김경종
22 KiB
elements can be used to model shell intersections and shell-to-solid transitions using compatible element idealizations without the use of special constraint equations. The features of generality and accuracy in the modeling of a shell structure can be especially important in the material and geometric nonlinear analysis of shell structures, since particularly in such analyses shell geometries must be accounted for accurately. We discuss the extension of the formulation to general nonlinear analysis in Section 6.5.2.
The underlying mathematical model of this pure displacement-based formulation corresponds to a general shell theory, referred to as the 'basic shell model', in D. Chapelle and K.J. Bathe [C, E], but the elements, as in the case of the beam elements, are not effective because they lock in shear and, when curved, in membrane actions. Since effective shell elements are enormously more difficult to obtain than plate elements, much research has been conducted. Among the proposed formulations, the MITC shell elements are quite effective, see the original formulations by E. N. Dvorkin and K.J. Bathe [A], K.J. Bathe and E.N. Dvorkin [B] and M. L. Bucalem and K.J. Bathe [A] on quadrilateral elements, and further work to develop triangular elements, P. S. Lee and K. J. Bathe [A], D. N. Kim and K. J. Bathe [B] and Y. Lee, P. S. Lee, and K. J. Bathe [A].
The first step in mixed interpolation is to write the complete strain tensor at an integration point as
\epsilon = \underbrace {\tilde {\epsilon} _ {r r} \mathbf {g} ^ {r} \mathbf {g} ^ {r} + \tilde {\epsilon} _ {s s} \mathbf {g} ^ {s} \mathbf {g} ^ {s} + \tilde {\epsilon} _ {r s} \left(\mathbf {g} ^ {r} \mathbf {g} ^ {s} + \mathbf {g} ^ {s} \mathbf {g} ^ {r}\right)} + \underbrace {\tilde {\epsilon} _ {r r} \left(\mathbf {g} ^ {r} \mathbf {g} ^ {t} + \mathbf {g} ^ {t} \mathbf {g} ^ {r}\right) + \tilde {\epsilon} _ {s t} \left(\mathbf {g} ^ {s} \mathbf {g} ^ {t} + \mathbf {g} ^ {t} \mathbf {g} ^ {s}\right)} \tag {5.122}
in-layer strains
transverse shear strains
where the \tilde{\epsilon}_{rr}, \tilde{\epsilon}_{ss}, \ldots , are the covariant strain components corresponding to the base vectors
\begin{array}{l} \mathbf {g} _ {r} = \frac {\partial \mathbf {x}}{\partial r}; \quad \mathbf {g} _ {s} = \frac {\partial \mathbf {x}}{\partial s}; \quad \mathbf {g} _ {t} = \frac {\partial \mathbf {x}}{\partial t} \\ \mathbf {x} = \left[ \begin{array}{l} x \\ y \\ z \end{array} \right] \tag {5.123} \\ \end{array}
and the \mathbf{g}^r, \mathbf{g}^s, \mathbf{g}' are the corresponding contravariant base vectors (see Section 2.4). We note that if we use indicial notation with i = 1, 2, 3 corresponding to r, s , and t , respectively, and r_1 = r, r_2 = s, r_3 = t , we can define
{ } ^ { 0 } \mathbf { g } _ { i } = \frac { \partial \mathbf { x } } { \partial r _ { i } } ; \quad { } ^ { 1 } \mathbf { g } _ { i } = \frac { \partial ( \mathbf { x } + \mathbf { u } ) } { \partial r _ { i } } \tag {5.124}
and then the covariant Green-Lagrange strain tensor components are
{ } _ { 0 } ^ { 1 } \tilde { \boldsymbol { \epsilon } } _ { i j } = \frac { 1 } { 2 } ( { } ^ { 1 } \mathbf { g } _ { i } \cdot { } ^ { 1 } \mathbf { g } _ { j } - { } ^ { 0 } \mathbf { g } _ { i } \cdot { } ^ { 0 } \mathbf { g } _ { j } ) \tag {5.125}
The strain components in (5.118) are the linear Cartesian components of the strain tensor given by (5.125) (see Example 2.28).
