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Elements
\begin{array}{l} \varepsilon^ {f} d L ^ {f} \approx \left[ \varepsilon + g G ^ {- 1} \left(\frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {n} _ {1}}{d S} - (1 + \varepsilon) \frac {d \mathbf {X}}{d S} \cdot \frac {d \mathbf {N} _ {1}}{d S}\right) \right. \\ + \left. h G ^ {- 1} \left(\frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {n} _ {2}}{d S} - (1 + \varepsilon) \frac {d \mathbf {X}}{d S} \cdot \frac {d \mathbf {N} _ {2}}{d S}\right) \right] d L, \\ \end{array}
where
G = \frac {d \mathbf {X}}{d S} \cdot \frac {d \mathbf {X}}{d S} \mathrm{and} \varepsilon = \frac {1}{2} (\lambda^ {2} - 1) = \frac {1}{2} \left[ \left(\frac {d l}{d L}\right) ^ {2} - 1 \right] \mathrm {is the Green^ {\prime} s strain of the beam axis}.
This simplification allows us to write the internal virtual work rate associated with axial stress as
\delta W _ {1} ^ {I} = \int_ {L ^ {f}} \int_ {A} \sigma^ {f} \delta \varepsilon^ {f} d A d L ^ {f} = \int_ {L} \int_ {A} \sigma^ {f} \delta \tilde {\varepsilon} ^ {f} d A d L,
where
\tilde {\varepsilon} ^ {f} = \varepsilon - g K _ {2} + h K _ {1},
with
\varepsilon = \frac { 1 } { 2 } ( \lambda ^ { 2 } - 1 ) as Green's strain of the beam axis,
K _ {2} = - G ^ {- 1} \left(\frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {n} _ {1}}{d S} - (1 + \varepsilon) \frac {d \mathbf {X}}{d S} \cdot \frac {d \mathbf {N} _ {1}}{d S}\right)
and
K _ {1} = - G ^ {- 1} \left(\frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {n} _ {2}}{d S} - (1 + \varepsilon) \frac {d \mathbf {X}}{d S} \cdot \frac {d \mathbf {N} _ {2}}{d S}\right).
Now the cross-sectional base vectors \mathbf { n } _ { 1 } and \mathbf { n } _ { 2 } are assumed to remain normal to the beam axis, so
{\frac {d \mathbf {x}}{d S}} \cdot \mathbf {n} _ {1} = {\frac {d \mathbf {x}}{d S}} \cdot \mathbf {n} _ {2} = 0.
Hence,
\frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {n} _ {1}}{d S} = - \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \cdot \mathbf {n} _ {1}, \mathrm{and} \frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {n} _ {2}}{d S} = - \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \cdot \mathbf {n} _ {2}.
So we have
K _ {2} = G ^ {- 1} \left(\mathbf {n} _ {1} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} - (1 + \varepsilon) \mathbf {N} _ {1} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}}\right)
Elements
and
K _ {1} = - G ^ {- 1} \left(\mathbf {n} _ {2} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} - (1 + \varepsilon) \mathbf {N} _ {2} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}}\right).
This defines the generalized strains associated with axial stretch. For torsional strain the internal virtual work rate is
\delta W _ {2} ^ {I} = \int_ {L} \int_ {A} \tau^ {f} r \delta e _ {1} d A d L,
where, as in the shear beams above,
e _ {1} = \mathbf {n} _ {2} \cdot \frac {d \mathbf {n} _ {1}}{d S} - \mathbf {N} _ {2} \cdot \frac {d \mathbf {N} _ {1}}{d S} \mathrm{and} r = (g ^ {2} + h ^ {2}) ^ {\frac {1}{2}}.
For computational simplicity the form of the torsional strains is taken to be
e _ {1} = \mathbf {n} _ {2} \cdot \frac {d \tilde {\mathbf {n}} _ {1}}{d S} - \mathbf {N} _ {2} \cdot \frac {d \mathbf {N} _ {1}}{d S},
where
\tilde {\mathbf {n}} _ {1} = \mathbf {C} (p _ {1} \pmb {\omega} ^ {1} + p _ {2} \pmb {\omega} ^ {2}) \cdot \mathbf {N} _ {1},
and
p _ {1} = \frac {1}{2} (1 - g),
p _ {2} = \frac {1}{2} (1 + g),
- 1 \leq g \leq 1.
