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12.3 FLAT ELEMENTS FOR SHELLS
A curved shell can be approximated as a faceted surface, formed by connecting flat triangular elements together at vertex nodes. If we elect to use three translational and three rotational d.o.f. per node, each triangular element has 18 d.o.f. Let a typical element lie in the xy plane of a local coordinate system xyz. Nodal d.o.f. are shown in Fig. 12.3-1. Let these d.o.f. be called \{d'\} and be arranged in the order ^{1} .
\{\mathbf {d} ^ {\prime} \} = \left[ \begin{array}{l l l l l l} \mathbf {u} _ {i} & \mathbf {v} _ {i} & \theta_ {z i} & \mathbf {w} _ {i} & \theta_ {x i} & \theta_ {y i} \end{array} \right] ^ {T} \tag {12.3-1}
where \left[u_{i}\right]=\left[u_{1}\quad u_{2}\quad u_{3}\right] , \left[v_{i}\right]=\left[v_{1}\quad v_{2}\quad v_{3}\right] , and so on. A plane element that includes “drilling freedoms” \theta_{z} among its d.o.f. is described in Section 8.6. Its 9 by 9 stiffness matrix, which we now call [k_{m}] , models membrane action in the shell. To this element we add a nine-d.o.f. triangular plate element, such as element DKT of Section 11.4. Its 9 by 9 stiffness matrix, which we now call [k_{b}] , models bending action in the shell. For the composite flat shell element, whose stiffness matrix in local xyz coordinates is called [k'] , we have
\left[ \mathbf {k} ^ {\prime} \right] \left\{\mathbf {d} ^ {\prime} \right\} = \left[ \begin{array}{c c} {\left[ \mathbf {k} _ {m} \right]} & {\left[ \mathbf {0} \right]} \\ {9 \times 9} & {9 \times 9} \\ {- - } & {- - } \\ {\left[ \mathbf {0} \right]} & {\left[ \mathbf {k} _ {b} \right]} \\ {9 \times 9} & {9 \times 9} \end{array} \right] \left\{ \begin{array}{l} \mathbf {u} _ {i} \\ \mathbf {v} _ {i} \\ \boldsymbol {\theta} _ {z i} \\ \mathbf {w} _ {i} \\ \boldsymbol {\theta} _ {x i} \\ \boldsymbol {\theta} _ {y i} \end{array} \right\} \tag {12.3-2}
This stiffness matrix is clearly analogous to that of a straight arch element, Eq. 12.2-6. Coordinate transformation of [k'] is required prior to assembly of elements so that a common set of d.o.f. is used at each structure node shared by two or
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z θz1 u1 v1 y 1 θz2 2 v2 3 θz3 u2 v3 u3 x
{α}
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z w₁ θᵧ₁ θₓ₁ y 1 w₂ 2 3 w₃ θₓ₃ θᵧ₂ θᵧ₃ θₓ₂ θᵧ₂ x (t)
(b)
Figure 12.3-1. Triangular element in a local xy plane. (a) D.o.f. associated with membrane action. (b) D.o.f. associated with bending action.
^{1} This ordering of d.o.f. is used only for convenience of notation. If this ordering is used, coefficients in [k] must be appropriately arranged.
more elements. Membrane and bending actions are not coupled within a single element, as evidenced by the 9 by 9 null matrices in Eq. 12.3-2. Nevertheless, the element works well enough to be competitive with curved elements [12.8]. (In Ref. 12.8, [k_{m}] is identical to the “basic triangular subelement” discussed in Section 8.6.)
