 25. It is clear that the results of the NS-DSG3 match well those of ES-DSG3 and Pb-2 Ritz models.
5.3.2 Simply supported rectangular plates subjected to biaxial compression
Fig. 21 The first eight mode shapes of the triangular square plate with t/a=0.001
λcr the critical buckling load. The material parameters are assumed: Young’s modulus E = 2.0 × 1011 , Poisson’s ratio ν = 0.3. 5.3.1 Simply supported rectangular plates subjected to uniaxial compression Let us first consider a plate with length a, width b and thickness t subjected to uniaxial compression. SSSS and CCCC boundary conditions are assumed. The geometry and a typical mesh are shown in Fig. 22a. Table 10 gives the convergence of the buckling load factor corresponding to meshes with 4×4, 8×8, 12×12 and 16×16 rectangular elements. Figure 23 plots the convergence of the normalized buckling load K h /K exact of the square plate with thickness ratio t/b = 0.01, where K h is the numerical buckling load and K exact is the analytical buckling load [80].
123
Consider the square plate subjected to biaxial compression shown in Fig. 22b. Table 14 gives the shear buckling factor of the square plate subjected to biaxial compression with three essential boundary conditions (SSSS, CCCC, SCSC) using 2 × 16 × 16 triangular elements. The relative error percentages compared with the analytical solutions [80] are given in a parentheses. The results of the NS-DSG3 element are more accurate than those of the DSG3 and slightly less accurate than those of the ES-DSG3.
5.3.3 Simply supported rectangular plates subjected to in-plane pure shear The final example is the simply supported plate subjected to in-plane shear shown in Fig. 22c. The shear buckling load factors K of this plate are calculated using a 16×16 mesh. The shear buckling factors with thickness-to-width ratio, t/b = 0.001 and various length-to-width ratios, a/b = 1.0; 2.0; 3.0; 4.0 are listed in Table 15. The NS-DSG3 element agrees well with the exact solution and other numerical models. Figure 26 illustrates the convergence of the shear buckling load. Figure 27 also presents the shear buckling modes of simply-supported rectangular plates. Table 16 gives the shear buckling factor of the square plate subjected to in-plane pure shear with three essential boundary conditions (SSSS, CCCC, SCSC). The NS-DSG3 element works well for these cases.
Comput Mech (2010) 46:679–701
697
Fig. 22 Rectangular plates: a Axial compression, b biaxial compression, c shear in-plane, d regular mesh
(a)
(b)
(c)
Plates type
Methods
Index mesh 4×4
SSSS
12 × 12
16 × 16
DSG3
7.5891
4.8013
4.3200
4.1590
ES-DSG3
4.7023
4.1060
4.0368
4.0170
NS-DSG3 CCCC
8×8
4.1313
4.0741
4.0396
4.0231
DSG3
31.8770
14.7592
11.9823
11.0446
ES-DSG3
14.7104
11.0428
10.3881
10.2106
NS-DSG3
11.4457
11.2947
10.7144
10.4473
Exact SSSS (DSG3) CCCC (DSG3) SSSS (ES−DSG3) CCCC (ES−DSG3) SSSS (NS−DSG3) CCCC (NS−DSG3)
3
Normalized buckling load Kh/Kexact
Table 10 The axial buckling load factors K h along the x axis of rectangular plates with length-to-width ratios a/b = 1 and thickness-to-width ratios t/b = 0.01
(d)
2.5
2
1.5
1
4
6 Conclusions A node-based smoothed finite element method (NS-FEM) with a stabilized discrete shear gap technique using triangular elements (NS-DSG3) has been formulated for static, free vibration and buckling analyses of Reissner–Mindlin plates. The method is based on the strain smoothing technique over smoothing domains associated with nodes of finite elements [42]. Transverse shear locking is solved with help of the discrete shear gap method and a stability of the NS-DSG3 is
6
8
10
12
14
16
mesh index N
Fig. 23 Normalized buckling load K h /K exact of a square plate with t/b = 0.01
ensured using the stabilization technique. The present element uses only three primitive DOFs at each vertex node without additional degrees of freedom. The NS-DSG3 is simple to implement into finite element programs using triangular meshes that can be generated with ease for complicated
123
698
Comput Mech (2010) 46:679–701
Table 11 The axial buckling load factors K h along the x axis of rectangular plates with length-to-width ratios a/b = 1 and thickness-to-width ratios t/b = 0.01
Plates type
Methods DSG3
ES-DSG3
NS-DSG3
Liew
Ansys
Tham
Timoshenko
SSSS
4.1590
4.0170
4.0008
3.9700
4.0634
4.00
4.00
(%)
3.97%
0.4%
0.02%
−0.75%
1.85%
–
–
11.0446
10.2106
10.4473
10.1501
10.1889
10.08
10.07
9.68%
1.4%
3.61%
0.8%
1.18%
0.1%
–
CCCC (%)
Table 12 The axial buckling load factors K h along the x axis of rectangular plates with various length-to-width ratios a/b = 1 and various thickness-to-width ratios
