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r₂ r₁ t g₃ g₂ g₁ 3 + 1 r₂ 4 + 5 r₁ + 2
The rebar is integrated using 2 \times 2 or 1 \times 1 Gauss points, depending on the order of the underlying element. The volume of integration at a Gauss point is
\Delta V = \frac {A _ {r}}{S _ {r}} \bigg | \frac {\partial \mathbf {X}}{\partial r _ {1}} \times \frac {\partial \mathbf {X}}{\partial r _ {2}} \bigg | W _ {N},
where A _ { r } is the cross-sectional area of each rebar, S _ { r } is the rebar spacing, W _ { N } is the Gauss weighting associated with the integration point, X is the position of the Gauss point, and
\frac {\partial \mathbf {X}}{\partial r _ {\alpha}} = \frac {\partial \mathbf {X}}{\partial g _ {i}} \frac {\partial g _ {i}}{\partial r _ {\alpha}}.
In these expressions all quantities are taken in the reference configuration, and so ABAQUS ignores changes in rebar cross-sectional area due to straining of the rebar and changes in the rebar spacing due to straining of the finite element in which the rebar is placed.
The strain in the rebar is
\varepsilon = \frac {1}{2} \ln \left(\frac {g}{G}\right),
where
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g = \frac {\partial \mathbf {x}}{\partial t} \cdot \frac {\partial \mathbf {x}}{\partial t} \qquad \text {and} \qquad \frac {\partial \mathbf {x}}{\partial t} = \frac {\partial \mathbf {x}}{\partial r _ {i}} \frac {\partial r _ {i}}{\partial t},
and G is the value of g in the original configuration.
For convenience we define s, a material coordinate that is distance measuring along the rebar in the current configuration:
d s = \sqrt {g} d t.
The first variation of strain is
\delta \varepsilon = \frac {\partial \mathbf {x}}{\partial s} \cdot \frac {\partial \delta \mathbf {u}}{\partial s},
and the second variation of strain is
d \delta \varepsilon = \frac {\partial \delta \mathbf {u}}{\partial s} \cdot \frac {\partial d \mathbf {u}}{\partial s} - 2 \frac {\partial \delta \mathbf {u}}{\partial s} \cdot \frac {\partial \mathbf {x}}{\partial s} \frac {\partial \mathbf {x}}{\partial s} \cdot \frac {\partial d \mathbf {u}}{\partial s}.
3.7.3 Rebar modeling in shell and membrane elements
The definition of rebar in shell or membrane elements is based on three geometric properties: the cross-sectional area of each individual rebar, the spacing between the rebars, and the orientation of the rebar with respect to the local coordinate system of the element. For shell elements the rebar definition also requires the distance from the midsurface to the rebar. In ABAQUS an equivalent "smeared" orthotropic layer is created based on these geometric properties and the elastic modulus of the rebar material. The equivalent rebar layer lies parallel to the midsurface of the element. For membrane elements this layer coincides with the plane of the element, and for shell elements this layer can be offset by an amount up to half of the shell's thickness. In geometrically linear analyses the geometric properties of the equivalent rebar layer remain constant. However, in geometrically nonlinear analyses each of these properties can change as a result of finite-strain effects.
The user has many options for defining which direction the rebar acts in the element. In each case an angle £ is determined between the reinforcement and one of the element's isoparametric coordinate directions (selected by the user), measured positive with the axis of rotation along the normal to the element. Let the unit vector T define the rebar direction at a point in the element. The isoparametric directions are given by the tangent vectors \mathbf { A } _ { \alpha } defined
\mathbf {A} _ {\alpha} = \frac {\partial \mathbf {X}}{\partial \xi^ {\alpha}} = \frac {\partial N ^ {A}}{\partial \xi^ {\alpha}} \mathbf {X} _ {A},
where X is the reference midsurface position, \xi ^ { \alpha } are the isoparametric coordinate functions ( \alpha = 1 or 2), N ^ { A } are the element's shape functions, and \mathbf { X } _ { A } are the element's reference nodal positions.