In the mixed interpolation, the objective is to interpolate the in-layer and transverse shear strain components independently and tie these interpolations to the usual displacement interpolations. The result is that the stiffness matrix is then obtained corresponding to only the same nodal point variables (displacements and section rotations) as are used for the displacement-based elements. Of course, the key is to choose in-layer and transverse
shear strain component interpolations, for the displacement interpolations used, such that the resulting element has an optimal predictive capability.
An attractive four-node element is the MITC4 shell element proposed by E. N. Dvorkin and K. J. Bathe [A] for which the in-layer strains are computed from the displacement interpolations (since the element is not curved and membrane locking is not present in the displacement-based element) and the covariant transverse shear strain components are interpolated and tied to the displacement interpolations as discussed for the plate element [see (5.101)]. The element performs quite well in out-of-plane bending (the plate bending) actions, and also in the in-plane (the membrane) actions if the incompatible modes as discussed in Example 4.28 are added to the basic four-node element displacement interpolations.
A significantly better predictive capability is obtained with higher-order elements, and Fig. 5.35 shows the interpolations and tying points for the 9-node and 16-node elements proposed by M. L. Bucalem and K. J. Bathe [A] and K. J. Bathe, P. S. Lee and J.-F. Hiller [A]. These elements are referred to as the MITC9 and MITC16 shell elements. Elements similarly formulated to the MITC9 shell element have been presented by H. C. Huang and E. Hinton [A], K. C. Park and G. M. Stanley [A] and J. Jang and P. M. Pinsky [A].
Figure 5.35 MITC shell elements; interpolations of strain components and tying points
Following our earlier discussion of the mixed interpolation of plate elements, in the formulation of these shell elements we use
\tilde {\epsilon} _ {i j} = \sum_ {k = 1} ^ {n _ {y}} h _ {k} ^ {i j} \left. \mathbf {B} _ {i j} ^ {D I} \right| _ {k} \hat {\mathbf {u}} \tag {5.126}
where n_{ij} denotes the number of tying points used by the strain component considered, h_{k}^{ij} is the interpolation function corresponding to the tying point k, and B_{ij}^{Dt}\big|_{k}\hat{u} is the strain component evaluated at the tying point k by the displacement assumption (by displacement interpolation). Note that with (5.126) only point tying and no integral tying (as in the higher-order MITC plate elements) is performed.
Unfortunately, a mathematical analysis of the MITC9 and MITC16 shell elements, as achieved for the plate elements summarized in Fig. 5.27, is not yet available but some valuable insight based on some mathematical analyses and test problems has been gained, see D. Chapelle and K. J. Bathe [B, C, D, E], P. S. Lee and K. J. Bathe [B, C], K. J. Bathe, F. Brezzi, and L. D. Marini [A], K. J. Bathe and P. S. Lee [A] and K. J. Bathe, D. Chapelle, and P. S. Lee [A].
As discussed in detail in these references, the analyses of well-chosen and stringent test problems is very important. In these benchmark tests, appropriate shell geometries, boundary conditions and loadings need to be used. The accuracy of the solutions obtained should be measured in appropriate norms (for example, a displacement at a point is not sufficient), and for decreasing shell thickness values.
Figure 5.36 Convergence curves of the MITC4 element for decreasing shell thickness (t/L = 10^{-2} , 10^{-3} , 10^{-4} ) in the solution of three shell test problems; L = 1.0, E = 1.0 \times 10^{11} , \nu = 1/3 . The shell surface is given by X^{2} + Z^{2} = 1 + Y^{2} ; the pressure loading is p(\theta) = \cos 2\theta . Only the shaded region in the figure is modeled; (a) free-free shell; (b) fixed-fixed shell; (c) fixed-free shell.