If [\mathbf{k}_m] pertains to the constant-strain triangle (Eq. 5.4-7), which does not use \theta_z d.o.f., Eq. 12.3-2 is replaced by
\left[ \mathbf {k} ^ {\prime} \right] \left\{\mathbf {d} ^ {\prime} \right\} = \left[ \begin{array}{c c c} \left[ \mathbf {k} _ {m} \right] & [ \mathbf {0} ] & \\ 6 \times 6 & 6 \times 3 & [ \mathbf {0} ] \\ - & - & - \\ [ \mathbf {0} ] & [ \mathbf {0} ] & 9 \times 9 \\ 3 \times 6 & 3 \times 3 & \\ - & - & - \\ & [ \mathbf {0} ] & [ \mathbf {k} _ {b} ] \\ & 9 \times 9 & 9 \times 9 \end{array} \right] \left\{ \begin{array}{l} \mathbf {u} _ {i} \\ \mathbf {v} _ {i} \\ \boldsymbol {\theta} _ {z i} \\ \mathbf {w} _ {i} \\ \boldsymbol {\theta} _ {x i} \\ \boldsymbol {\theta} _ {y i} \end{array} \right\} \tag {12.3-3}
Again there is no coupling on the element level between [k_{m}] and [k_{b}] . In addition, no stiffness is associated with \theta_{z} d.o.f. This means that the structure stiffness matrix will be singular if all elements connected to any node happen to be coplanar. This potential difficulty can be avoided by modifying element matrices [k^{\prime}] as follows. Replace the on-diagonal null matrix in Eq. 12.3-3 by the 3 by 3 matrix in the following equation, which causes element-normal nodal rotations \theta_{z} to produce corresponding moments M_{z} [12.9],
\left\{ \begin{array}{l} M _ {z 1} \\ M _ {z 2} \\ M _ {z 3} \end{array} \right\} = \alpha E V \left[ \begin{array}{c c c} 1. 0 & - 0. 5 & - 0. 5 \\ - 0. 5 & 1. 0 & - 0. 5 \\ - 0. 5 & - 0. 5 & 1. 0 \end{array} \right] \left\{ \begin{array}{l} \theta_ {z 1} \\ \theta_ {z 2} \\ \theta_ {z 3} \end{array} \right\} \tag {12.3-4}
where E is elastic modulus, V is element volume, and \alpha is a number such as 0.3 or less [12.9]. The added matrix provides each \theta_z d.o.f. with a fictitious stiffness but offers no resistance to the mode \theta_{z1} = \theta_{z2} = \theta_{z3} or to any other rigid-body motion.
Another way to avoid singularity is to eliminate rotation about a normal to the shell from the list of global d.o.f. at each node. Thus the element has 15 d.o.f. rather than 18.
Numerical tests show that the element of Eq. 12.3-2 is more accurate than the element of Eq. 12.3-3, and that reducing the latter element to 15 d.o.f. further degrades its performance [12.8].
Difficulties with spurious bending moments, noted in connection with straight arch elements, can also occur when flat elements are used to model a shell.
12.4 SHELLS OF REVOLUTION
A shell of revolution resembles a solid of revolution in that elements are symmetric with respect to an axis and node points are cross sections of nodal circles. An element meridional cross section resembles an arch element, whose behavior and pitfalls are discussed in Section 12.2.
Geometry and notation are shown in Fig. 12.4-1. Circumferential displacement v may be nonzero because initially we make no restriction that loads and dis-
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Nodal circle 1 w v,θ σ_H σ_x τ_SH L u Nodal circle 2
(a)
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s = - \frac{L}{2} w u φ s s = \frac{L}{2} φ r₁ φ₁ 1 P Rₛ r 2 r₂ φ₂
(b)
Figure 12.4-1. (a) Shell of revolution finite element. (b) Meridian of the element. Displacements u, v, and w are mutually orthogonal. R_{s} is the radius of curvature of the meridian.
placements must be axially symmetric. The following geometric relations can be written:
R _ {\theta} = \frac {r}{\cos \phi} \quad R _ {s} = - \frac {d s}{d \phi} \quad \sin \phi = \frac {d r}{d s} \quad \cos \phi = - \frac {d z}{d s} \tag {12.4-1}
R_{\theta} and R_{s} are principal radii of curvature. R_{s} is considered negative if its center lies “outside” the shell (Fig. 12.1-1d). The center of an arc R_{\theta} d \theta lies on the axis of revolution, but the center of an arc R_{s} d \phi may not. Both centers lie on the same normal to the shell. Both radii can vary with s. Distance r and angle \phi can be expressed in terms of s by integrating the second and third of Eqs. 12.4-1. Thus, if R_{s} is assumed constant over the element [12.10],
\phi = \phi_ {1} - \frac {1}{R _ {s}} \left(\frac {L}{2} + s\right) \quad \text { and } \quad r = r _ {1} + R _ {s} (\cos \phi - \cos \phi_ {1}) \tag {12.4-2}
The equation for r fails if the meridian is straight, but then r can be linearly interpolated between r_{1} and r_{2} in terms of s. Stresses shown in Fig. 12.4-1a may vary with s and with \theta . They may also vary with distance z from the shell mid-surface because of bending action.