The reference or initial rebar angle, £, is calculated from the inner product between the rebar unit
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vector, T, and the user-selected isoparametric direction, \mathbf { A } _ { \alpha } , where ® is given as the value of the ISODIRECTION parameter on the *REBAR option:
\Theta = \cos^ {- 1} \left\{\frac {< \mathbf {T} , \mathbf {A} _ {\alpha} >}{\| \mathbf {A} _ {\alpha} \|} \right\}, \quad \mathrm{nosumon} \alpha .
Both the rebar direction, T, and the user-selected isoparametric direction, \mathbf { A } _ { \alpha } , lie in a tangent plane parallel to the midsurface. The in-plane unit vector perpendicular to the rebar direction, P, is defined by rotating T through 9 0 ^ { \circ } around the normal to the midsurface, N. The normal direction, N, is found as
\mathbf {N} = \frac {\mathbf {A} _ {1} \times \mathbf {A} _ {2}}{\| \mathbf {A} _ {1} \times \mathbf {A} _ {2} \|}.
As the rebar-reinforced element deforms, the rebars change in length and spacing. The smeared rebar layer assumption implies that the deformation of the rebar layer is determined from the deformation gradient F of the underlying element. Following from this assumption, the rebar stretch \lambda _ { r } is
\lambda_ {r} = \left\| \mathbf {F} \cdot \mathbf {T} \right\| = \left\| \mathbf {t} \right\|,
where \mathbf { t } = \mathbf { F } \cdot \mathbf { T } is the deformed rebar material fiber. Since the deformation gradient maps material lines that are etched into the reference body into the deformed configuration, the length change of these material lines defines the stretch. The rebar logarithmic strain, \varepsilon _ { r } , , is
\varepsilon_ {r} = \ln \lambda_ {r} = \ln \| \mathbf {t} \|.
The rebar spacing stretch \lambda _ { p } is the stretch in the plane of the rebar in the direction perpendicular to the rebar. To determine the spacing stretch \lambda _ { p } , use the fact that the unit normal, p, perpendicular to the deformed rebar direction, t, in the plane of the rebar is
\mathbf {p} = \frac {\mathbf {F} ^ {- T} \cdot \mathbf {P}}{\| \mathbf {F} ^ {- T} \cdot \mathbf {P} \|}.
It is easily verified that p is a unit vector. To see that it is perpendicular to t, take the inner product:
< \mathbf {p}, \mathbf {t} > = \frac {1}{\| \mathbf {F} ^ {- T} \cdot \mathbf {P} \|} < \mathbf {F} ^ {- T} \cdot \mathbf {P}, \mathbf {F} \cdot \mathbf {T} > = \frac {1}{\| \mathbf {F} ^ {- T} \cdot \mathbf {P} \|} < \mathbf {P}, \mathbf {T} > = 0.
The spacing stretch, \lambda _ { p } { } _ { : } , can be defined as the component along p of the deformation of the direction P perpendicular to the reference rebar direction. Since the deformation of P is F ¢ P, the spacing stretch follows from
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\lambda_ {p} = < \mathbf {p}, \mathbf {F} \cdot \mathbf {P} > = \frac {1}{\| \mathbf {F} ^ {- T} \cdot \mathbf {P} \|} < \mathbf {F} ^ {- T} \cdot \mathbf {P}, \mathbf {F} \cdot \mathbf {P} > = \frac {< \mathbf {P} , \mathbf {P} >}{\| \mathbf {F} ^ {- T} \cdot \mathbf {P} \|}.
Since P is a unit vector,
\lambda_ {p} = \frac {1}{\| \mathbf {F} ^ {- T} \cdot \mathbf {P} \|}.