This assumes a linear interpolation of rotation ! along the beam. Thus, the generalized strain measures for these beams are
" axial strain,
K _ { 1 } and K _ { 2 } the beam curvature change measures, and
e _ { 1 } the torsional strain.
With these measures, the internal virtual work rate can be written
\delta W ^ {I} = \int_ {L} \left[ \delta \varepsilon \int_ {A} \sigma^ {f} d A - \delta K _ {2} \int_ {A} g \sigma^ {f} d A + \delta K _ {1} \int_ {A} h \sigma^ {f} d A + \delta e _ {1} \int_ {A} \tau^ {f} r d A \right] d L.
Internal virtual work rate Jacobian
For the Jacobian matrix of the overall Newton method, Equation 2.1.1-3, the variation of this internal work rate with respect to nodal displacement variations must be formed. Proceeding as in the shear beams above, the constitutive theory is written as
\left\{ \begin{array}{l} \partial \sigma^ {f} \\ \partial \tau^ {f} \end{array} \right\} = \left[ \begin{array}{l} H \end{array} \right] \left\{ \begin{array}{l} d \varepsilon^ {f} \\ d \gamma^ {f} \end{array} \right\},
and so
\partial \delta W ^ {I} = \int_ {L} \left[ \left\lfloor \delta \varepsilon \delta K _ {1} \delta K _ {2} \delta e _ {1} \right\rfloor [ B ] \left\{ \begin{array}{l} d \varepsilon \\ d K _ {1} \\ d K _ {2} \\ d e _ {1} \end{array} \right\} + \int_ {A} \sigma^ {f} d A d \varepsilon + \int_ {A} h \sigma^ {f} d A d \delta K _ {1} \right.
\left. - \int_ {A} g \sigma^ {f} d A d \delta K _ {2} + \int_ {A} r \tau^ {f} d A d \delta \varepsilon_ {1} \right] d L,
where [ \left[ B \right] is the same as for the shear beams.
First variations of strains
Taking the variations of the above strain definitions gives directly
\delta \varepsilon = G ^ {- 1} \frac {d \mathbf {x}}{d S} \cdot \frac {d \delta \mathbf {u}}{d S},
\delta K _ {2} = G ^ {- 1} \left(\delta \mathbf {n} _ {1} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} + \mathbf {n} _ {1} \cdot \frac {d ^ {2} \delta \mathbf {u}}{d S ^ {2}} - \delta \varepsilon \mathbf {N} _ {1} \cdot \frac {d ^ {2} \mathbf {X}}{d S ^ {2}}\right)
\delta K _ {1} = G ^ {- 1} \left(\delta \mathbf {n} _ {2} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} + \mathbf {n} _ {2} \cdot \frac {d ^ {2} \delta \mathbf {u}}{d S ^ {2}} - \delta \varepsilon \mathbf {N} _ {2} \cdot \frac {d ^ {2} \mathbf {X}}{d S ^ {2}}\right)
and
\delta e _ {1} = \delta \mathbf {n} _ {2} \cdot \frac {d \tilde {\mathbf {n}} _ {1}}{d S} + \mathbf {n} _ {2} \cdot \frac {d \delta \tilde {\mathbf {n}} _ {1}}{d S}.
In these expressions we need \delta \mathbf { n } _ { 1 } and \delta \mathbf { n } _ { 2 } as well as \delta \tilde { \mathbf { n } } _ { 1 } ; these are now derived. From the expression for \tilde { \mathbf { n } } _ { 1 } , namely
\tilde {\mathbf {n}} _ {1} = \mathbf {C} (p _ {1} \pmb {\omega} ^ {1} + p _ {2} \pmb {\omega} ^ {2}) \cdot \mathbf {N} _ {1},
another ancillary vector \overline { { \mathbf { n } } } _ { 1 } , normal to the tangent, is defined by
\overline {{\mathbf {n}}} _ {1} = \tilde {\mathbf {n}} _ {1} - \tilde {\mathbf {n}} _ {1} \cdot \mathbf {t t}
so that
\mathbf {n} _ {1} = \overline {{\mathbf {n}}} _ {1} / | \overline {{\mathbf {n}}} _ {1} |.