Loads without axial symmetry can be treated by superposition; that is, an analysis is performed for each Fourier harmonic of loading and the results of all harmonics are superposed. In essence, the method is that used for solids of revolution, as described in Section 10.5.
Assembly of elements must allow for the possibility that meridians of adjacent elements may meet at a cusp—for example, where a cylindrical shell is joined to a conical cap. Accordingly, it is appropriate to transform from element d.o.f. (Fig. 12.4-2a) to a convenient set of global d.o.f. (Fig. 12.4-2b). In general, one should infer that all these d.o.f. represent amplitudes of nodal displacements and rota-
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φ₁ w₁ φ₁ u₁ 1 β₁ w₂ φ₂ 2 β₂ u₂ φ₂ (a)
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W₁ β₁ 1 U₁ 2 W₂ β₂ U₂ (b)
Figure 12.4-2. D.o.f. at nodes, in (a) local, and (b) global directions. D.o.f. v_{1} and v_{2} , not shown, are perpendicular to the paper.
tions, as in a problem without axial symmetry analyzed by Fourier series. If axial symmetry prevails, circumferential nodal displacements v_{i} are zero, and the remaining nodal displacements and rotations are \theta -independent motions. The transformation of d.o.f. at a typical node i is
\left\{ \begin{array}{l} u _ {i} \\ w _ {i} \\ v _ {i} \\ \beta_ {i} \end{array} \right\} = \left[ \begin{array}{c c c c} \sin \phi_ {i} & - \cos \phi_ {i} & 0 & 0 \\ \cos \phi_ {i} & \sin \phi_ {i} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \left\{ \begin{array}{l} U _ {i} \\ W _ {i} \\ v _ {i} \\ \beta_ {i} \end{array} \right\} \tag {12.4-3}
D.o.f. v_{i} and \beta_{i} require no transformation. By writing Eq. 12.4-3 for node 1 and then for node 2, one constructs an 8 by 8 transformation matrix [T], which yields an element matrix [k] = [T]^{T}[k^{\prime}][T] , ready for assembly into the structure. If the problem is axially symmetric, d.o.f. v_{i} do not appear, and [T] is 6 by 6.
Axial Symmetry. If material properties, support conditions, and loads are all independent of \theta , then displacements and stresses are also independent of \theta . Thus v = \tau_{s\theta} = 0 in Fig. 12.4-1. (Here we exclude pure torsion, for which v \neq 0 and \tau_{s\theta} \neq 0 , and vibration and buckling, whose displacement modes may lack axial symmetry despite axial symmetry of material properties, supports, and loads.)
For axial symmetry, the strain-displacement relations are
\epsilon_ {m s} = \frac {d u}{d s} + \frac {w}{R _ {s}} \quad \epsilon_ {m \theta} = \frac {u \sin \phi + w \cos \phi}{r} \tag {12.4-4}
\kappa_ {s} = \frac {d}{d s} \left(\frac {u}{R _ {s}}\right) - \frac {d ^ {2} w}{d s ^ {2}} \quad \kappa_ {\theta} = \frac {\sin \phi}{r} \left(\frac {u}{R _ {s}} - \frac {d w}{d s}\right)
where \epsilon_{ms} and \epsilon_{m\theta} are membrane strains of the shell midsurface, and \kappa_{s} and \kappa_{\theta} are curvature changes of the midsurface. Subscripts s and \theta refer to meridional and circumferential directions, respectively. Transverse shear strain \gamma_{zs} is assumed to be negligible because the shell is thin. The formulation of a thin shell element proceeds in a way very similar to the formulation of a thin arch element. First, one writes displacement fields for u and w that depend on nodal values of u, w, and rotation \beta = w_{,s} . These fields are substituted into Eqs. 12.4-4, and the results into the strain energy expression