The final angle µ that the rebar direction makes with respect to the user-selected isoparametric direction is
\theta = \cos^ {- 1} \left\{\frac {< \mathbf {t} , \mathbf {a} _ {\alpha} >}{\| \mathbf {t} \| \| \mathbf {a} _ {\alpha} \|} \right\}, \quad \mathrm{nosumon} \alpha .
The rebar rotation or change in rebar angle, ¢µ, is the difference between the final angle and the original angle:
\Delta \theta = \theta - \Theta .
ABAQUS reports the current angle, µ, and the change in the rebar angle, ¢µ, for each rebar definition at each integration location of the element.
The equivalent thickness of the smeared layer is equal to the area of the rebar divided by the rebar's spacing; ABAQUS assumes that the volume of the rebar remains constant throughout the analysis. This assumption implies that the area and spacing of the rebar may change as a result of finite-strain effects. The rebar's area and spacing in the deformed configuration are defined as follows:
A _ {r} = \frac {A _ {r} ^ {0}}{\lambda_ {r}} \quad \mathrm{and} \quad S _ {r} = S _ {r} ^ {0} \lambda_ {p},
where
\begin{array}{l} A _ {r} ^ {0} = \text { original rebar cross - sectional area and } \\ S _ {r} ^ {0} = \text { original spacing of rebar. } \\ \end{array}
In shell elements the rebar layer can be defined initially at a distance above or below the midsurface. In shell elements that permit finite strain, the shell's thickness can change as a function of the in-plane deformation. In ABAQUS/Standard this behavior is defined with the POISSON parameter on the *SHELL SECTION option. In ABAQUS/Explicit this behavior is based on the actual material properties through the shell's thickness. To account for the change in the shell's thickness, the rebar layer's distance from the midsurface is scaled by the thickness stretch.
3.8 Hydrostatic fluid elements
3.8.1 Hydrostatic fluid elements
ABAQUS includes a family of elements that can be used to represent fluid-filled cavities under hydrostatic conditions. These elements provide the coupling between the deformation of the fluid-filled structure and the pressure exerted by the contained fluid on the boundary of the cavity. In ABAQUS/Explicit the fluid must be compressible and the pressure is calculated from the cavity volume. In ABAQUS/Standard the fluid inside the cavity can be compressible or incompressible, with the fluid volume given as a function of the fluid pressure, p ; the fluid temperature, \theta ; and the fluid mass, m, in the cavity:
\overline {{V}} = \overline {{V}} (p, \theta , m).
We refer to the incompressible case as a "hydraulic" fluid and to the compressible case as a "pneumatic" fluid. The volume, { \overline { { V } } } _ { ; } , derived from the fluid pressure and temperature should equal the actual volume, V , of the cavity. In ABAQUS/Standard this is achieved by augmenting the virtual work expression for the structure with the constraint equation
V - \overline {{V}} = 0
and the virtual work contribution due to the cavity pressure:
\delta \Pi^ {*} = \delta \Pi - p \delta V - \delta p (V - \overline {{V}}),
where \delta \Pi ^ { * } is the augmented virtual work expression and ±¦ is the virtual work expression for the structure without the cavity. The negative signs imply that an increase in the cavity volume releases energy from the fluid. This represents a mixed formulation in which the structural displacements and fluid pressure are primary variables. The rate of the augmented virtual work expression is obtained as
d \delta \Pi^ {*} = d \delta \Pi - p d \delta V - d p \delta V - (d V - d \overline {{V}}) \delta p
= d \delta \Pi - p d \delta V - d p \delta V - d V \delta p + \frac {d \overline {{V}}}{d p} d p \delta p.
Here, - p d \delta V represents the pressure load stiffness, and d \overline { { V } } / d p is the volume-pressure compliance of the fluid.
Since the pressure is the same for all elements in the cavity, the augmented virtual work expression can be written as the sum of the expressions for the individual elements:
\delta \Pi^ {*} = \delta \Pi - p \sum_ {e} \delta V ^ {e} - \delta p \biggl [ \sum_ {e} V ^ {e} - \sum_ {e} \overline {{V}} ^ {e} \biggr ]
= \sum_ {e} \left[ \delta \Pi^ {e} - p \delta V ^ {e} - \delta p (V ^ {e} - \overline {{V}} ^ {e}) \right].