In addition,
\mathbf {t} = g ^ {- \frac {1}{2}} \frac {d \mathbf {x}}{d S}
so that
\delta \mathbf {t} = g ^ {- \frac {1}{2}} (\mathbf {I} - \mathbf {t t}) \cdot \frac {d \delta \mathbf {u}}{d S}.
Since \left( \mathbf { t } _ { 1 } ~ \mathbf { n } _ { 1 } ~ \mathbf { n } _ { 2 } \right) form an orthonormal triad,
Equation 3.5.3-2
\begin{array}{l} \delta \mathbf {n} _ {1} = \delta \mathbf {n} _ {1} \cdot \left(\mathbf {n} _ {2} \mathbf {n} _ {2} + \mathbf {t t}\right) \\ = \delta \mathbf {n} _ {1} \cdot \mathbf {n} _ {2} \mathbf {n} _ {2} + \delta \mathbf {n} _ {1} \cdot \mathbf {t t} \\ = \delta \mathbf {n} _ {1} \cdot \mathbf {n} _ {2} \mathbf {n} _ {2} - g ^ {- \frac {1}{2}} \mathbf {n} _ {1} \cdot \frac {d \delta \mathbf {u}}{d S} \mathbf {t}, \\ \end{array}
because \mathbf { n } _ { 1 } \cdot \mathbf { t } = 0 . From the definition of \mathbf { n } _ { 1 } , it is straightforward to show that
\delta \mathbf {n} _ {1} = \frac {1}{| \overline {{\mathbf {n}}} _ {1} |} (\mathbf {I} - \mathbf {n} _ {1} \mathbf {n} _ {1}) \cdot \delta \overline {{\mathbf {n}}}.
So
\begin{array}{l} \delta \mathbf {n} _ {1} \cdot \mathbf {n} _ {2} = \frac {1}{| \overline {{\mathbf {n}}} _ {1} |} \mathbf {n} _ {2} \cdot \delta \mathbf {n} _ {1} \\ = \frac {1}{| \overline {{\mathbf {n}}} _ {1} |} \mathbf {n} _ {2} \cdot \left[ \delta \tilde {\mathbf {n}} _ {1} - (\delta \tilde {\mathbf {n}} _ {1} \cdot \mathbf {t}) \mathbf {t} - (\tilde {\mathbf {n}} _ {1} \cdot \mathbf {t}) \delta \mathbf {t} \right] \\ = \frac {1}{| \overline {{\mathbf {n}}} _ {1} |} \left[ \mathbf {n} _ {2} \cdot \delta \tilde {\mathbf {n}} _ {1} - g ^ {- \frac {1}{2}} (\tilde {\mathbf {n}} _ {1} \cdot \mathbf {t}) \mathbf {n} _ {2} \cdot \frac {d \delta \mathbf {u}}{d S} \right] \\ \end{array}
and
\delta \tilde { \mathbf { n } } _ { 1 } = \delta \pmb { \omega } \times \tilde { \mathbf { n } } _ { 1 } with ! interpolated linearly.
Thus,
Elements
\delta \mathbf {n} _ {1} \cdot \mathbf {n} _ {2} = \frac {1}{| \overline {{\mathbf {n}}} _ {1} |} \left(\delta \pmb {\omega} \cdot \tilde {\mathbf {n}} _ {1} \times \mathbf {n} _ {2} - g ^ {- \frac {1}{2}} \tilde {\mathbf {n}} _ {1} \cdot \mathbf {t n} _ {2} \cdot \frac {d \delta \mathbf {u}}{d S}\right).
We can also write
\tilde {\mathbf {n}} _ {1} \times \mathbf {n} _ {2} = \tilde {\mathbf {n}} _ {1} \mathbf {t} - \tilde {\mathbf {n}} _ {1} \cdot \mathbf {t} \mathbf {n} _ {1}
and
\left| \overline {{\mathbf {n}}} _ {1} \right| = \overline {{\mathbf {n}}} _ {1} \cdot \mathbf {n} _ {1} = \tilde {\mathbf {n}} _ {1} \cdot \mathbf {n} _ {1}.
