김경종
471 lines
31 KiB
Markdown
471 lines
31 KiB
Markdown
<!-- source-page: 341 -->
|
||
|
||
|
||
|
||
<details>
|
||
<summary>flowchart</summary>
|
||
|
||
```mermaid
|
||
graph TD
|
||
w1 -->|w1x1| 1
|
||
w1 -->|w1y1| 2
|
||
w2 -->|w2x2| 2
|
||
w2 -->|w2y2| 3
|
||
w3 -->|w3x3| 3
|
||
w3 -->|w3y3| 3
|
||
1 --> w1
|
||
2 --> w3
|
||
3 --> w3
|
||
```
|
||
</details>
|
||
|
||
(a)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>flowchart</summary>
|
||
|
||
```mermaid
|
||
graph TD
|
||
w1["1"] -->|w_{n6}| 6["6"]
|
||
w1 -->|w_{n4}| 4["4"]
|
||
w2["2"] -->|w_{n5}| 5["5"]
|
||
w2 -->|w_{n4}| 4
|
||
w3["3"] -->|w_{n5}| 5
|
||
w3 -->|w_{n4}| 4
|
||
w1 -->|w_{n6}| 6
|
||
```
|
||
</details>
|
||
|
||
(b)
|
||
Figure 11.2-2. Triangular Kirchhoff plate elements. (a) Nine-d.o.f. element. (b) Six-d.o.f., constant-curvature element. The symbol $\odot$ indicates an arrow directed out of the paper.
|
||
|
||
Equation 11.2-6a omits the $xy$ term. The resulting element is therefore incapable of passing a constant-twist patch test, and is considered unacceptable. Equation 11.2-6b leads to an element that lacks geometric isotropy, has poor convergence properties, and for certain element shapes has a singular transformation matrix $[\mathbf{A}]$ in the relation $\{\mathbf{d}\} = [\mathbf{A}]\{\mathbf{a}\}$ .
|
||
|
||
Eventually such difficulties were overcome, often by using subelements, by abandoning strict Kirchhoff plate theory, or by using variational principles other than stationary potential energy. An interesting result of these efforts is the element of Fig. 11.2-2b, which appears to be the simplest possible Kirchhoff element. It can represent only rigid-body motion (constant w, $w_{,x}$ , or $w_{,y}$ ) and constant-curvature states (constant $w_{,xx}$ , $w_{,yy}$ , or $w_{,xy}$ ). Its six d.o.f. are translations w at corners and normal slopes $w_{,n}$ at midsides, where n is a direction normal to the side.
|
||
|
||
After nodal d.o.f. have been obtained by solution of the structural equations, element curvatures $\{\kappa\}$ are obtained from Eq. 11.2-3, then bending moments from Eq. 11.1-7, and finally stresses from Eqs. 11.1-2.
|
||
|
||
Mindlin Elements. Nodal d.o.f. consist of lateral deflections $w_{i}$ and rotations $\theta_{xi}$ and $\theta_{yi}$ of midsurface normals. The corresponding deflections and rotations within an element are obtained by independent shape function interpolations:
|
||
|
||
$$
|
||
w = \sum N _ {i} w _ {i} \quad \theta_ {x} = \sum N _ {i} \theta_ {x i} \quad \theta_ {y} = \sum N _ {i} \theta_ {y i} \tag {11.2-7}
|
||
$$
|
||
|
||
Equations 11.1-4 and 11.2-7 yield strains. We see that the field for w is coupled to the fields for $\theta_{x}$ and $\theta_{y}$ only via the shear strains $\gamma_{yz}$ and $\gamma_{zx}$ . Using the strains of Eq. 11.1-4, one can write an expression for strain energy and from it obtain an element stiffness matrix. Mindlin elements are considered in more detail in Section 11.3.
|
||
|
||
Discrete Kirchhoff Elements. Initially, Eqs. 11.2-7 are again used to express displacements and rotations within an element. However, more d.o.f. are included than are to appear in the final element. Strains of importance in Kirchhoff theory
|
||
|
||
<!-- source-page: 342 -->
|
||
|
||
$(\epsilon_x, \epsilon_y, \text{and } \gamma_{xy} \text{ only})$ are evaluated from Eqs. 11.1-4. Accordingly, element strains and strain energy depend on $\theta_x$ and $\theta_y$ but are independent of $w$ . Next, using $\gamma_{yz} = \gamma_{zx} = 0$ in Eqs. 11.1-4, we impose the “Kirchhoff constraints” $w_{,y} = \theta_y$ and $w_{,x} = \theta_x$ at certain points. These points are sufficient in number to eliminate the “excess” d.o.f. Thus $w$ becomes coupled to the rotations $\theta_x$ and $\theta_y$ , and only nodal d.o.f. used for interelement connections remain. Discrete Kirchhoff elements are considered further in Section 11.4.