Moreover, since the temperature is the same for all elements in the cavity, the fluid volume can be
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calculated for each element individually:
\overline {{V}} ^ {e} = \overline {{V}} ^ {e} (p, \theta , m ^ {e}),
where m ^ { e } is the element mass. Note that in the solution, the actual volume of the element may be different from the element volume:
V ^ {e} - \overline {{V}} ^ {e} \neq 0.
The total fluid volume will match the volume of the cavity, however.
Hydraulic fluid with thermal expansion
In ABAQUS/Standard the fluid is incompressible by default and the fluid volume, { \overline { { V } } } , is dependent upon temperature but independent of the fluid pressure:
\overline {{V}} = \overline {{V}} (\theta , m), \qquad \frac {d \overline {{V}}}{d p} = 0.
If compressibility is introduced, the fluid volume depends upon both the temperature and pressure:
\overline {{V}} = \overline {{V}} (p, \theta , m), \quad \frac {d \overline {{V}}}{d p} = - \frac {m}{\rho_ {R} K},
where K is the fluid bulk modulus and \rho _ { R } is the reference fluid density at zero pressure and the initial temperature.
The total fluid mass in the cavity is the sum of the fluid masses of the elements making up the cavity:
m = \sum_ {e} m ^ {e}.
The mass of a fluid element in the cavity, m ^ { e } , is calculated from the initial fluid density, \rho ( p _ { I } , \theta _ { I } ) , and the initial element volume, V _ { I } ^ { e } :
m ^ {e} = \rho (p _ {I}, \theta_ {I}) V _ {I} ^ {e},
where p _ { I } is the initial fluid pressure and \theta _ { I } is the initial temperature. The initial fluid density follows from the user-defined reference density, \rho _ { R } . :
\rho (p _ {I}, \theta_ {I}) = \frac {\rho_ {R}}{(1 - p _ {I} / K)}.
The fluid density at the current pressure and temperature, \rho ( p , \theta ) , is obtained as
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\rho (p, \theta) = \rho_ {R} \left[ 1 + 3 \alpha (\theta) \left(\theta - \theta_ {0}\right) - 3 \alpha \left(\theta_ {I}\right) \left(\theta_ {I} - \theta_ {0}\right) - p / K \right] ^ {- 1},
where \theta _ { 0 } is the reference temperature for the coefficient of thermal expansion and \alpha ( \theta ) is the mean (secant) coefficient of thermal expansion, and it is assumed that | \rho / \rho _ { R } - 1 | < < 1 .
Thus, the fluid volume at the current pressure and temperature is
\overline {{V}} (p, \theta) = m / \rho (p, \theta).
This volume can be calculated on an element by element basis:
\overline {{V}} (p, \theta) = \sum_ {e} \overline {{V}} ^ {e} (p, \theta) = \sum_ {e} m ^ {e} / \rho (p, \theta) = m / \rho (p, \theta).
Fluid can be added to or removed from the cavity. The amount of fluid added is given as the change in (fluid) mass \Delta m . Consequently, the change of the fluid volume at the current cavity temperature is
\Delta \overline {{{V}}} (p, \theta) = \Delta m / \rho (p, \theta).