U = \frac {1}{2} \int_ {- L / 2} ^ {L / 2} \{\epsilon \} ^ {T} \left[ \begin{array}{c c} \mathbf {E} _ {M} & \mathbf {0} \\ \mathbf {0} & \mathbf {D} _ {K} \end{array} \right] \{\epsilon \} 2 \pi r d s \tag {12.4-5}
where, with C = Et / (1 - \nu^2) and D = Et^3 / 12(1 - \nu^2) for an isotropic material,
\left[ \mathbf {E} _ {M} \right] = C \left[ \begin{array}{l l} 1 & \nu \\ \nu & 1 \end{array} \right] \quad \left[ \mathbf {D} _ {K} \right] = D \left[ \begin{array}{l l} 1 & \nu \\ \nu & 1 \end{array} \right] \quad \{\epsilon \} = \left\{ \begin{array}{l} \epsilon_ {m s} \\ \epsilon_ {m \theta} \\ K _ {s} \\ K _ {\theta} \end{array} \right\} \tag {12.4-6}
Equation 12.4-5 reduces to the form U = \{d\}^{T}[k]\{d\}/2 , from which one identifies the element stiffness matrix [k]. After numerical values of nodal d.o.f. have been calculated, strains and curvature changes follow from Eqs. 12.4-4. Membrane forces and bending moments in the shell are
\left\{ \begin{array}{l} N _ {s} \\ N _ {\theta} \end{array} \right\} = \frac {E t}{1 - \nu^ {2}} \left[ \begin{array}{l l} 1 & \nu \\ \nu & 1 \end{array} \right] \left\{ \begin{array}{l} \epsilon_ {m s} \\ \epsilon_ {m \theta} \end{array} \right\} \quad \text { and } \quad \left\{ \begin{array}{l} M _ {s} \\ M _ {\theta} \end{array} \right\} = \frac {E t ^ {3}}{1 2 (1 - \nu^ {2})} \left[ \begin{array}{l l} 1 & \nu \\ \nu & 1 \end{array} \right] \left\{ \begin{array}{l} \kappa_ {s} \\ \kappa_ {\theta} \end{array} \right\} \tag {12.4-7}
Stresses a distance z from the midsurface are obtained by use of Eq. 12.1-2, with x = s for meridional stress and x = \theta for circumferential stress (see Fig. 12.4-3b).
Equations 12.4-4 simplify if R_{s} is infinite; for example, if the element is conical. Use of conical elements to model a doubly curved shell is analogous to use of straight elements to model an arch. If R_{s} is infinite and \phi = 0 , the element becomes cylindrical.
A Mindlin Axisymmetric Shell Element. This element is similar to the Mindlin beam element discussed in Section 9.4. We consider a conical element, Fig. 12.4-3. Thus \phi is independent of s within a single element and R_{s} is infinite. As before, u and w are midsurface displacements, respectively parallel and normal to a meridian. Let \beta represent the rotation of a line that was normal to the midsurface of the undeformed shell. Transverse shear strain is \gamma_{zs} = (dw/ds) -
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1 z t φ ξ = 2s/L 2 u1 w1 L/2 β1 s ξ w2 u2 β2 (a)
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ds M_H r dθ Q_s M_s N_s (b)
Figure 12.4-3. (a) Cross section of a conical shell element. Distance z is measured from the midsurface. (b) A differential element, bounded by meridians and parallels, and viewed normal to the shell. Membrane forces N_{s} and N_{\theta} , transverse shear force Q_{s} , and bending moments M_{s} and M_{\theta} are shown.
\beta . If \gamma_{zs} is small, then (dw/ds) \approx \beta . Thus, with the addition of \gamma_{zs} , and with R_s infinite, Eqs. 12.4-4 become
\{\boldsymbol {\epsilon} \} = \left\{ \begin{array}{l} \epsilon_ {m s} \\ \epsilon_ {m \theta} \\ \kappa_ {s} \\ \kappa_ {\theta} \\ \gamma_ {z s} \end{array} \right\} = \left[ \begin{array}{c c c} d / d s & 0 & 0 \\ (\sin \phi) / r & (\cos \phi) / r & 0 \\ 0 & 0 & - d / d s \\ 0 & 0 & - (\sin \phi) / r \\ 0 & d / d s & - 1 \end{array} \right] \left\{ \begin{array}{l} u \\ w \\ \beta \end{array} \right\} \tag {12.4-8}
where, as in Eq. 12.2-15b, \beta has taken the place of dw/ds.