Table 13 The axial buckling load factors K h along the x axis of rectangular plates with various length-to-width ratios and various thicknessto-width ratios
t/b Plates type
a/b t/b
Methods
Methods DSG3
ES-DSG3 NS-DSG3 Meshfree [58] Ritz [81]
DSG3 ES-DSG3 NS-DSG3 RPIM [73] Ritz [81] SSSS
0.05 6.0478 5.9873
6.0381
6.0405
6.0372
0.1
5.3555 5.3064
5.3638
5.3116
5.4777
9.5586
0.2
3.7524 3.7200
3.7811
3.7157
3.9963
3.8005
0.05 3.9786 3.9412
3.9687
3.9293
3.9444
0.1
3.7692 3.7402
3.7870
3.7270
3.7865
0.2
3.1493 3.1263
3.1739
3.1471
3.2637
3.9786 3.9412
3.9536
3.9464
3.9444
0.05 CCCC
9.8284 9.5426
9.6633
9.5819
CFCF
3.8365 3.7654
3.8214
3.8187
SSSS
3.7692 3.7702
3.7563
3.7853
3.7873
CCCC
8.2670 8.2674
8.2753
8.2931
8.2921
CFCF
3.4594 3.4966
3.4652
3.5138
3.5077
0.1
0.5
1.0
1.5 7 DSG3 (t/b=0.05) DSG3 (t/b=0.1) DSG3 (t/b=0.2) ES−DSG3 (t/b=0.05) ES−DSG3 (t/b=0.1) ES−DSG3 (t/b=0.2) NS−DSG3 (t/b=0.05) NS−DSG3 (t/b=0.1) NS−DSG3 (t/b=0.2) Pb−2−Ritz (t/b=0.05) Pb−2−Ritz (t/b=0.1) Pb−2−Ritz (t/b=0.2)
Buckling axial factor K
6.5 6 5.5 5 4.5
2.0
2.5
0.05 4.3930 4.2852
4.3230
4.2116
4.2570
0.1
4.0604 3.9844
4.0368
3.8982
4.0250
0.2
3.2014 3.1461
3.1558
3.1032
3.3048
0.05 4.1070 3.9811
4.0323
3.8657
3.9444
0.1
3.8539 3.7711
3.8116
3.6797
3.7865
0.2
3.2023 3.1415
3.2208
3.0783
3.2637
0.05 4.3577 4.1691
4.2523
3.9600
4.0645
0.1
4.0644 3.8924
3.9347
3.7311
3.8683
0.2
3.2393 3.1234
3.1811
3.0306
3.2421
4 3.5 3 0.5
1
1.5
2
2.5
a/b
Fig. 24 Convergence of axial buckling load K h of SSSS plate with various length-to-width ratios and various thickness-to-width ratios
problem domains. Numerical results showed that the NSDSG3 is shear-locking free, stable and is superior to the original DSG3 element. Furthermore, the present formulation also exhibits good agreement compared with several published methods in the literature. As observed from numerical experiments, it is useful to note that the NS-DSG3 can produce an upper bound solution in the elastic energy for static analyses, while the ES-DSG3 produces results of comparable accuracy, but underestimates the elastic energy. Both methods could be used in tandem to
123
(a)
(b)
(c)
(d)
Fig. 25 Axial buckling modes of simply-supported rectangular plates with thickness-to-width ratios t/b = 0.01 and various length-to-width ratios a/b = 1; 1.5; 2.0; 2.5
Comput Mech (2010) 46:679–701 Table 14 The biaxial buckling load factors K h of rectangular plates with length-to-width ratios a/b = 1, thickness-to-width ratios t/b = 0.01 and various boundary conditions
699
Plates type
Methods DSG3 (%)
ES-DSG3 (%)
Tham [82]
Timoshenko [80]
SSSS
2.0549 (2.74)
2.0023 (0.11)
2.0138 (0.69)
2.00
2.00
CCCC
5.6419 (6.20)
5.3200 (0.20)
5.3537 (0.82)
5.61
5.31
SCSC
4.0108 (4.72)
3.8332 (0.08)
3.8531 (0.60)
3.83
3.83
Table 15 The shear buckling load factors K h of simply supported rectangular plates with various length-to-width ratios, choose t/b = 0.01 a/b
NS-DSG3 (%)
Methods DSG3
ES-DSG3
NS-DSG3
Meshfree [58]
Exact [83]
1.0
9.5195
9.2830
9.3807
9.3962
9.34
2.0
6.7523
6.4455
6.4599
6.3741
6.34
3.0
6.5129
5.8830
5.7380
5.7232
5.784
4.0
6.3093
5.6732
5.4972
5.4367
5.59
Table 16 The shear buckling load factors K h of rectangular plates with length-to-width ratios a/b = 1, thickness-to-width ratios t/b = 0.01 and various boundary conditions Plates type
Methods DSG3
ES-DSG3 NS-DSG3 Tham [82] Timoshenko [80]
SSSS
9.5195 9.2830
9.3807
9.40
9.33
CCCC
15.6397 14.6591
15.0281
14.58
14.66
SCSC
13.1652 12.5533 12.7296
12.58
12.58
10 DSG3 ES−DSG3 NS−DSG3 Meshfree Exact
9.5
Buckling shear factor K
9
provide an estimate of the global error in energy for general problems where the exact energy is unknown. However, a rigorous numerical analysis of both methods would be required.
8.5 8
Acknowledgments Stephane Bordas would like to thank the support of the Royal Academy of Engineering and of the Leverhulme Trust for his Senior Research Fellowship entitled “Towards the next generation surgical simulators” as well as the support of EPSRC under grants EP/G069352/1 Advanced discretisation strategies for “atomistic” nano CMOS simulation and EP/G042705/1 Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method.
7.5 7 6.5 6 5.5 5
1
1.5
2
2.5
3
3.5
4
a/b
Fig. 26 Normalized shear buckling load K h /K exact of a square plate with t/b = 0.01
(a)
(b)
(c)
(d)
Fig. 27 Shear buckling mode of simply-supported rectangular plates with various length-to-width ratios a/b = 1.0; 2.0; 3.0; 4.0 using the NS-DSG3
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