Figure 5.36 gives the geometry and loading of a shell structure, and Table 5.4 gives the test cases considered. In these problems the membrane and bending behaviors of shell elements are tested when a shell of negative Gaussian curvature is analyzed; indeed, such structural problems should be solved as stringent tests of a shell element for its reliability and accuracy. In the solutions, the s-norm is used, which measures the accuracy of the predicted stresses, see K. J. Bathe and P. S. Lee [A]. Figure 5.36 shows that in these analyses, with the uniform meshes used, the MITC4 shell element performs very well, but such excellent behavior cannot always be expected.
Finally, we should note an important point, namely that the MITC formulated shell elements can directly be extended to geometrically nonlinear analyses by simply using the appropriate stress and strain measures, discussed in Chapter 6.
Table 5.4 Shell test cases with the geometry and loading of Figure 5.36
| Boundary conditions | Category and boundary layer |
| Free – free shell | Bending-dominated; boundary layer = $0.5\sqrt{t}$ and does not require special meshing |
| Fixed – fixed shell | Membrane-dominated; boundary layer = $6\sqrt{t}$ and requires fine mesh* |
| Free – fixed shell | Mixed membrane-bending state; boundary layer at fixed end to be specially meshed |
* An equal number of element layers as in the rest of the domain is appropriate
Boundary Conditions
The plate elements presented in this section are based on Reissner-Mindlin plate theory, in which the transverse displacement and section rotations are independent variables. This assumption is fundamentally different from the kinematic assumption used in Kirchhoff plate theory, in which the transverse displacement is the only independent variable (M. L. Bucalem and K. J Bathe [B]). Hence, whereas in Kirchhoff plate theory all boundary conditions are written only in terms of the transverse displacement (and of course its derivatives), in the Reissner-Mindlin theory all boundary conditions are written in terms of the transverse displacement and the section rotations (and their derivatives). Since the section rotations are used as additional kinematic variables, the actual physical condition of a support can also be modeled more accurately.
As an example, consider the support conditions at the edge of the thin structure shown in Fig. 5.37. If this structure were modeled as a three-dimensional continuum, the element idealization might be as shown in Fig. 5.38(a), and then the boundary conditions would be
text_image
z, w y, v x, u L h (small) Rigidly built-in h/L << 1
Figure 5.37 Knife-edge support for thin structure
text_image
z, w y, v x, u i + 2 i i + 1 For all nodes on this line, uk = vk = wk = 0, k = ..., i, i + 1, i + 2, ...
(a) Three-dimensional model using 27-node elements
text_image
z, w y θy x θx i + 2 i + 1 θy i θx For all nodes on this line, wk = 0, k = ..., i, i + 1, i + 2, ...
(b) Plate model using plate elements
Figure 5.38 Three-dimensional and plate models for problem in Fig. 5.37
those given in the figure. Of course, such a model would be inefficient and impractical because the finite element discretization would have to be very fine for an accurate solution (recall that the three-dimensional elements would display the shear locking phenomenon).
Employing Reissner-Mindlin plate theory, the thin structure is represented using the assumptions given in (5.88) and Fig. 5.25. The boundary conditions are that the transverse displacement is restrained to zero but the section rotations are free; see Fig. 5.38(b). Surely, these conditions represent the physical situation as closely as possible consistent with the assumptions of the theory.
We note, on the other hand, that using Kirchhoff plate theory, the transverse displacement and edge rotation given by \partial w/\partial x would both be zero, and therefore the finite element model would also have to impose \theta_{y}=0 . Hence, in summary, the edge conditions in Fig. 5.37 would be modeled as follows in a finite element solution.