Equations 12.4-5 and 12.4-6 are again applicable if \{\epsilon\} is taken from Eq. 12.4-8 and the material property matrix in Eq. 12.4-5 is augmented by a shear stiffness term 5Gt/6 —that is, if the material property matrix is
\left[ \begin{array}{c c c} \mathbf {E} _ {M} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {D} _ {K} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & 5 G t / 6 \end{array} \right] \tag {12.4-9}
in which the factor 5/6 accounts for replacement of the true parabolic variation of \gamma_{\mathrm{zf}} through the thickness by a uniform transverse shear strain.
In what follows we describe a specific element that uses linear interpolations for u and \beta and a quadratic interpolation for w . Membrane locking is avoided because the meridian is straight. Shear locking is avoided by making \gamma_{zs} constant over the element, thus invoking only one penalty constraint as the element becomes thin. In the transverse shear constraint, the coefficient 5Gt/6 in Eq. 12.4-9 plays the role of a penalty number and may require adjustment according to guidelines offered at the end of Section 9.4. The element was suggested by Tessler [12.11].
The displacement field is taken as
\left\{ \begin{array}{l} u \\ w \\ \beta \end{array} \right\} = \left[ \begin{array}{c c c c c c c} N _ {1} & N _ {2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & N _ {1} & N _ {2} & N _ {3} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & N _ {1} & N _ {2} \end{array} \right] \{\mathbf {d} \} \tag {12.4-10}
where \{d\} = \left[u_{1} \quad u_{2} \quad w_{1} \quad w_{2} \quad w_{3r} \quad \beta_{1} \quad \beta_{2}\right]^{T} , in which w_{3r} is the transverse displacement at the element center s = 0, measured relative to the average value (w_{1} + w_{2})/2 . With \xi = 2s/L , the shape functions are
N _ {1} = \frac {1}{2} (1 - \xi) \quad N _ {2} = \frac {1}{2} (1 + \xi) \quad N _ {3} = 1 - \xi^ {2} \tag {12.4-11}
Shear locking could be avoided by use of one Gauss point at s = 0 to integrate stiffness terms associated with \gamma_{zs} . Then internal d.o.f. w_{3r} would be eliminated by static condensation after a 7 by 7 stiffness matrix [k] is formulated. However, the same result can be produced more efficiently by eliminating w_{3r} at the outset via a requirement that \gamma_{zs} be constant over the element. It is possible to satisfy this requirement because the w field is one degree higher than the \beta field. A constant \gamma_{zs} implies that
\frac {d \gamma_ {z s}}{d s} = \frac {d ^ {2} w}{d s ^ {2}} - \frac {d \beta}{d s} = 0 \tag {12.4-12}
Equations 12.4-8 through 12.4-12 yield w_{3r} = (\beta_{1} - \beta_{2})L/8 . With this substitution, Eq. 12.4-10 becomes
\left\{ \begin{array}{l} u \\ w \\ \beta \end{array} \right\} = \left[ \begin{array}{c c c c c c} N _ {1} & N _ {2} & 0 & 0 & 0 & 0 \\ 0 & 0 & N _ {1} & N _ {2} & \overline {{{N}}} _ {3} & - \overline {{{N}}} _ {3} \\ 0 & 0 & 0 & 0 & N _ {1} & N _ {2} \end{array} \right] \{\mathbf {d} \} \tag {12.4-13}
where \{\mathbf{d}\} = \lfloor u_1, u_2, w_1, w_2, \beta_1, \beta_2 \rfloor^T , and, with \xi = 2s / L ,
N _ {1} = \frac {1}{2} (1 - \xi) \quad N _ {2} = \frac {1}{2} (1 + \xi) \quad \overline {{{N}}} _ {3} = \frac {L}{8} (1 - \xi^ {2}) \tag {12.4-14}
We see that w is quadratic in s and that the w associated with \overline{N}_{3} and \beta_{1} = -\beta_{2} is identical to the lateral displacement of a beam whose end rotations are of equal magnitude but opposite sign. One can easily show that Eq. 12.4-13 yields a \gamma_{zs} that is the same as \gamma_{zs} at s = 0 from Eq. 12.4-10, which is the result desired.
Equations 12.4-8 and 12.4-13 are used to formulate the element stiffness matrix. If the element is cylindrical and its stiffness matrix is to be integrated exactly, membrane contributions require three Gauss points (because \epsilon_{\theta} is quadratic in s), curvature contributions require two Gauss points, and the transverse-shear contribution requires one Gauss point. Numerical tests of the element show very good accuracy (e.g., Fig. 12.4-4).