Ideal gas
In this case the fluid is compressible, and the volume is a function of the pressure and the temperature in the cavity:
\overline {{V}} = \overline {{V}} (p, \theta , m),
where, as before, the total fluid mass in the cavity is the sum of the masses of the elements in the cavity. The fluid is assumed to behave like an ideal gas; hence, the density of the fluid in the cavity can be calculated as
\rho (p, \theta) = \rho_ {R} \frac {(\theta_ {R} - \theta_ {A}) (p + p _ {A})}{(\theta - \theta_ {A}) (p _ {R} + p _ {A})},
where \theta _ { R } and p _ { R } are the temperature and pressure at the reference density \rho _ { R } \mathrm { ~ , ~ } \theta _ { A } is the temperature at absolute zero, and p _ { A } is the ambient pressure. The current fluid volume can again be calculated on an element by element basis:
\overline {{V}} (p, \theta) = \sum_ {e} \overline {{V}} ^ {e} (p, \theta) = \sum_ {e} m ^ {e} / \rho (p, \theta) = m / \rho (p, \theta).
The corresponding volume-pressure compliance is
\frac {d \overline {{V}}}{d p} = - \frac {m}{\rho^ {2}} \frac {d \rho}{d p} = - \frac {m}{\rho_ {R}} \frac {(\theta - \theta_ {A}) (p _ {R} + p _ {A})}{(\theta_ {R} - \theta_ {A}) (p + p _ {A}) ^ {2}}.
Fluid can be added to or removed from the cavity. The amount of fluid added is again given as the change in (fluid) mass. Consequently, the change of the fluid volume at the current cavity temperature is
\Delta \overline {{V}} (p, \theta) = \Delta m / \rho (p, \theta).
Volume calculation
The hydrostatic fluid elements appear as surface elements that cover the cavity boundary, but they are actually volume elements when the cavity reference node is accounted for. Figure 3.8.1-1 depicts the 4-node hydrostatic fluid volume element, F3D4. The dashed lines indicate that the element is actually pyramidal in shape.
Figure 3.8.1-1 F3D4 hydrostatic fluid element.
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cavity reference node 1 2 3 4
The volume, V ^ { e } , of each element must be calculated. The coordinates of any point on the base of the pyramid element can be found by
\mathbf {x} = \sum_ {N} N ^ {N} (g, h) \mathbf {x} ^ {N},
where N ^ { N } are the interpolation functions for the base of the pyramid ( ``Solid isoparametric quadrilaterals and hexahedra,'' Section 3.2.4), expressed in terms of parametric coordinates g and h ; ; \mathbf { x } ^ { N } are the nodal coordinates; and the summation extends over all nodes on the base. For three-dimensional elements the Jacobian on the surface is calculated as
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\frac {\partial \mathbf {x}}{\partial g} = \sum_ {N} \frac {\partial N ^ {N}}{\partial g} \mathbf {x} ^ {N}, \quad \frac {\partial \mathbf {x}}{\partial h} = \sum_ {N} \frac {\partial N ^ {N}}{\partial h} \mathbf {x} ^ {N}.
The normal to the element face, n, multiplied by an infinitesimal area, dA, of the element face is, hence, obtained as
\mathbf {n} d A = \left(\frac {\partial \mathbf {x}}{\partial g} \times \frac {\partial \mathbf {x}}{\partial h}\right) d g d h.
The infinitesimal volume, dV , associated with this infinitesimal area is
d V = \frac {1}{3} \left(\mathbf {x} _ {R} - \mathbf {x}\right) \cdot \mathbf {n} d A,
where \mathbf { x } _ { R } is the position of the cavity reference node. The volume of the element, V ^ { e } , is then obtained by integration. For a quadrilateral base this yields
V ^ {e} = \int_ {V ^ {e}} d V = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} \frac {1}{3} \left(\mathbf {x} _ {R} - \mathbf {x}\right) \cdot \left(\frac {\partial \mathbf {x}}{\partial g} \times \frac {\partial \mathbf {x}}{\partial h}\right) d g d h.
The integration boundaries will be different for a triangular base. Introducing the relative position, \overline { { \mathbf { x } } } = \mathbf { x } - \mathbf { x } _ { R } , this becomes
V ^ {e} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} - \frac {1}{3} \overline {{\mathbf {x}}} \cdot \left(\frac {\partial \overline {{\mathbf {x}}}}{\partial g} \times \frac {\partial \overline {{\mathbf {x}}}}{\partial h}\right) d g d h.