|
||
|
||
Finite Strips. The finite strip method [11.7] exploits the “semianalytical” method described in Section 10.5. Imagine that the plate of Fig. 11.2-3 is simply supported along edges y = 0 and y = b. Lateral displacement w can be taken as
|
||
|
||
$$
|
||
w = \sum_ {n} \left[ \mathrm{N} \right] \{\overline {{{\mathrm{d}}}} \} _ {n} \sin \frac {n \pi y}{b} \tag {11.2-8}
|
||
$$
|
||
|
||
where $[N]$ represents standard cubic shape functions, Fig. 3.13-2, and $\{\overline{d}\}_{n}$ contains amplitudes of displacement and rotation along nodal lines 1–1 and 2–2 for mode n. Thus Eq. 11.2-8 expresses the superposition of several solutions, each associated with a single Fourier harmonic of loading. This superposition method replaces division of the plate into elements in the y direction. The finite strip method can also be applied to folded plates and to box beams, either straight or curved. Advantages include modest needs for computer resources and input data. Disadvantages include an inability to cope with arbitrary shapes and arbitrary boundary conditions.
|
||
|
||
Nodal Loads. Consistent element nodal loads $\{r_{e}\}$ , Eq. 4.1-6, include terms such as
|
||
|
||
$$
|
||
\int_ {A} \left\lfloor \mathbf {N} \right] ^ {T} q d A \quad \int \left\lfloor \mathbf {N} _ {, x} \right] ^ {T} \overline {{M}} _ {x} d y \quad \text { and } \quad \int \left\lfloor \mathbf {N} _ {, x} \right] ^ {T} \overline {{M}} _ {x y} d x \tag {11.2-9}
|
||
$$
|
||
|
||
where $q$ is the intensity of distributed lateral load, and $\overline{M}_x$ and $\overline{M}_{xy}$ are prescribed boundary values of moment loads. The respective shape function matrices in Eqs. 11.2-9 are associated with lateral displacement, rotation about the $y$ axis along a $y$ -parallel edge that carries $\overline{M}_x$ , and rotation about the $y$ axis along an $x$ -parallel edge that carries $\overline{M}_{xy}$ .
|
||
|
||
As a simpler but less accurate alternative to Eq. 4.1-6, a distributed load on a plate element can be “lumped” by assigning equal fractions of the total force on the element to its translational d.o.f. Correct answers are approached with mesh refinement.
|
||
|
||
For Kirchhoff elements, the first of Eqs. 11.2-9 produces nodal moments as
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
z,w
|
||
b
|
||
1
|
||
y
|
||
2
|
||
L
|
||
1
|
||
2
|
||
x
|
||
</details>
|
||
|
||
Figure 11.2-3. A thin rectangular plate divided into finite strips. A typical strip is shaded.
|
||
|
||
<!-- source-page: 343 -->
|
||
|
||
well as nodal forces, because $w = [N]\{d\}$ describes a field dependent on nodal rotations as well as nodal translations. For Mindlin elements, from the first of Eqs. 11.2-7, $w = [N]\{w\}$ describes a field dependent on only nodal translations, so Eq. 11.2-9 yields only nodal forces. If elements have no internal nodes, a one-element, simply supported, uniformly loaded Mindlin plate would then yield zero deflection, which is unreasonable. Engineers may therefore be willing to forgo mathematical consistency and use in Eq. 11.2-9 a substitute field that provides nodal moments, or they may adopt some other ad hoc strategy in order to improve accuracy in a coarse mesh.
|
||
|
||
In Kirchhoff plate theory, nodal loads produced by initial curvatures $\{\kappa_{0}\}$ (e.g., Eq. 11.1-11) are
|
||
|
||
$$
|
||
\left\{\mathbf {r} _ {e} \right\} = \int_ {A} \left[ \mathbf {B} \right] ^ {T} \left[ \mathbf {D} _ {K} \right] \left\{\kappa_ {0} \right\} d A \tag {11.2-10}
|
||
$$
|
||
|
||
In Mindlin plate theory, the last two entries in the product $[D_{M}]\{\kappa_{0}\}$ are zero. Accordingly, [B] can be truncated to three rows, so that [B]{d} yields only $\theta_{x,x}$ , $\theta_{y,y}$ , and $\theta_{x,y} + \theta_{y,x}$ (see Eq. 11.3-4). Then Eq. 11.2-10 yields nodal moments produced by $\{\kappa_{0}\}$ , but no z-direction nodal forces.
|
||
|
||
# 11.3 MINDLIN PLATE ELEMENTS
|
||
|
||
Mindlin plate elements account for bending deformation and for transverse shear deformation. Accordingly, they may be used to analyze thick plates as well as thin plates. When used for thin plates, however, they may be less accurate than Kirchhoff elements, which do not allow transverse shear deformation.
|
||
|
||
Typical Mindlin plate elements are shown in Fig. 11.3-1. For convenience of notation, but not because of any demand of finite element plate theory, we will assume that all three d.o.f. shown in Fig. 11.3-1c are present at every node. Rotations $\theta_{x}$ and $\theta_{y}$ are shown by two-headed arrows according to the right-hand rule. These are rotations of a line that was normal to the midsurface of the undeformed plate. Note that $\theta_{x} \neq w_{,x}$ and $\theta_{y} \neq w_{,y}$ unless we approach the thin-plate limit, in which case $\gamma_{zx} = \gamma_{yz} = 0$ (see Fig. 11.1-3). A special form of Mindlin plate element is the Mindlin beam element, which the reader may wish to review (see Eqs. 9.4-1 and 9.4-2 and Fig. 9.4-1).