Another Option for Cylindrical Shells. A cylindrical shell that is not circular or not symmetrically loaded can be analyzed by applying Fourier series to a substitute toroidal shell. If the given shell is slightly bent to form a toroidal shell of large radius (like an inner tube for a bicycle tire), the axial coordinate of the cylindrical shell becomes the circumferential coordinate of the toroidal shell. Series analysis of the toroidal shell creates several repetitions of geometry and loading around the circumference. A typical repetition provides a satisfactory model of the original cylindrical shell if enough series terms are used.
line
| s (in.) | M_s (in-lb/in.) | | ------- | -------------- | | 5 | 0 | | 6 | 0 | | 7 | 0 | | 8 | 5 | | 9 | 20 | | 10 | 0 |Figure 12.4-4. Meridional bending moment M_{s} in a cylindrical shell with open ends under uniform internal radial pressure [12.11]. The shell from s = 5 in. to s = 10 in. is spanned by eight identical elements. Finite element results are shown at element midpoints.
12.5 ISOPARAMETRIC GENERAL SHELL ELEMENTS
Summary. A shell of general shape can be modeled by three-dimensional solid elements that (typically) have a thickness dimension considerably smaller than their other dimensions, as in Fig. 12.5-1a. But even for a very thick shell, three nodes along thickness-direction lines supply more d.o.f. than needed. Elimination of the middle nodes yields the element of Fig. 12.5-1b, in which thickness-direction strain \epsilon_{3} is modeled as constant through the thickness. Here the subscript 3 indicates a direction normal to the shell midsurface. As the element becomes even thinner, stiffness coefficients associated with \epsilon_{3} become far larger than other stiffness coefficients. This circumstance invites numerical difficulties. The difficulty can be avoided by constraining adjacent thickness-direction nodes to have the same thickness-direction displacement. Thus five d.o.f. are associated with each pair of thickness-direction nodes in Fig. 12.5-1b. These five d.o.f. can be attached to a single node whose d.o.f. are three translations and two rotations. These five d.o.f. define the motion of a thickness-direction line that remains straight but not necessarily normal to the shell midsurface after deformation. Thus the final element has midsurface nodes only. In formulating the element stiffness matrix, one uses a material property matrix [E] that corresponds to the plane stress condition \sigma_{3} = 0 .
The element need not have eight nodes, as in Fig. 12.5-1c; other popular forms have four or nine nodes. All are Mindlin-type elements, and are therefore able to account for transverse shear deformation. They can be regarded as more general forms of the plate elements discussed in Section 11.3. As is the case with other plate and shell elements, Mindlin shell elements may encounter problems attributable to mechanisms and locking. These problems may be provoked (or avoided) by the choice of element geometry, type and number of d.o.f., shape functions, and numerical integration scheme. Pertinent references include [12.6,12.7, 12.9,12.12-12.14].
Some details are presented in what follows. No particular element or number of nodes is assumed.
Geometry. At a typical node i , Fig. 12.5-2a, one can write a thickness-direction vector \mathbf{V}_{3i} ,
\mathbf {V} _ {3 i} = t _ {i} \left\{ \begin{array}{l} \ell_ {3 i} \\ m _ {3 i} \\ n _ {3 i} \end{array} \right\}, \quad \text { where } \quad \left\{ \begin{array}{l} \ell_ {3 i} \\ m _ {3 i} \\ n _ {3 i} \end{array} \right\} = \frac {1}{t _ {i}} \left\{ \begin{array}{l} x _ {j} - x _ {k} \\ y _ {j} - y _ {k} \\ z _ {j} - z _ {k} \end{array} \right\} \tag {12.5-1}
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ξ ξ η
(a)
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ξ η ξ
(b)
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ξ ξ̃ η
(c)
Figure 12.5-1. (a) A 20-node, 60-d.o.f. solid element. (b) Elimination of four mid-edge nodes yields a 16-node, 48-d.o.f. element. (c) Further constraint yields an 8-node, 40-d.o.f. shell element.