Combining terms appropriately,
\delta \mathbf {n} _ {1} \cdot \mathbf {n} _ {2} = \delta \pmb {\omega} \cdot \mathbf {t} - \frac {\tilde {\mathbf {n}} _ {1} \cdot \mathbf {t}}{\tilde {\mathbf {n}} _ {1} \cdot \mathbf {n} _ {1}} \left[ \delta \pmb {\omega} \cdot \mathbf {n} _ {1} + g ^ {- \frac {1}{2}} \mathbf {n} _ {2} \cdot \frac {d \delta \mathbf {u}}{d S} \right].
Hence, from Equation 3.5.3-2
\delta \mathbf {n} _ {1} = \delta \pmb {\omega} \cdot \mathbf {t} \mathbf {n} _ {2} - \frac {\tilde {\mathbf {n}} _ {1} \cdot \mathbf {t}}{\tilde {\mathbf {n}} _ {1} \cdot \mathbf {n} _ {1}} \left[ \delta \pmb {\omega} \cdot \mathbf {n} _ {1} + g ^ {- \frac {1}{2}} \mathbf {n} _ {2} \cdot \frac {d \delta \mathbf {u}}{d S} \right] \mathbf {n} _ {2} - g ^ {- \frac {1}{2}} \mathbf {n} _ {1} \cdot \frac {d \delta \mathbf {u}}{d S} \mathbf {t}.
In a similar manner one can show that
\delta \mathbf {n} _ {2} = - \delta \pmb {\omega} \cdot \mathbf {t} \mathbf {n} _ {1} - \frac {\tilde {\mathbf {n}} _ {1} \cdot \mathbf {t}}{\tilde {\mathbf {n}} _ {1} \cdot \mathbf {n} _ {1}} \left[ \delta \pmb {\omega} \cdot \mathbf {n} _ {1} + g ^ {- \frac {1}{2}} \mathbf {n} _ {1} \cdot \frac {d \delta \mathbf {u}}{d S} \right] \mathbf {n} _ {1} - g ^ {- \frac {1}{2}} \mathbf {n} _ {2} \cdot \frac {d \delta \mathbf {u}}{d S} \mathbf {t}.
The first variations of strain become
\delta \varepsilon = G ^ {- 1} \frac {d \mathbf {x}}{d S} \cdot \frac {d \delta \mathbf {u}}{d S}
\begin{array}{l} \delta K _ {2} = G ^ {- 1} \left[ \mathbf {n} _ {2} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \left(\mathbf {t} - \frac {\tilde {\mathbf {n}} _ {1} \cdot \mathbf {t}}{\tilde {\mathbf {n}} _ {1} \cdot \mathbf {n} _ {1}} \mathbf {n} _ {1}\right) \cdot \delta \pmb {\omega} \right. \\ - g ^ {- \frac {1}{2}} \bigg (\mathbf {t} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \mathbf {n} _ {1} + \frac {\tilde {\mathbf {n}} _ {1} \cdot \mathbf {t}}{\tilde {\mathbf {n}} _ {1} \cdot \mathbf {n} _ {1}} \mathbf {n} _ {2} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \bigg) \mathbf {n} _ {2} \cdot \frac {d \delta \mathbf {u}}{d S} \\ \left. + \mathbf {n} _ {1} \frac {d ^ {2} \delta \mathbf {u}}{d S ^ {2}} - \delta \varepsilon \mathbf {N} _ {1} \cdot \frac {d ^ {2} \mathbf {X}}{d S ^ {2}} \right] \\ \end{array}
\delta K _ {1} = G ^ {- 1} \left[ \mathbf {n} _ {1} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \left(\mathbf {t} - \frac {\tilde {\mathbf {n}} _ {1} \cdot \mathbf {t}}{\tilde {\mathbf {n}} _ {1} \cdot \mathbf {n} _ {1}} \mathbf {n} _ {1}\right) \cdot \delta \boldsymbol {\omega} \right.