|
||
|
||
Stiffness Matrix. The starting point for formulating an element stiffness matrix is an expression for strain energy U. With A the area of the plate midsurface,
|
||
|
||
$$
|
||
U = \frac {1}{2} \int_ {A} \int_ {- t / 2} ^ {t / 2} \{\boldsymbol {\epsilon} \} ^ {T} [ \mathrm{E} ] \{\boldsymbol {\epsilon} \} d z d A \tag {11.3-1}
|
||
$$
|
||
|
||
where $\{\pmb{\epsilon}\}^T = \left\lfloor \epsilon_x \quad \epsilon_y \quad \gamma_{xy} \quad \gamma_{yz} \quad \gamma_{zx} \right\rfloor$ , and the individual strains are stated in terms of displacements by Eqs. 11.1-4. Integration through the thickness yields
|
||
|
||
$$
|
||
U = \frac {1}{2} \int_ {A} \{\boldsymbol {\kappa} \} ^ {T} [ \mathbf {D} _ {M} ] \{\boldsymbol {\kappa} \} d A \tag {11.3-2}
|
||
$$
|
||
|
||
<!-- source-page: 344 -->
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
y
|
||
4
|
||
η
|
||
ξ
|
||
3
|
||
1
|
||
2
|
||
x
|
||
</details>
|
||
|
||
(a)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
y
|
||
4
|
||
7
|
||
η
|
||
3
|
||
8
|
||
9
|
||
ξ
|
||
6
|
||
1
|
||
5
|
||
2
|
||
x
|
||
</details>
|
||
|
||
(b)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>flowchart</summary>
|
||
|
||
```mermaid
|
||
graph TD
|
||
w_i((w_i)) -->|θ_yi| x
|
||
w_i -->|θ_xi| i
|
||
i --> y
|
||
x --> y
|
||
```
|
||
</details>
|
||
|
||
(c)
|
||
Figure 11.3-1. (a) Bilinear element, top view. (b) Quadratic element, top view. (c) Notation and sign convention for d.o.f. at a typical node i. The symbol ⊙ indicates an arrow directed out of the paper.
|
||
|
||
where $[D_{M}]$ and $\{\kappa\}$ are defined in Eq. 11.1-12. If the d.o.f. of Fig. 11.3-1c are present at every node, the same shape functions $N_{i}$ are used to interpolate w, $\theta_{x}$ , and $\theta_{y}$ from nodal values of these quantities, that is,
|
||
|
||
$$
|
||
\left\{ \begin{array}{l} w \\ \theta_ {x} \\ \theta_ {y} \end{array} \right\} = \sum_ {i = 1} ^ {N} \left[ \begin{array}{c c c} N _ {i} & 0 & 0 \\ 0 & N _ {i} & 0 \\ 0 & 0 & N _ {i} \end{array} \right] \left\{ \begin{array}{l} w _ {i} \\ \theta_ {x i} \\ \theta_ {y i} \end{array} \right\} \quad \text { or } \quad \left\{\mathbf {u} \right\} = \left[ \begin{array}{l} \mathbf {N} \\ 3 \times 3 N \end{array} \right] \left\{\mathbf {d} \right\} \tag {11.3-3}
|
||
$$
|
||
|
||
where $N$ is the number of nodes per element, and $\{\mathbf{d}\} = \left[w_1 \quad \theta_{x1} \quad \theta_{y1} \ldots w_N \quad \theta_{xN} \quad \theta_{yN}\right]^T$ . Curvatures $\{\kappa\}$ stated in Eq. 11.1-12 are
|
||
|
||
$$
|
||
\{\boldsymbol {\kappa} \} = \left\{ \begin{array}{c} \theta_ {x, x} \\ \theta_ {y, y} \\ \theta_ {x, y} + \theta_ {y, x} \\ \theta_ {y} - w _ {, y} \\ \theta_ {x} - w _ {, x} \end{array} \right\} = [ \partial ] \{\mathbf {u} \}, \quad \text { where } \quad [ \partial ] = \left[ \begin{array}{c c c} 0 & \partial / \partial x & 0 \\ 0 & 0 & \partial / \partial y \\ 0 & \partial / \partial y & \partial / \partial x \\ - \partial / \partial y & 0 & 1 \\ - \partial / \partial x & 1 & 0 \end{array} \right] \tag {11.3-4}
|
||
$$
|
||
|
||
Equations 11.3-3 and 11.3-4 yield
|
||
|
||
$$
|
||
\{\boldsymbol {\kappa} \} = \underset {5 \times 3 N} {[ \mathbf {B} ]} \{\mathbf {d} \}, \quad \text { where } \quad [ \mathbf {B} ] = [ \partial ] [ \mathbf {N} ] = \left[ \begin{array}{c c c c} 0 & N _ {1, x} & 0 & \dots \\ 0 & 0 & N _ {1, y} & \dots \\ 0 & N _ {1, y} & N _ {1, x} & \dots \\ - N _ {1, y} & 0 & N _ {1} & \dots \\ - N _ {1, x} & N _ {1} & 0 & \dots \end{array} \right] \tag {11.3-5}