Using three-dimensional elements:
\text { on the edge: } \quad u = v = w = 0 \tag {5.127}
Using Reissner-Mindlin plate theory-based elements (e.g., the MITC elements in Fig. 5.27):
\text { on the edge: } \quad w = 0; \quad \theta_ {x} \text { and } \theta_ {y} \text { are left free } \tag {5.128}
Using Kirchhoff plate theory-based elements (e.g., the elements in Example 4.18):
\text { on the edge: } \quad w = \theta_ {y} = 0; \quad \theta_ {x} \text { is left free } \tag {5.129}
where in Kirchhoff plate theory,
\theta_ {y} = - \frac {\partial w}{\partial x} \tag {5.130}
Of course, we could also visualize a physical support condition that, in addition to the rigid knife-edge support in Fig. 5.37, prevents the section rotation \beta_{x} . In this case we also would set \theta_{y} to zero when using the Reissner-Mindlin plate theory-based elements, and we would set all u-displacements on the face of the plate equal to zero when using the three-dimensional elements.
The boundary condition in (5.128) is referred to as the “soft” boundary condition for a simple support, whereas when \theta_{y} is also set to zero, the boundary condition is of the “hard” type. Similar possibilities also exist when the plate edge is “clamped”, i.e., when the edge is also restrained against the rotation \theta_{x} . In this case we clearly have w = 0 and \theta_{x} = 0 on the plate edge. However, again a choice exists regarding \theta_{y} : in the soft boundary condition \theta_{y} is left free, and in the hard boundary condition \theta_{y} = 0 . In practice, we usually use the soft boundary conditions, but of course, depending on the actual physical situation, the hard boundary condition is also employed.
The important point is that when the Reissner-Mindlin plate theory-based elements are used, the boundary conditions on the transverse displacement and rotations are not necessarily the same as when Kirchhoff plate theory is being used and must be chosen to model appropriately the actual physical situation.
The same observations hold of course for the use of the shell elements presented earlier, for which the section rotations are also independent variables (and are not given by the derivatives of the transverse displacement).
Since the Reissner-Mindlin theory contains more variables for describing the plate behavior than the Kirchhoff theory, various interesting questions arise regarding a comparison of these theories and the convergence of results based on the Reissner-Mindlin theory to those based on the Kirchhoff theory. These questions have been addressed, for example, by K. O. Friedrichs and R. F. Dressler [A], E. Reissner [C], B. Häggblad and K. J. Bathe [A], and D. N. Arnold and R. S. Falk [A]. A main result is that when the Reissner-Mindlin theory is used, boundary layers along plate edges develop for specific boundary conditions when the thickness/length ratio of the plate becomes very small. These boundary layers represent the actual physical situation more realistically than the Kirchhoff plate theory does. Hence, the plate and shell elements presented in this section are not only attractive for computational reasons but can also be used to represent the actual situations in nature more accurately. Some numerical results and comparisons using the Kirchhoff and Reissner-Mindlin plate theories are given by B. Häggblad and K. J. Bathe [A] and K. J. Bathe, N. S. Lee, and M. L. Bucalem [A].
5.4.3 Exercises
5.32. Consider the beam of constant cross-sectional area in Fig. 5.19. Derive from (4.7), using the assumptions in Fig. 5.18, the virtual work expression in (5.58).
5.33. Consider the cubic displacement-based isoparametric beam element shown. Construct all matrices needed for the evaluation of the stiffness and mass matrices (but do not perform any integrations to evaluate these matrices).
text_image
L w₁ w₃ w₄ w₂ Thickness b θ₁ θ₃ θ₄ θ₂ h
5.34. Consider the 3-node isoparametric displacement-based beam element used to model the cantilever beam problem in Fig. 5.20. Show analytically that excellent results are obtained when node 3 is placed exactly at the midlength of the beam, but that the results deteriorate when this node is shifted from that position.