+ g ^ {- \frac {1}{2}} \left(\mathbf {t} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \mathbf {n} _ {1} + \frac {\tilde {\mathbf {n}} _ {1} \cdot \mathbf {t}}{\tilde {\mathbf {n}} _ {1} \cdot \mathbf {n} _ {1}} \mathbf {n} _ {1} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}}\right) \mathbf {n} _ {2} \cdot \frac {d \delta \mathbf {u}}{d S}
- \left. \mathbf {n} _ {2} \frac {d ^ {2} \delta \mathbf {u}}{d S ^ {2}} - \delta \varepsilon \mathbf {N} _ {2} \cdot \frac {d ^ {2} \mathbf {X}}{d S ^ {2}} \right]
and
\delta e _ {1} = \left(\frac {d \tilde {\bf n} _ {1}}{d S} \cdot {\bf n} _ {1} \frac {\tilde {\bf n} _ {1} \cdot {\bf t}}{\tilde {\bf n} _ {1} \cdot {\bf n} _ {1}} - \frac {d \tilde {\bf n} _ {1}}{d S} {\bf t}\right) {\bf n} _ {1} \cdot \delta {\pmb \omega}
+ g ^ {- \frac {1}{2}} \left[ \frac {\tilde {\mathbf {n}} _ {1} \cdot \mathbf {t}}{\tilde {\mathbf {n}} _ {1} \cdot \mathbf {n} _ {1}} \frac {d \tilde {\mathbf {n}} _ {1}}{d S} \cdot \mathbf {n} _ {1} - \frac {d \tilde {\mathbf {n}} _ {1}}{d S} \cdot \mathbf {t} \right] \mathbf {n} _ {2} \cdot \frac {d \delta \mathbf {u}}{d S}
+ \tilde {\mathbf {n}} _ {1} \times \mathbf {n} _ {2} \cdot \frac {d \delta \pmb {\omega}}{d S}
so that
\mathbf {n} _ {1} = \overline {{\mathbf {n}}} _ {1} / | \overline {{\mathbf {n}}} _ {1} |.
In addition,
\mathbf {t} = g ^ {- \frac {1}{2}} \frac {d \mathbf {x}}{d S}
so that
\delta \mathbf {t} = g ^ {- \frac {1}{2}} (\mathbf {I} - \mathbf {t t}) \cdot \frac {d \delta \mathbf {u}}{d S},
since \left( \mathbf { t } _ { 1 } ~ \mathbf { n } _ { 1 } ~ \mathbf { n } _ { 2 } \right) form an orthonormal triad,
\delta \mathbf {n} _ {1} = \delta \mathbf {n} _ {1} \cdot \left(\mathbf {n} _ {2} \mathbf {n} _ {2} + \mathbf {t t}\right)
= \delta \mathbf {n} _ {1} \cdot \mathbf {n} _ {2} \mathbf {n} _ {2} + \delta \mathbf {n} _ {1} \cdot \mathbf {t t}
= \delta \mathbf {n} _ {1} \cdot \mathbf {n} _ {2} \mathbf {n} _ {2} - g ^ {- \frac {1}{2}} \mathbf {n} _ {1} \cdot \frac {d \delta \mathbf {u}}{d S} \mathbf {t},
because
\mathbf {n} _ {1} \cdot \mathbf {t} = 0.
Second variations of strains
The second variation of the axial strain is simply
d \delta \varepsilon = G ^ {- 1} \frac {d \delta \mathbf {u}}{d S} \cdot \frac {d d \mathbf {u}}{d S}.
To compute the second variations of bending strain, we need expressions for d \delta \mathbf { n } _ { 1 } ; d \delta \mathbf { n } _ { 2 } . These are obtained by approximating
\begin{array}{l} \tilde {\mathbf {n}} _ {1} \approx \mathbf {n} _ {2} \mathbf {t} \cdot \delta \pmb {\omega} - \mathbf {t} \mathbf {n} _ {1} \cdot \frac {d \delta \mathbf {u}}{d S} \\ \tilde {\mathbf {n}} _ {2} \approx - \mathbf {n} _ {1} \mathbf {t} \cdot \delta \boldsymbol {\omega} - \mathbf {t n} _ {1} \cdot \frac {d \delta \mathbf {u}}{d S} \\ \tilde {\mathbf {t}} \approx \delta \pmb {\omega} \cdot \mathbf {n} _ {2} - \delta \pmb {\omega} \cdot \mathbf {n} _ {1} \mathbf {n} _ {2} \\ \end{array}
from which
d \tilde {\mathbf {n}} _ {1} = d \mathbf {n} _ {2} \mathbf {t} \cdot \delta \pmb {\omega} + \mathbf {n} _ {2} d \mathbf {t} \cdot \delta \pmb {\omega} - d \mathbf {t} \mathbf {n} _ {1} \cdot \frac {d \delta \mathbf {u}}{d S} - \mathbf {t} d \mathbf {n} _ {1} \cdot \frac {d \delta \mathbf {u}}{d S}.