|
||
$$
|
||
|
||
And finally, from Eqs. 11.3-2 and 11.3-5,
|
||
|
||
$$
|
||
U = \frac {1}{2} \{\mathbf {d} \} ^ {T} [ \mathbf {k} ] \{\mathbf {d} \}, \quad \text { where } \quad \underset {3 N \times 3 N} {[ \mathbf {k} ]} = \int_ {A} [ \mathbf {B} ] ^ {T} [ \mathbf {D} _ {M} ] [ \mathbf {B} ] d A \tag {11.3-6}
|
||
$$
|
||
|
||
If the plate is rectangular, shape functions $N_{i}$ can be expressed in terms of x and y. Then dA = dx dy. If the plate is of more general shape, as shown in Fig. 11.3-1, the $N_{i}$ can be expressed in terms of isoparametric coordinates $\xi$ and $\eta$ . Then dA = J d $\xi$ d $\eta$ , where J is the Jacobian determinant. For bilinear and quadratic elements respectively, shape functions $N_{i}$ are given by Eqs. 6.3-2 and Table
|
||
|
||
<!-- source-page: 345 -->
|
||
|
||
6.6-1. Shape function derivatives, needed in [B], are determined by the usual transformation (e.g, Eq. 6.3-7):
|
||
|
||
$$
|
||
N _ {i, x} = \Gamma_ {1 1} N _ {i, \xi} + \Gamma_ {1 2} N _ {i, \eta} \quad \text { and } \quad N _ {i, y} = \Gamma_ {2 1} N _ {i, \xi} + \Gamma_ {2 2} N _ {i, \eta} \tag {11.3-7}
|
||
$$
|
||
|
||
Mindlin plate elements can be regarded as special forms of solid elements. For example, the bilinear plate element of Fig. 11.3-1a closely resembles the trilinear solid element of Fig. 6.7-1, if the solid element is made thin in one direction. The solid element has twice as many d.o.f. as the plate element. In addition, nodes of the solid that lie on a midsurface-normal line define the thickness-direction strain, $\epsilon_{z}$ , which is ignored in plate-bending theory. If $\epsilon_{z}$ were included in the formulation, nodes that span the thickness would be coupled by stiffness coefficients that become very large in comparison with bending stiffnesses as the plate becomes thin. The discrepancy may lead to numerical difficulty of a type discussed in Section 18.2. In summary, considerations of economy and robustness indicate that solid elements should not be used to model plates.
|
||
|
||
Quadrature Rule and Locking. One can regard the stiffness matrix of a Mindlin plate element as being composed of a bending stiffness $[k_{b}]$ and a transverse shear stiffness $[k_{s}]$ . From Eq. 11.3-6, with $[B] = [B_{b}] + [B_{s}]$ ,
|
||
|
||
$$
|
||
[ \mathbf {k} ] = \underbrace {\int_ {A} [ \mathbf {B} _ {b} ] ^ {T} [ \mathbf {D} _ {M} ] [ \mathbf {B} _ {b} ] d A} _ {[ \mathbf {k} _ {b} ]} + \underbrace {\int_ {A} [ \mathbf {B} _ {s} ] ^ {T} [ \mathbf {D} _ {M} ] [ \mathbf {B} _ {s} ] d A} _ {[ \mathbf {k} _ {s} ]} \tag {11.3-8}
|
||
$$
|
||
|
||
$[B_{b}]$ is associated with in-plane strains $\epsilon_{x}, \epsilon_{y}$ , and $\gamma_{xy}$ and is obtained by setting rows 4 and 5 of [B] to zero. $[B_{s}]$ is associated with transverse shear strains $\gamma_{yz}$ and $\gamma_{zx}$ , and is obtained by setting rows 1, 2, and 3 of [B] to zero. The cross product terms $[B_{b}]^{T}[D_{M}][B_{s}]$ and $[B_{s}]^{T}[D_{M}][B_{b}]$ are zero because of the distribution of zeros in $[B_{b}]$ , $[B_{s}]$ , and $[D_{M}]$ . Bending stiffness $[k_{b}]$ mobilizes only the $[D_{K}]$ portion of $[D_{M}]$ , and transverse shear stiffness $[k_{s}]$ mobilizes only the $G_{yz}t$ and $G_{zx}t$ terms in $[D_{M}]$ .