text_image
w₁ = 0 θ₁ = 0 L w₃ θ₃ w₂ θ₂ Beam of constant cross section
5.35. Consider the two-node beam element shown. Specialize the expressions (5.71) to (5.86) to this case.
text_image
1.0 1 s Constant thickness b 30° 2 1.5 y x L Action in x-y plane
5.36. Consider the three-node beam element shown. Specialize the expressions (5.71) to (5.86) to this case.
text_image
Constant thickness b 0V_s^2 1.6 0V_s^3 s r 2 3 0V_s^1 1 45° y 0.8 x L Action in x-y plane
5.37. Consider the cantilever beam shown. Idealize this structure by one two-node mixed interpolated beam element and analyze the response. First, neglect warping effects. Next, introduce warping displacements using the warping displacement function w_{w} = xy(x^{2} - y^{2}) and assuming a linear variation in warping along the element axis.
text_image
y z L T h y z x h
Young's modulus E Shear modulus G
5.38. Consider the two-node mixed interpolated beam element shown. Derive all expressions needed to calculate the stiffness matrix, mass matrix, and nodal force vector for the degrees of freedom indicated. However, do not perform any integrations.
text_image
Load on beam is p/unit length in z-direction on centerline of beam Node 1 Node 2 L h h b x y z u1 v1 θx1 θy1 θz1 θx2 u2 v2 θy2 θz2
5.39. Consider the plane stress element shown and evaluate the strain-displacement matrix of this element (called B_{pl} ).
Also consider the two-node displacement-based isoparametric beam element shown and evaluate the strain-displacement matrix (called B_{b} ).
Derive from B_{pl} , using the appropriate kinematic constraints, the strain-displacement matrix of the degenerated plane stress element (called \tilde{B}_{pl} ) for the degrees of freedom used in the beam element. Show explicitly that
\int_ {V} \mathbf {B} _ {b} ^ {T} \mathbf {C} _ {b} \mathbf {B} _ {b} d V = \int_ {V} \tilde {\mathbf {B}} _ {p l} ^ {T} \tilde {\mathbf {C}} \tilde {\mathbf {B}} _ {p l} d V
with C_{b} and \tilde{C} to be determined by you.
text_image
t 2 s 1 v₁ u₁ r 3 4 L
Plane stress element (unit thickness) Young's modulus E Poisson's ratio v
text_image
2 L θ₁ v₁ 1 u₁
Beam element of depth t and unit thickness
5.40. Consider the problem of an infinitely long, thin plate, rigidly clamped on two sides as shown. Calculate the stiffness matrix of a two-node plane strain beam element to be used to analyze the plate. [Use the mixed interpolation of (5.68) and (5.69).]
text_image
Young's modulus E Poisson's ratio v w1 w2 θ1 θ2 h L ∞
5.41. Consider the axisymmetric shell element shown. Construct the strain-displacement matrix assuming mixed interpolation with a constant transverse shear strain. Also, establish the corresponding stress-strain matrix to be used in the evaluation of the stiffness matrix.
text_image
Young's modulus E Poisson's ratio ν L = 10 v2 θ2 u2 h v1 θ1 30° u1 20
5.42. Assume that in Example 5.28 axisymmetric conditions are being considered. Construct the strain-displacement matrix of the transition element. Assume that the axis of revolution, i.e. the y axis, is at distance R from node 3.
5.43. Use a computer program to analyze the curved beam shown for the deformations and internal stresses.
(a) Use displacement-based discretizations of, first, four-node plane stress elements, and then, eight-node plane stress elements.
(b) Use discretizations of, first, two-node beam elements, and then, three-node beam elements. Compare the calculated solutions with the analytical solution and increase the fineness of your meshes until an accurate solution is obtained.
text_image
h = 1.0 E = 200,000 v = 0.3 Unit thickness R = 100 90° y x P
5.44. Perform the analysis in Exercise 5.43 but assume axisymmetric conditions; i.e., assume that the figure in Exercise 5.43 shows the cross section of an axisymmetric shell with the centerline at x = 0, and that P is a line load per unit length.
5.45. Consider the four-node plate bending element in Example 5.29. Assume that w_{1} = 0.1 and \theta_{y}^{1} = 0.01 and that all other nodal point displacements and rotations are zero. Plot the curvatures \kappa and transverse shear strains \gamma as a function of r, s over the midsurface of the element.