Using these expressions, the second variations of bending strains are written as
\begin{array}{l} d \delta K _ {1} = G ^ {- 1} \left[ \mathbf {n} _ {2} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \mathbf {t} \cdot d \pmb {\omega} \mathbf {t} \cdot \delta \pmb {\omega} \right. \\ - \mathbf {t} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \left(\mathbf {n} _ {1} \cdot \frac {d d \mathbf {u}}{d S} \mathbf {t} \cdot \delta \boldsymbol {\omega} + \mathbf {n} _ {1} \cdot \frac {d \delta \mathbf {u}}{d S} \mathbf {t} \cdot d \boldsymbol {\omega}\right) \\ + \mathbf {n} _ {1} \cdot \frac {d ^ {2} d \mathbf {u}}{d S ^ {2}} \mathbf {t} \cdot \delta \boldsymbol {\omega} + \mathbf {t} \cdot \frac {d ^ {2} d \mathbf {u}}{d S ^ {2}} \mathbf {n} _ {2} \cdot \frac {d \delta \mathbf {u}}{d S} \\ + \mathbf {n} _ {1} \cdot \frac {d ^ {2} \delta \mathbf {u}}{d S ^ {2}} \mathbf {t} \cdot d \boldsymbol {\omega} + \mathbf {t} \cdot \frac {d ^ {2} \delta \mathbf {u}}{d S ^ {2}} \mathbf {n} _ {2} \cdot \frac {d d \mathbf {u}}{d S} \\ + d \delta \varepsilon \mathbf {N} _ {2} \cdot \frac {d ^ {2} \mathbf {X}}{d S ^ {2}} \\ + \mathbf {n} _ {1} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \left(\mathbf {n} _ {2} \cdot d \boldsymbol {\omega} \mathbf {n} _ {1} \cdot \delta \boldsymbol {\omega} - \mathbf {n} _ {1} \cdot d \boldsymbol {\omega} \mathbf {n} _ {2} \cdot \delta \boldsymbol {\omega} + \mathbf {n} _ {2} \cdot \frac {d \delta \mathbf {u}}{d S} \mathbf {n} _ {2} \cdot d \boldsymbol {\omega}\right) \\ - \mathbf {n} _ {2} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \left(\mathbf {n} _ {2} \cdot \frac {d \delta \mathbf {u}}{d S} \mathbf {n} _ {1} \cdot d \pmb {\omega}\right) \\ \left. - \mathbf {t} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \left(\mathbf {n} _ {2} \cdot \frac {d d \mathbf {u}}{d S} \mathbf {n} _ {2} \mathbf {t} \cdot \frac {d \delta \mathbf {u}}{d S}\right) \right] \\ \end{array}
and
Elements
\begin{array}{l} d \delta K _ {2} = G ^ {- 1} \left[ - \mathbf {n} _ {1} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \mathbf {t} \cdot \delta \boldsymbol {\omega} \mathbf {t} \cdot d \boldsymbol {\omega} \right. \\ - \mathbf {t} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \left(\mathbf {n} _ {2} \cdot \frac {d \delta \mathbf {u}}{d S} \mathbf {t} \cdot d \boldsymbol {\omega} + \mathbf {n} _ {2} \cdot \frac {d d \mathbf {u}}{d S} \mathbf {t} \cdot \delta \boldsymbol {\omega}\right) \\ + \mathbf {n} _ {2} \cdot \frac {d ^ {2} d \mathbf {u}}{d S ^ {2}} \mathbf {t} \cdot \delta \boldsymbol {\omega} - \mathbf {t} \cdot \frac {d ^ {2} d \mathbf {u}}{d S ^ {2}} \mathbf {n} _ {1} \cdot \frac {d \delta \mathbf {u}}{d S} \\ + \mathbf {n} _ {2} \cdot \frac {d ^ {2} \delta \mathbf {u}}{d S ^ {2}} \mathbf {t} \cdot d \boldsymbol {\omega} - \mathbf {t} \cdot \frac {d ^ {2} \delta \mathbf {u}}{d S ^ {2}} \mathbf {n} _ {1} \cdot \frac {d d \mathbf {u}}{d S} \\ - d \delta \varepsilon \mathbf {N} _ {1} \cdot \frac {d ^ {2} \mathbf {X}}{d S ^ {2}} \\ + \mathbf {n} _ {2} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \left(\mathbf {n} _ {2} \cdot d \boldsymbol {\omega} \mathbf {n} _ {1} \cdot \delta \boldsymbol {\omega} - \mathbf {n} _ {1} \cdot d \boldsymbol {\omega} \mathbf {n} _ {2} \cdot \delta \boldsymbol {\omega} + \mathbf {n} _ {1} \cdot \frac {d \delta \mathbf {u}}{d S} \mathbf {n} _ {1} \cdot d \boldsymbol {\omega}\right) \\ - \mathbf {n} _ {1} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \left(\mathbf {n} _ {1} \cdot \frac {d \delta \mathbf {u}}{d S} \mathbf {n} _ {2} \cdot d \boldsymbol {\omega}\right) \\ \left. - \mathbf {t} \cdot \frac {d ^ {2} \mathbf {x}}{d S ^ {2}} \left(\mathbf {n} _ {1} \cdot \frac {d d \mathbf {u}}{d S} \mathbf {n} _ {1} \mathbf {t} \cdot \frac {d \delta \mathbf {u}}{d S}\right) \right]. \\ \end{array}
For the torsional strain contribution to the initial stress matrix, we approximate
\delta e _ {2} \approx \mathbf {t} \cdot \frac {d \delta \pmb {\omega}}{d S}
d \delta e _ {1} \approx d \mathbf {t} \cdot \frac {d \delta \pmb {\omega}}{d S} = d \pmb {\omega} \times \mathbf {t} \cdot \frac {d \delta \pmb {\omega}}{d S}.
This matrix is again unsymmetric.
Interpolation
In the virtual work equation the strains include second derivatives of displacement. For this reason continuity of rotation as well as displacement is needed so that the Hermitian polynomial interpolation functions are the minimum interpolation order needed. These are used here. The Hermite cubic is written in terms of the function value and its derivative at the ends of the interval:
u (g) = N _ {1} (g) u ^ {1} + N _ {2} (g) u ^ {2} + N _ {3} (g) \frac {d u ^ {1}}{d g} + N _ {4} (g) \frac {d u ^ {2}}{d g}, - 1 < g < 1,
with node 1 at g = - 1 and node 2 at g = + 1 .
These functions are used in ABAQUS to interpolate the components of displacement u and the initial position vector x, so that the elements are basically isoparametric. In addition, rotation of \left( \mathbf { n } _ { 1 } ~ \mathbf { n } _ { 2 } \right) about the beam axis, \phi _ { a } , is interpolated linearly. This interpolation is unsatisfactory for the user, because the nodal variables are
(u _ {x} u _ {y} u _ {z} \frac {d u _ {x}}{d S} \frac {d u _ {y}}{d S} \frac {d u _ {z}}{d S} \phi_ {a}).
Elements
The last four of these variables are difficult to work with; furthermore, making them the same in all elements sharing the same node causes excessive constraint of axial stretch in these elements, especially if the beam axis is not continuous through the node, as in a frame structure or " \mathrm { T " } junction.
To avoid this difficulty, the following procedure is adopted. At a node the tangent to the beam axis is
\mathbf {t} = \frac {d \mathbf {x}}{d l} = \frac {d \mathbf {x}}{d S} \frac {d S}{d l},
so
{\frac {d \mathbf {x}}{d S}} = {\frac {d l}{d S}} \mathbf {t}.