|
||
|
||
The splitting of [k] into components $[k_{b}]$ and $[k_{s}]$ is also seen in Eq. 9.4-4 of Section 9.4, where it is argued that each integration point used to evaluate $[k_{s}]$ imposes one constraint on transverse shear strain $\gamma_{zx}$ of the two-node Mindlin beam element, and may produce locking of the mesh if the beam is thin and too many integration points are used to evaluate $[k_{s}]$ . Similar considerations apply to Mindlin plate elements. However, each integration point used for $[k_{s}]$ brings two constraints to a Mindlin plate element, one associated with $\gamma_{yz}$ and the other with $\gamma_{zx}$ . Locking of Mindlin plate elements caused by too many transverse shear constraints can be avoided by adopting a reduced or selective integration rule to generate [k]. Or, one can redefine the transverse shear interpolation; see [11.17].
|
||
|
||
Various Mindlin plate elements are possible. Some are summarized in Table 11.3-1. Typical behavior is reported in Fig. 11.3-2. “Full integration” is sufficient to avoid element mechanisms. The stiffness matrix of a rectangular element with midside nodes is integrated exactly by full integration.
|
||
|
||
The bilinear element responds properly to pure bending with either reduced or selective integration. With full (2 by 2) integration and pure bending, parasitic shear strains appear at the Gauss points (Fig. 11.3-3a). As the element becomes
|
||
|
||
<!-- source-page: 346 -->
|
||
|
||
TABLE 11.3-1. DATA FOR SELECTED MINDLIN PLATE ELEMENTS.
|
||
|
||
<table><tr><td rowspan="2">Element Type</td><td colspan="3">Integration Rule</td><td rowspan="2">ShearConstraints</td><td rowspan="2">Number ofMechanisms</td></tr><tr><td>Type</td><td> $[k_b]$ </td><td> $[k_s]$ </td></tr><tr><td rowspan="3"></td><td>Bilinear:</td><td>Reduced</td><td> $1 \times 1$ </td><td> $1 \times 1$ </td><td>2</td></tr><tr><td>4 nodes,</td><td>Selective</td><td> $2 \times 2$ </td><td> $1 \times 1$ </td><td>2</td></tr><tr><td>12 d.o.f.</td><td>Full</td><td> $2 \times 2$ </td><td> $2 \times 2$ </td><td>4</td></tr><tr><td rowspan="3"></td><td>Quadratic:</td><td>Reduced</td><td> $2 \times 2$ </td><td> $2 \times 2$ </td><td>8</td></tr><tr><td>9 nodes,</td><td>Selective</td><td> $3 \times 3$ </td><td> $2 \times 2$ </td><td>8</td></tr><tr><td>27 d.o.f.</td><td>Full</td><td> $3 \times 3$ </td><td> $3 \times 3$ </td><td>18</td></tr><tr><td rowspan="3"></td><td>Serendipity:</td><td>Reduced</td><td> $2 \times 2$ </td><td> $2 \times 2$ </td><td>8</td></tr><tr><td>8 nodes,</td><td>Selective</td><td> $3 \times 3$ </td><td> $2 \times 2$ </td><td>8</td></tr><tr><td>24 d.o.f.</td><td>Full</td><td> $3 \times 3$ </td><td> $3 \times 3$ </td><td>18</td></tr><tr><td rowspan="3"></td><td>Heterosis:</td><td></td><td></td><td></td><td></td></tr><tr><td>9 nodes,</td><td>Selective</td><td> $3 \times 3$ </td><td> $2 \times 2$ </td><td>8</td></tr><tr><td>26 d.o.f.</td><td>(Ref. 11.8.</td><td colspan="3">D.o.f. at the center node are $\theta_x$ and $\theta_y$ only.)</td></tr></table>
|
||
|
||
|
||
|
||
<details>
|
||
<summary>line</summary>
|
||
|
||
| L_T/t | (computed w_t) ÷ (theoretical w_t) |
|
||
|-------|-----------------------------------|
|
||
| 10 | 1.15 |
|
||
| 20 | 1.05 |
|
||
| 30 | 1.02 |
|
||
| 50 | 1.00 |
|
||
| 100 | 0.98 |
|
||
| 200 | 0.96 |
|
||
| 300 | 0.94 |
|
||
| 500 | 0.90 |
|
||
| 1000 | 0.85 |
|
||
| 10^6 | 0.80 |
|
||
</details>
|
||
|
||
Figure 11.3-2. Center deflection of a uniformly loaded clamped square plate of side length $L_{T}$ and thickness t. An 8 by 8 mesh is used in all cases. Thin plates correspond to large $L_{T}/t$ . Transverse shear deformation becomes significant for small $L_{T}/t$ . Integration rules, from Table 11.3-1, are reduced (R), selective (S), and full (F) [11.9].
|
||
|
||
<!-- source-page: 347 -->
|
||
|
||
thin, its stiffness is due almost entirely to parasitic shear. Thus, if fully integrated, a bilinear Mindlin plate element exhibits almost no bending deformation: that is, the mesh “locks.”
