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Then, following the same procedure as for \mathrm { d } \delta b _ { \alpha } ,
\begin{array}{l} \mathrm{d} \delta b = \delta \pmb {\phi} \cdot \left(\mathbf {n} _ {2} \frac {d \mathbf {n} _ {1}}{d S} + \frac {d \mathbf {n} _ {1}}{d S} \mathbf {n} _ {2}\right) \cdot \delta \pmb {\phi} - 2 \left(\mathbf {n} _ {2} \cdot \frac {d \mathbf {n} _ {1}}{d S}\right) (\delta \pmb {\phi} \cdot \mathrm{d} \pmb {\phi}) \\ + \frac {1}{2} \frac {d \delta \boldsymbol {\phi}}{d S} \cdot (\mathbf {n} _ {1} \mathbf {n} _ {2} + \mathbf {n} _ {2} \mathbf {n} _ {1}) \cdot \mathrm{d} \boldsymbol {\phi} + \frac {1}{2} \delta \boldsymbol {\phi} \cdot (\mathbf {n} _ {1} \mathbf {n} _ {2} + \mathbf {n} _ {2} \mathbf {n} _ {1}) \cdot \frac {d \mathrm{d} \boldsymbol {\phi}}{d S} \\ - \delta \pmb {\phi} \cdot \mathbf {n} _ {1} \mathbf {n} _ {2} \cdot \frac {d \mathrm{d} \pmb {\phi}}{d S} + \left(\mathbf {n} _ {2} \cdot \frac {d \mathbf {n} _ {1}}{d S}\right) (\delta \pmb {\phi} \cdot \mathrm{d} \pmb {\phi}) - \delta \pmb {\phi} \cdot \frac {d \mathbf {n} _ {1}}{d S} \mathbf {n} _ {2} \cdot \mathrm{d} \pmb {\phi} \\ - \mathrm{d} \boldsymbol {\phi} \cdot \mathbf {n} _ {1} \mathbf {n} _ {2} \cdot \frac {d \mathrm{d} \boldsymbol {\phi}}{d S} + \left(\mathbf {n} _ {2} \cdot \frac {d \mathbf {n} _ {1}}{d S}\right) (\mathrm{d} \boldsymbol {\phi} \cdot \delta \boldsymbol {\phi}) - \mathrm{d} \boldsymbol {\phi} \cdot \frac {d \mathbf {n} _ {1}}{d S} \mathbf {n} _ {2} \cdot \delta \boldsymbol {\phi} \\ = \frac {1}{2} \left[ \frac {d \delta \pmb {\phi}}{d S} \cdot (\mathbf {n} _ {1} \mathbf {n} _ {2} - \mathbf {n} _ {2} \mathbf {n} _ {1}) \cdot \mathrm{d} \pmb {\phi} - \delta \pmb {\phi} \cdot (\mathbf {n} _ {1} \mathbf {n} _ {2} - \mathbf {n} _ {2} \mathbf {n} _ {1}) \cdot \frac {d \mathrm{d} \pmb {\phi}}{d S} \right]. \\ \end{array}
Finally, we note
\mathrm{d} \delta \chi = 0.
The resulting second variations are summarized as
\begin{array}{l} \mathrm{d} \delta \lambda = \lambda^ {- 1} \frac {d \mathrm{d} \mathbf {x}}{d S} \cdot (\mathbf {I} - \mathbf {s s}) \cdot \frac {d \delta \mathbf {x}}{d S}, \\ + \frac {1}{2} \delta \pmb {\phi} \cdot (\mathbf {n} _ {\alpha} \mathbf {t} + \mathbf {t n} _ {\alpha}) \cdot \dot {\pmb {\phi}}, \\ \mathrm{d} \delta b _ {\alpha} = \frac {1}{2} \epsilon_ {\alpha} ^ {\beta} \bigg [ \frac {d \delta \pmb {\phi}}{d S} \cdot (\mathbf {n} _ {\beta} \mathbf {t} - \mathbf {t n} _ {\beta}) \cdot \mathrm{d} \pmb {\phi} - \delta \pmb {\phi} \cdot (\mathbf {n} _ {\beta} \mathbf {t} - \mathbf {t n} _ {\beta}) \cdot \frac {d \mathrm{d} \pmb {\phi}}{d S} \bigg ], \\ \mathrm{d} \delta b = \frac {1}{2} \left[ \frac {d \delta \pmb {\phi}}{d S} \cdot (\mathbf {n} _ {1} \mathbf {n} _ {2} - \mathbf {n} _ {2} \mathbf {n} _ {1}) \cdot \mathrm{d} \pmb {\phi} - \delta \pmb {\phi} \cdot (\mathbf {n} _ {1} \mathbf {n} _ {2} - \mathbf {n} _ {2} \mathbf {n} _ {1}) \cdot \frac {d \mathrm{d} \pmb {\phi}}{d S} \right], \\ \mathrm{d} \delta \chi = 0. \\ \end{array}
\mathrm{d} \delta \gamma_ {\alpha} = - \lambda^ {- 2} \frac {d \delta \mathbf {x}}{d S} \cdot (\mathbf {n} _ {\alpha} \mathbf {s} + \mathbf {s n} _ {\alpha}) \cdot \frac {d \mathrm{d} \mathbf {x}}{d S} + \lambda^ {- 1} \epsilon_ {\alpha} ^ {\beta} \left(\frac {d \delta \mathbf {x}}{d S} \cdot \mathbf {n} _ {\beta} \mathbf {t} \cdot \mathrm{d} \pmb {\phi} + \delta \pmb {\phi} \cdot \mathbf {n} _ {\beta} \mathbf {t} \cdot \frac {d \mathrm{d} \mathbf {x}}{d S}\right)
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The gradient of the current position of a point in the section with respect to the coordinate S is
\frac {\partial \hat {\mathbf {x}}}{\partial S} = \frac {d \mathbf {x}}{d S} + \frac {d f}{d S} S ^ {\alpha} \mathbf {n} _ {\alpha} + f S ^ {\alpha} \frac {d \mathbf {n} _ {\alpha}}{d S} + \frac {d w}{d S} \psi \mathbf {t} + w \psi \frac {d \mathbf {t}}{d S}.