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ζ = +1 j V3i P i ti k ζ = 0 ζ = -1
(a)
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z w_i V_{3i} i v_i y \beta_i v_{2i} u_i V_{1i} \alpha_i x
(b)
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j αi (ζ t_i/2) P βi (ζ t_i/2) ζ t_i/2 i V_{2i} V_{1i}
(c)
Figure 12.5-2. (a) Typical node i, with thickness-direction vector V_{3i} . (b) Orthogonal vectors at node i. Directions of rotational d.o.f. \alpha_{i} and \beta_{i} are given by the right-hand rule. Translational d.o.f. u_{i} , v_{i} , and w_{i} are in the Cartesian directions x, y, and z. (c) Displacements of an arbitrary point P on V_{3i} owing to small nodal rotations.
in which \ell_{3i} , m_{3i} , and n_{3i} are direction cosines of the line kij. The Cartesian coordinates of an arbitrary point in the element are
\left\{ \begin{array}{l} x \\ y \\ z \end{array} \right\} = \sum N _ {i} \left\{ \begin{array}{l} x _ {i} \\ y _ {i} \\ z _ {i} \end{array} \right\} + \sum N _ {i} \zeta \frac {t _ {i}}{2} \left\{ \begin{array}{l} \ell_ {3 l} \\ m _ {3 i} \\ n _ {3 i} \end{array} \right\} \tag {12.5-2}
where x_{i} = (x_{j} + x_{k})/2 , and so on, and shape functions N_{i} are functions of \xi and \eta but are independent of \zeta . For example, the N_{i} are given by Eqs. 6.3-2 for a four-node element, by Eqs. 6.6-1 for an eight-node element, and by Table 6.6-1 for a nine-node element. To define element geometry, one can either supply the Cartesian coordinates of all nodes j and k, or supply x_{i} , y_{i} , z_{i} , t_{i} , and the direction cosines of V_{3i} for all nodes i.
Vectors V_{1i} and V_{2i} in Fig. 12.5-2 are perpendicular to each other and to V_{3i} . Thus V_{1i} and V_{2i} are tangent to the midsurface, but they are not required to have a particular relation to Cartesian coordinate directions. V_{1i} and V_{2i} are used to define the directions of nodal rotation d.o.f. \alpha_{i} and \beta_{i} , which are shared by all elements that share node i. (It is possible that the set of \alpha_{i} and \beta_{i} directions will differ from node to node.) One can define V_{1i} as a principal material direction if the material is orthotropic. Or, one can define a midsurface-tangent vector e_{1i} whose components are \Delta x = \Sigma N_{i,\xi} x_{i} \Delta \xi , \Delta y = \Sigma N_{i,\xi} y_{i} \Delta \xi , and \Delta z = \Sigma N_{i,\xi} z_{i} \Delta \xi , each evaluated at the node i in question, and with \Delta \xi a small number. A similar vector e_{2i} can be defined using an increment \Delta \eta . Then e_{3i} = e_{1i} \times e_{2i} and V_{3i} = t_{i} e_{3i} / e_{3i} . Finally, V_{1i} = e_{1i} and V_{2i} = V_{3i} \times V_{1i} . Direction cosines of V_{1i} and V_{2i} are given by dividing each vector by its magnitude. To avoid input data errors, if the three vectors are supplied as data rather than being calculated within the program, one should ensure that V_{1i} , V_{2i} , and V_{3i} are precisely orthogonal.
For later use, we define the following matrix of direction cosines.
[ \boldsymbol {\mu} _ {i} ] = \left[ - \frac {\mathbf {V} _ {2 i}}{V _ {2 i}} \quad \frac {\mathbf {V} _ {1 i}}{V _ {1 i}} \right] = \left[ \begin{array}{l l} - \ell_ {2 i} & \ell_ {1 i} \\ - m _ {2 i} & m _ {1 i} \\ - n _ {2 i} & n _ {1 i} \end{array} \right] \tag {12.5-3}
where V_{1i} and V_{2i} are the magnitudes of \mathbf{V}_{1i} and \mathbf{V}_{2i} .