|
||
|
||
The nine-node quadratic element, when integrated with any number of Gauss points, can properly represent pure bending. As seen in Fig. 11.3-3b, because lateral deflection w can vary quadratically, zero-shear conditions impose the constraint of pure bending rather than no bending. The latter element in Fig. 11.3-3b displays linearly varying bending. Here w = 0, not w cubic in x as for a standard beam or a Kirchhoff plate, because the quadratic plate element can display only a quadratic variation of w. Accordingly, linearly varying bending is represented by w = 0 and a quadratic variation of $\theta_{x}$ . It can be shown that this state is properly represented only when $[k_{s}]$ is integrated with a 2 by 2 rule (Ref. 12.6; see also Problem 11.20).
|
||
|
||
The “serendipity” element has eight nodes and uses the quadratic shape functions of Eqs. 6.6-1. It is an unreliable element: as shown by Fig. 11.3-2, its accuracy is acceptable only for small values of $L_{T}/t$ , regardless of quadrature rule.
|
||
|
||
The upper curves in Fig. 11.3-2 become horizontal, indicating apparent convergence, but may diverge for very large values of $L_{T}/t$ . This difficulty has nothing to do with locking. Rather, it is caused by the penalty matrix $[k_{s}]$ becoming numerically so large that it overwhelms matrix $[k_{b}]$ , as discussed in Section 9.4. Divergence can be avoided by basing $[k_{s}]$ on a value of t that is arbitrarily increased, if necessary, so that
|
||
|
||
$$
|
||
\frac {5}{1 + \nu} \left(\frac {L _ {T}}{t}\right) ^ {2} < 1 0 ^ {p / 2} \tag {11.3-9}
|
||
$$
|
||
|
||
where p is the approximate number of decimal digits per computer word (see Eq. 9.4-11). The actual value of t is used to compute matrix $[k_{b}]$ . With this adjustment, transverse shear deformation is misrepresented only when it is so small as not to matter anyway.
|
||
|
||
The “heterosis” element [11.8] is the best of the elements summarized in Table 11.3-1. It does not exhibit locking, erratic convergence characteristics, or mech-
|
||
|
||
|
||
Figure 11.3-3. (a) Edge view of bilinear element, shown deformed, in pure bending. (b) Edge view of quadratic element, shown deformed, in pure bending and in linearly varying bending.
|
||
|
||
<!-- source-page: 348 -->
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
y
|
||
x
|
||
</details>
|
||
|
||
In-plane twist mode $w = 0, \theta_{,r} = -y, \theta_{y} = x$
|
||
(a)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
y
|
||
x
|
||
</details>
|
||
|
||
w-hourglass mode $w = xy, \theta_{x} = \theta_{y} = 0$
|
||
(b)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
y
|
||
x
|
||
</details>
|
||
|
||
$\theta_{x}$ -hourglass mode $w = 0,\theta_{x} = xy,\theta_{y} = 0$
|
||
(c)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
y
|
||
x
|
||
</details>
|
||
|
||
$\theta_y$ -hourglass mode $w = 0, \theta_x = 0, \theta_y = xy$
|
||
(d)
|
||
Figure 11.3-4. Mechanisms of the four-node bilinear Mindlin element with reduced integration (one point for all terms) [11.4].
|
||
|
||
anisms. Other good Mindlin plate elements can be formulated by hybrid methods [9.8].
|
||
|
||
Mechanisms. An element having one or more mechanisms is not a foolproof element. Occasional numerical disaster is possible in the hands of a user who is unaware or inattentive. Mechanisms of elements in Table 11.3-1 resemble those of the corresponding plane elements, discussed in Section 6.12.
|
||
|
||
Mechanisms of the bilinear element with reduced integration are shown in Fig. 11.3-4. With selective integration, only two of these mechanisms remain possible: the in-plane twist mode and the w-hourglass mode. The in-plane twist mode, Fig. 11.3-4a, is not communicable between adjacent elements, so that a mesh of two or more elements cannot have this mechanism. Control of mechanisms is discussed in [6,11-6,13,13,49,13,52-13,54].
|
||
|
||
Under both reduced and selective integration, the quadratic and serendipity elements have the mechanism described by Eqs. 6.12-3 (for Mindlin plates, substitute $-z\theta_{x}$ for u and $-z\theta_{y}$ for v). This mechanism cannot be communicated between elements. Three additional mechanisms are possible in the nine-node quadratic element under reduced integration. One comes from Eq. 6.12-2, and another from Eq. 6.12-2 with u and v interchanged. The third mechanism is u = v = 0, $w = 3\xi^{2}\eta^{2} - \xi^{2} - \eta^{2}$ . The latter mechanism is not possible in the heterosis element because the $\xi^{2}\eta^{2}$ term is not present in the w field.
|
||
|
||
Stress Computation. When element nodal d.o.f. $\{d\}$ are known, Eq. 11.3-5 yields curvatures $\{\kappa\}$ , Eq. 11.1-12 yields moments and shears, and Eqs. 11.1-2 yields stresses. Transverse shear stresses may be greatly in error except at Gauss points of the selective integration rule appropriate to $[k_{s}]$ . (Even at these points, accuracy may be poor unless thickness t used in $[k_{s}]$ has been adjusted according to Eq. 11.3-9.) Thus, in the bilinear element, transverse shear stresses $\tau_{yz}$ and $\tau_{zx}$ should be calculated at the element center, and these values assumed to prevail throughout the element. With the remaining elements in Table 11.3-1, it usually is good strategy to calculate stresses at Gauss points of a 2 by 2 rule, then extrapolate to other locations in the element as required. Extrapolation of stresses is discussed in Section 6.13.