Now suppose we store as degrees of freedom at the node,
(u _ {x}, u _ {y}, u _ {z}, \omega_ {x}, \omega_ {y}, \omega_ {z}, d l / d S),
where \pmb { \omega } is the rotation definition introduced above. Since the initial geometry and hence T, the initial direction of the beam axis, is known, \mathbf { t } = \mathbf { C } \cdot \mathbf { T } ; where C is the rotation matrix defined by \omega , and hence
Equation 3.5.3-3
\frac {d \mathbf {x}}{d S} = \frac {d l}{d S} \mathbf {C} \cdot \mathbf {T}
is defined by these variables and the initial geometry. Furthermore, \phi _ { a } is directly available from ! and t. Thus, the above set is a satisfactory set of nodal variables. To eliminate the unwanted axial strain constraint, in ABAQUS the stretch d l / d S at the node of each such element is taken as an internal variable, local to the element (a third internal node is created for this purpose, and so it is not shared with neighboring elements.)
It should be remarked that the above transformation (Equation 3.5.3-3) is nonlinear. This leads to some complications--for example, the d'Alembert forces no longer have the simple form M _ { M N } \ddot { \overline { { u } } } ^ { N } ; rather, a matrix \overline { { { M } } } _ { M N } = Q ^ { M P } M _ { P Q } Q ^ { Q N } replaces M _ { M N } where Q ^ { M P } and Q ^ { Q N } use the transformation (Equation 3.5.3-3) and its appropriate time derivatives. The resulting Jacobian is nonsymmetric; ABAQUS ignores the nonsymmetric terms.
Integration
The cross-section integration has already been discussed in the context of the shear beams--it is the same for these beams. Along the beam axis, the integration schemes are as described below.
Stiffness and internal forces
Three Gauss points are used. Two Gauss points are not sufficient because the torsional strain is not independent.
Elements
Mass and distributed loads
Three Gauss points are used. Rotary inertia is not included in the mass, except for rotation of the section about the beam axis, where it is included to avoid singularity in perfectly straight beams.
3.5.4 Hybrid beam elements
The hybrid beam elements in ABAQUS/Standard are designed to handle very slender situations, where the axial stiffness of the beam is very large compared to the bending stiffness; and so a mixed method, where axial force is treated as an independent unknown, is required. For the shear beams mixed elements are provided where the transverse shear forces are also treated as independent unknowns. This section discusses the basis of these mixed methods.
Axial and bending behavior
The internal virtual work of the beam can be written
\delta W _ {1} ^ {I} = \int_ {L} (N \delta \varepsilon + M _ {1} \delta K _ {1} + M _ {2} \delta K _ {2} + M _ {3} \delta e _ {1}) d L.
Alternatively, we can introduce an independent axial force variable, \tilde { N } , and write
\delta W _ {2} ^ {I} = \int_ {L} (\tilde {N} \delta \varepsilon + M _ {1} \delta K _ {1} + M _ {2} \delta K _ {2} + M _ {3} \delta e _ {1} + \delta \lambda (N - \tilde {N}) d L,
where \delta \lambda is a Lagrange multiplier introduced to impose the constraint N = \tilde { N } : A linear combination of these expressions is
\delta W _ { C } ^ { I } = \rho \delta W _ { 1 } ^ { I } + ( 1 - \rho ) \delta W _ { 2 } ^ { I } ; where \rho is a parameter that will be de¯ned later.
Then
\delta W _ {C} ^ {I} = \int_ {L} \Big [ (\rho N + (1 - \rho) \tilde {N}) \delta \varepsilon + M _ {1} \delta K _ {1} + M _ {2} \delta K _ {2} + M _ {3} \delta e _ {1} + (1 - \rho) \delta \lambda (N - \tilde {N}) \Big ] d L.
The contribution of this term to the Newton scheme is then
\begin{array}{l} \int_ {L} \left[ (\rho d N + (1 - \rho) d \tilde {N}) \delta \varepsilon + d M _ {1} \delta K _ {1} + d M _ {2} \delta K _ {2} + d M _ {3} \delta e _ {1} + (1 - \rho) \delta \lambda (d N - d \tilde {N}) \right. \\ \left. + \overline {{N}} d \delta \varepsilon + M _ {1} d \delta K _ {1} + M _ {2} d \delta K _ {2} + M _ {3} d \delta e _ {1} \right] d L \\ = - \int_ {L} \left[ \overline {{N}} \delta \varepsilon + M _ {1} \delta K _ {1} + M _ {2} \delta K _ {2} + M _ {3} \delta e _ {1} + (1 - \rho) \delta \lambda (N - \tilde {N}) \right] d L, \\ \end{array}
where