The 3 by 3 Jacobian matrix [J], defined by Eq. 6.7-2, contains terms such as
x _ {, \xi} = \sum N _ {i, \xi} \left(x _ {i} + \zeta t _ {i} \ell_ {3 i} / 2\right)
x _ {, \eta} = \sum N _ {i, \eta} \left(x _ {i} + \zeta t _ {i} \ell_ {3 i} / 2\right) \tag {12.5-4}
x _ {, \zeta} = \sum N _ {i} \left(t _ {i} \ell_ {3 i} / 2\right)
Displacements and Strains. The displacement of a point P on vector V_{3i} , Fig. 12.5-2, consists of the displacement of node i plus the displacement relative to node i created by rotation of V_{3i} . The relative displacement components, shown in Fig. 12.5-2c, must be resolved into x, y, and z components before being added to the displacements of node i. Thus, for example, point P has the x-direction displacement
u _ {P} = u _ {i} - \alpha_ {i} \left(\zeta \frac {t _ {i}}{2}\right) \ell_ {2 i} + \beta_ {i} \left(\zeta \frac {t _ {i}}{2}\right) \ell_ {1 i} \tag {12.5-5}
in which nodal rotations \alpha_{i} and \beta_{i} are presumed small. Displacements of an arbitrary point in the element are
\left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\} = \sum N _ {i} \left\{ \begin{array}{l} u _ {i} \\ v _ {i} \\ w _ {i} \end{array} \right\} + \sum N _ {i} \zeta \frac {t _ {i}}{2} [ \boldsymbol {\mu} _ {i} ] \left\{ \begin{array}{l} \alpha_ {i} \\ \beta_ {i} \end{array} \right\} \tag {12.5-6}
Following standard isoparametric procedure, we express strains in terms of displacement derivatives,
\left[ \begin{array}{l l l l l l} \epsilon_ {x} & \epsilon_ {y} & \epsilon_ {z} & \gamma_ {x y} & \gamma_ {y z} & \gamma_ {z x} \end{array} \right] ^ {T} = [ \mathbf {H} ] \left[ \begin{array}{l l l l l l} u _ {, x} & u _ {, y} & u _ {, z} & v _ {, x} \dots w _ {, z} \end{array} \right] ^ {T} \tag {12.5-7}
\left\{ \begin{array}{c} u _ {, x} \\ u _ {, y} \\ u _ {, z} \\ v _ {, x} \\ \vdots \\ w _ {, z} \end{array} \right\} = \left[ \begin{array}{c c c} \mathbf {J} ^ {- 1} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {J} ^ {- 1} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {J} ^ {- 1} \end{array} \right] \left\{ \begin{array}{c} u _ {, \xi} \\ u _ {, \eta} \\ u _ {, \zeta} \\ v _ {, \xi} \\ \vdots \\ w _ {, \zeta} \end{array} \right\} \tag {12.5-8}
where [H] is stated in Eq. 6.7-5 and [J]^{-1} is the inverse of the 3 by 3 Jacobian matrix [J]. All six strains are included in Eq. 12.5-7 because the shell midsurface has no particular orientation with respect to Cartesian coordinates xyz. The condition \sigma_{3}=0 will be introduced subsequently via the stress–strain relation. From Eq. 12.5-6 we obtain
\left\{ \begin{array}{c} u, _ {\xi} \\ u, _ {\eta} \\ u, _ {\zeta} \\ v, _ {\xi} \\ \vdots \\ w, _ {\zeta} \end{array} \right\} = \sum \left[ \begin{array}{c c c c c} N _ {i, \xi} & 0 & 0 & - \zeta t _ {i} N _ {i, \xi} \ell_ {2 i} / 2 & \zeta t _ {i} N _ {i, \xi} \ell_ {1 i} / 2 \\ N _ {i, \eta} & 0 & 0 & - \zeta t _ {i} N _ {i, \eta} \ell_ {2 i} / 2 & \zeta t _ {i} N _ {i, \eta} \ell_ {1 i} / 2 \\ 0 & 0 & 0 & - t _ {i} N _ {i} \ell_ {2 i} / 2 & t _ {i} N _ {i} \ell_ {1 i} / 2 \\ 0 & N _ {i, \xi} & 0 & - \zeta t _ {i} N _ {i, \xi} m _ {2 i} / 2 & \zeta t _ {i} N _ {i, \xi} m _ {1 i} / 2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & - t _ {i} N _ {i} n _ {2 i} / 2 & t _ {i} N _ {i} n _ {1 i} / 2 \end{array} \right] \left\{ \begin{array}{c} u _ {i} \\ v _ {i} \\ w _ {i} \\ \alpha_ {i} \\ \beta_ {i} \end{array} \right\} \tag {12.5-9}