|
||
|
||
# 11.4 A TRIANGULAR DISCRETE KIRCHHOFF ELEMENT
|
||
|
||
The element to be discussed was published in 1969 [11.10]. It was reexamined over ten years later and found to remain among the best elements for analysis of
|
||
|
||
<!-- source-page: 349 -->
|
||
|
||
thin plates [11,11]. The element is currently known as DKT, for discrete Kirchhoff triangle. Explicit expressions [11,12] and Fortran coding [11,13] for the element are available. Details of element formulation involve lengthy expressions. The following is a summary. Its essential step, that is, the enforcement of zero transverse shear strain at specific points, is also used in the formulation of other discrete Kirchhoff elements.
|
||
|
||
The starting point is a straight-sided element with corner and midside nodes (Fig. 11.4-1a). Rotations $\theta_{x}$ and $\theta_{y}$ of a midsurface-normal line are each interpolated from nodal rotations $\theta_{xi}$ and $\theta_{yi}$ , where i runs from 1 to 6, using a complete quadratic polynomial:
|
||
|
||
$$
|
||
\theta_ {x} = \sum N _ {i} \theta_ {x i} \quad \text { and } \quad \theta_ {y} = \sum N _ {i} \theta_ {y i} \tag {11.4-1}
|
||
$$
|
||
|
||
Here the $N_{i}$ are given by Eqs. 5.3-5. There are a total of twelve d.o.f. in Eq. 11.4-1. Lateral deflection w along each edge is assumed to be cubic in an edge-tangent coordinate s. Thus, along side 2–3 for example, the rotation $w_{ss}$ at midside node 5 is
|
||
|
||
$$
|
||
w _ {, s 5} = - \frac {3}{2 L _ {2 3}} w _ {2} - \frac {1}{4} w _ {, s 2} + \frac {3}{2 L _ {2 3}} w _ {3} - \frac {1}{4} w _ {, s 3} \tag {11.4-2}
|
||
$$
|
||
|
||
Two similar equations are written for the remaining two sides. When nodal values of $w_{,s}$ are replaced by nodal values of $w_{,x}$ and $w_{,y}$ by coordinate transformation, there are a total of nine d.o.f. in these three equations for $w_{,s}$ (vertex-node values of w, $w_{,x}$ , and $w_{,y}$ ). Accordingly, in Eq. 11.4-1 and the three rotation equations such as 11.4-2, there are a total of 21 d.o.f.
|
||
|
||
We seek a nine-d.o.f. element that has the nodal d.o.f. shown in Fig. 11.4-1b. Accordingly, the twelve d.o.f. $\theta_{xi}$ and $\theta_{yi}$ at nodes 1 through 6 must be expressed in terms of $w_{i}$ , $w_{,xi}$ , and $w_{,yi}$ at only the corner nodes. Constraints used for this purpose are as follows.
|
||
|
||
1. Transverse shear strains $\gamma_{yz}$ and $\gamma_{zx}$ vanish at corners 1, 2, and 3. Thus, from Eqs. 11.1-4,
|
||
|
||
$$
|
||
\theta_ {x i} = w _ {, x i} \quad \text { and } \quad \theta_ {y i} = w _ {, y i} \quad \text { for } \quad i = 1, 2, 3 \tag {11.4-3}
|
||
$$
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
s
|
||
3
|
||
i
|
||
w_i
|
||
θ_yi
|
||
θ_xi
|
||
y, v
|
||
6
|
||
ξ_2
|
||
ξ_1
|
||
ξ_3
|
||
1
|
||
4
|
||
2
|
||
n
|
||
z, w
|
||
x, u
|
||
</details>
|
||
|
||
(a)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
y
|
||
1
|
||
3
|
||
2
|
||
z
|
||
x
|
||
w_i
|
||
w_yi
|
||
w_xi
|
||
</details>
|
||
|
||
(b)
|
||
Figure 11.4-1. Development of a discrete Kirchhoff triangle. (a) Initial element and its d.o.f. Area coordinates $\xi_{1}$ , $\xi_{2}$ , and $\xi_{3}$ are shown. (b) Final nine-d.o.f. element and its d.o.f.
|
||
|
||
<!-- source-page: 350 -->
|
||
|
||
2. Transverse shear strain $\gamma_{sz}$ vanishes at nodes 4, 5, and 6, where s is an edge-tangent coordinate. Thus
|
||
|
||
$$
|
||
\theta_ {s i} = w _ {, s i} \quad \text { for } \quad i = 4, 5, 6 \tag {11.4-4}
|
||
$$
|
||
|
||
Use of Eq. 11.4-4 requires coordinate transformation operations.