The variation in the element volume is readily obtained as
\delta V ^ {e} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} - \frac {1}{3} \left[ \delta \overline {{{\bf x}}} \cdot \left(\frac {\partial \overline {{{\bf x}}}}{\partial g} \times \frac {\partial \overline {{{\bf x}}}}{\partial h}\right) + \overline {{{\bf x}}} \cdot \left(\frac {\partial \delta \overline {{{\bf x}}}}{\partial g} \times \frac {\partial \overline {{{\bf x}}}}{\partial h} + \frac {\partial \overline {{{\bf x}}}}{\partial g} \times \frac {\partial \delta \overline {{{\bf x}}}}{\partial h}\right) \right] d g d h.
This expression includes contributions to the volume change due to variation in the "sides" of the pyramid element. Hence, the equivalent forces will also include the effect of the pressure on these sides. The pressure on the sides will be balanced by the pressure on the side of the adjacent pyramid element; hence, we only need to calculate the contribution from the pressure on the "base" of the pyramid. This can be done by separating the contributions by using partial integration, which yields
\begin{array}{l} \delta V _ {b} ^ {e} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} - \delta \overline {{{\mathbf {x}}}} \cdot \left(\frac {\partial \overline {{{\mathbf {x}}}}}{\partial g} \times \frac {\partial \overline {{{\mathbf {x}}}}}{\partial h}\right) d g d h - \int_ {- 1} ^ {+ 1} \left[ \overline {{{\mathbf {x}}}} \cdot \left(\delta \overline {{{\mathbf {x}}}} \times \frac {\partial \overline {{{\mathbf {x}}}}}{\partial h} + \overline {{{\mathbf {x}}}} \times \frac {\partial \delta \overline {{{\mathbf {x}}}}}{\partial h}\right) \right] _ {g = - 1} ^ {g = 1} d h \\ - \int_ {- 1} ^ {+ 1} \left[ \overline {{{\mathbf {x}}}} \cdot \left(\frac {\partial \delta \overline {{{\mathbf {x}}}}}{\partial g} \times \overline {{{\mathbf {x}}}} + \frac {\partial \overline {{{\mathbf {x}}}}}{\partial g} \times \delta \overline {{{\mathbf {x}}}}\right) \right] _ {h = - 1} ^ {h = 1} d g. \\ \end{array}
The last two integrals represent the contributions on the sides of the pyramid; hence, the resulting expression is
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\delta V _ {b} ^ {e} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} - \delta \overline {{\mathbf {x}}} \cdot \left(\frac {\partial \overline {{\mathbf {x}}}}{\partial g} \times \frac {\partial \overline {{\mathbf {x}}}}{\partial h}\right) d g d h.
This equation can readily be obtained directly (see ``Pressure load stiffness for continuum elements,'' Section 3.2.10).
The second variation of the expression for the volume is
d \delta V _ {b} ^ {e} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} - \frac {1}{3} \Bigg [ \delta \overline {{{{\mathbf {x}}}}} \cdot \left(\frac {\partial d \overline {{{{\mathbf {x}}}}}}{\partial g} \times \frac {\partial \overline {{{{\mathbf {x}}}}}}{\partial h} + \frac {\partial \overline {{{{\mathbf {x}}}}}}{\partial g} \times \frac {\partial d \overline {{{{\mathbf {x}}}}}}{\partial h}\right) + d \overline {{{{\mathbf {x}}}}} \cdot \left(\frac {\partial \delta \overline {{{{\mathbf {x}}}}}}{\partial g} \times \frac {\partial \overline {{{{\mathbf {x}}}}}}{\partial h} + \frac {\partial \overline {{{{\mathbf {x}}}}}}{\partial g} \times \frac {\partial \delta \overline {{{{\mathbf {x}}}}}}{\partial h}\right) +
\overline {{{\mathbf {x}}}} \cdot \left(\frac {\partial \delta \overline {{{\mathbf {x}}}}}{\partial g} \times \frac {\partial d \overline {{{\mathbf {x}}}}}{\partial h} + \frac {\partial d \overline {{{\mathbf {x}}}}}{\partial g} \times \frac {\partial \delta \overline {{{\mathbf {x}}}}}{\partial h}\right) \Biggr ] d g d h.