|
||
|
||
3. Normal slopes vary linearly along each edge. Thus
|
||
|
||
$$
|
||
\theta_ {n 4} = \frac {1}{2} (w, _ {n 1} + w, _ {n 2}) \quad \theta_ {n 5} = \frac {1}{2} (w, _ {n 2} + w, _ {n 3}) \quad \theta_ {n 6} = \frac {1}{2} (w, _ {n 3} + w, _ {n 1}) \tag {11.4-5}
|
||
$$
|
||
|
||
After the foregoing constraints have been applied, the twelve nodal d.o.f. $\theta_{xi}$ and $\theta_{yi}$ in Eqs. 11.4-1 are expressed in terms of the nine nodal d.o.f. $w_{i}$ , $w_{,xi}$ , and $w_{,yi}$ at the corners. Symbolically, these relations are
|
||
|
||
$$
|
||
\left[ \begin{array}{l l l l l l l l} \theta_ {x 1} & \theta_ {y 1} & \theta_ {x 2} & \dots & \theta_ {y 6} \end{array} \right] ^ {T} = \underset {1 2 \times 9} {[ \mathbf {T} ]} \left[ \begin{array}{l l l l l l l} w _ {1} & w _ {, x 1} & w _ {, y 1} & w _ {2} & \dots & w _ {, y 3} \end{array} \right] ^ {T} \tag {11.4-6}
|
||
$$
|
||
|
||
To generate the element stiffness matrix, we can begin with Eqs. 11.4-1. Strains are stated as in Mindlin plate theory, Eqs. 11.1-4, but now $\gamma_{yz}$ and $\gamma_{zx}$ are ignored. Thus one considers only the strains $\epsilon_{x} = -z\theta_{x,x}$ , $\epsilon_{y} = -z\theta_{y,y}$ , and $\gamma_{xy} = -z(\theta_{x,y} + \theta_{y,x})$ , that is,
|
||
|
||
$$
|
||
\{\epsilon \} = - z [ \partial ] \left\{ \begin{array}{l} \theta_ {x} \\ \theta_ {y} \end{array} \right\}, \quad \text { where } \quad [ \partial ] = \left[ \begin{array}{c c} \partial / \partial x & 0 \\ 0 & \partial / \partial y \\ \partial / \partial y & \partial / \partial x \end{array} \right] \tag {11.4-7}
|
||
$$
|
||
|
||
Equations 11.4-1 and 11.4-7 yield the strains $\{\epsilon\} = \left[\epsilon_x \quad \epsilon_y \quad \gamma_{xy}\right]^T$ as
|
||
|
||
$$
|
||
\{\boldsymbol {\epsilon} \} = - z [ \partial ] \underbrace {\left[ \begin{array}{c c c c c c c} N _ {1} & 0 & N _ {2} & 0 & \dots & N _ {6} & 0 \\ 0 & N _ {1} & 0 & N _ {2} & \dots & 0 & N _ {6} \end{array} \right]} _ {[ \mathbf {B} _ {\theta} ]} \underbrace {\left[ \begin{array}{c c c c c} \theta_ {x 1} & \theta_ {y 1} & \dots & \theta_ {y 6} \end{array} \right] ^ {T}} _ {\{\mathbf {d} _ {\theta} \}} \tag {11.4-8}
|
||
$$
|
||
|
||
Equations 5.2-7 must be used in forming $[B_{\theta}]$ . After integration through the thickness, the strain energy expression $U = \frac{1}{2} \int \{\epsilon\}^{T}[E]\{\epsilon\} \, dV$ becomes
|
||
|
||
$$
|
||
U = \frac {1}{2} \left\{\mathbf {d} _ {\theta} \right\} ^ {T} \left[ \mathbf {k} _ {\theta} \right] \left\{\mathbf {d} _ {\theta} \right\}, \quad \text { where } \quad \left[ \mathbf {k} _ {\theta} \right] _ {1 2 \times 1 2} = \int_ {A} \left[ \mathbf {B} _ {\theta} \right] ^ {T} \left[ \mathbf {D} _ {K} \right] \left[ \mathbf {B} _ {\theta} \right] d A \tag {11.4-9}
|
||
$$
|
||
|
||
Matrix $[D_{K}]$ is the rigidity matrix of Kirchhoff plate theory, for example, Eq. 11.1-10. Matrix $[k_{\theta}]$ operates on the twelve rotational d.o.f. used in Eq. 11.4-1. It can be converted to a 9 by 9 matrix [k], which operates on the standard Kirchhoff d.o.f. at the corners, by applying Eq. 11.4-6:
|
||
|
||
$$
|
||
[ \mathbf {k} ] = [ \mathbf {T} ] ^ {T} [ \mathbf {k} _ {\theta} ] [ \mathbf {T} ] \tag {11.4-10}
|
||
$$
|
||
|
||
Transformations of this type are explained in Section 7.4. However, more efficient coding results if Eq. 11.4-6 is substituted into Eq. 11.4-8, so that the strain-displacement relation contains nine columns rather than twelve. Thus [k] is produced directly.
|