Similar expressions can be obtained for planar and axisymmetric fluid elements. For a planar element the volume is readily obtained as
V ^ {e} = - \frac {1}{2} \mathbf {e} _ {3} \cdot \int_ {- 1} ^ {+ 1} \overline {{\mathbf {x}}} \times \frac {\partial \overline {{\mathbf {x}}}}{\partial g} d g,
where \mathbf { e } _ { 3 } is the unit vector in the direction perpendicular to the plane. Similarly, for axisymmetric elements
V ^ {e} = - \frac {2}{3} \pi \mathbf {e} _ {\theta} \cdot \int_ {- 1} ^ {+ 1} \left(\overline {{\mathbf {x}}} \times \frac {\partial \overline {{\mathbf {x}}}}{\partial g}\right) \left(\overline {{\mathbf {x}}} \cdot \mathbf {e} _ {r}\right) d g.
The first and second variations are obtained in the same way as for three-dimensional elements.
The integrations can be carried out analytically. For instance, for element type F3D4 the above expressions yield, after some manipulation,
V ^ {e} = \frac {1}{1 2} \left[ (\overline {{\mathbf {x}}} ^ {3} - \overline {{\mathbf {x}}} ^ {1}) \cdot (\overline {{\mathbf {x}}} ^ {2} \times \overline {{\mathbf {x}}} ^ {4}) + (\overline {{\mathbf {x}}} ^ {4} - \overline {{\mathbf {x}}} ^ {2}) \cdot (\overline {{\mathbf {x}}} ^ {3} \times \overline {{\mathbf {x}}} ^ {1}) \right],
where \overline { { \mathbf { x } } } ^ { N } , N = 1 ; 4 denote the relative nodal coordinates. The first variation (involving the base only) is equal to
\delta V _ {b} ^ {e} = \frac {1}{1 2} \Big [ \delta \overline {{{{\mathbf {x}}}}} ^ {1} \cdot [ (\overline {{{{\mathbf {x}}}}} ^ {2} - \overline {{{{\mathbf {x}}}}} ^ {4}) \times (2 \overline {{{{\mathbf {x}}}}} ^ {1} - \overline {{{{\mathbf {x}}}}} ^ {2} - \overline {{{{\mathbf {x}}}}} ^ {3}) ] + \delta \overline {{{{\mathbf {x}}}}} ^ {2} \cdot [ (\overline {{{{\mathbf {x}}}}} ^ {3} - \overline {{{{\mathbf {x}}}}} ^ {1}) \times (2 \overline {{{{\mathbf {x}}}}} ^ {2} - \overline {{{{\mathbf {x}}}}} ^ {3} - \overline {{{{\mathbf {x}}}}} ^ {4}) ] +
\delta \overline {{{\mathbf {x}}}} ^ {3} \cdot [ (\overline {{{\mathbf {x}}}} ^ {4} - \overline {{{\mathbf {x}}}} ^ {2}) \times (2 \overline {{{\mathbf {x}}}} ^ {3} - \overline {{{\mathbf {x}}}} ^ {4} - \overline {{{\mathbf {x}}}} ^ {1}) ] + \delta \overline {{{\mathbf {x}}}} ^ {4} \cdot [ (\overline {{{\mathbf {x}}}} ^ {1} - \overline {{{\mathbf {x}}}} ^ {3}) \times (2 \overline {{{\mathbf {x}}}} ^ {4} - \overline {{{\mathbf {x}}}} ^ {1} - \overline {{{\mathbf {x}}}} ^ {2}) ] \Big ],
and the second variation of the volume is