We will keep only terms up to order \frac { h } { \ell } . We assume that \frac { d f } { d S } \ll \frac { 1 } { \ell } 1 , and--hence--the second term can be neglected. We also assume that w \psi = O ( \frac { h ^ { 2 } } { \ell } ) ; and--since \frac { d \mathbf { t } } { d S } = O ( \frac { 1 } { \ell } ) --we can neglect the last term as well. However, the warping function may vary rapidly near the ends where warping is constrained.
Hence, \frac { d w } { d S } \psi = O ( \frac { h } { \ell } ) and should be preserved. With those approximations,
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\frac {\partial \hat {\mathbf {x}}}{\partial S} \approx \frac {d \mathbf {x}}{d S} + f S ^ {\alpha} \frac {d \mathbf {n} _ {\alpha}}{d S} + \frac {d w}{d S} \psi \mathbf {t} = \lambda \mathbf {s} + f S ^ {\alpha} \epsilon_ {\alpha} ^ {\beta} (- b _ {\beta} \mathbf {t} + b \mathbf {n} _ {\beta}) + \chi \psi \mathbf {t}.
The gradients with respect to S ^ { \alpha } are
\frac {\partial \hat {\mathbf {x}}}{\partial S ^ {\alpha}} = f \mathbf {n} _ {\alpha} + w \frac {\partial \psi}{\partial S ^ {\alpha}} \mathbf {t}.
Correspondingly, in the original configuration
\frac {\partial \hat {\mathbf {X}}}{\partial S} = \frac {d \mathbf {X}}{d S} + S ^ {\alpha} \frac {d \mathbf {N} _ {\alpha}}{d S} = \mathbf {T} + S ^ {\alpha} \epsilon_ {\alpha} ^ {\beta} (- B _ {\beta} \mathbf {T} + B \mathbf {N} _ {\beta}),
\frac {\partial \hat {\mathbf {X}}}{\partial S ^ {\alpha}} = \mathbf {N} _ {\alpha}.
The above relations are readily inverted to yield
\frac {\partial S}{\partial \hat {\mathbf {X}}} = (1 - S ^ {\gamma} \epsilon_ {\gamma} ^ {\delta} B _ {\delta}) ^ {- 1} \mathbf {T},
\frac {\partial S ^ {\alpha}}{\partial \hat {\mathbf {X}}} = \mathbf {N} _ {\alpha} - (1 - S ^ {\gamma} \epsilon_ {\gamma} ^ {\delta} B _ {\delta}) ^ {- 1} S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} B \mathbf {T}.
The deformation gradient then becomes
\begin{array}{l} \mathbf {F} = \frac {\partial \hat {\mathbf {x}}}{\partial S} \frac {\partial S}{\partial \hat {\mathbf {X}}} + \frac {\partial \hat {\mathbf {x}}}{\partial S ^ {\alpha}} \frac {\partial S ^ {\alpha}}{\partial \hat {\mathbf {X}}} \\ = (1 - S ^ {\gamma} \epsilon_ {\gamma} ^ {\delta} B _ {\delta}) ^ {- 1} \bigg [ \lambda \mathbf {s} \mathbf {T} + (- f S ^ {\alpha} \epsilon_ {\alpha} ^ {\beta} b _ {\beta} + \chi \psi) \mathbf {t} \mathbf {T} + f S ^ {\alpha} \epsilon_ {\alpha} ^ {\beta} b \mathbf {n} _ {\beta} \mathbf {T} \bigg ] \\ + (1 - S ^ {\alpha} \epsilon_ {\alpha} ^ {\beta} B _ {\beta}) f \mathbf {n} _ {\alpha} \mathbf {N} _ {\alpha} + (1 - S ^ {\alpha} \epsilon_ {\alpha} ^ {\beta} B _ {\beta}) w \frac {d \psi}{d S ^ {\eta}} \mathbf {t} \mathbf {N} _ {\eta} \\ - S ^ {\alpha} \epsilon_ {\alpha} ^ {\beta} B f \mathbf {n} _ {\beta} \mathbf {T} - S ^ {\alpha} \epsilon_ {\alpha} ^ {\beta} B w \frac {\partial \psi}{\partial S ^ {\beta}} \mathbf {t} \left. \mathbf {T} \right]. \\ \end{array}
We define the initial length ratio R as
R = 1 - S ^ {\alpha} \epsilon_ {\alpha} ^ {\beta} B _ {\beta}.
In the expression enclosed within square brackets above, terms of order h ^ { 2 } / \ell ^ { 2 } can again be neglected. However, it is assumed that the section may have low resistance to torsion and that, hence, the warping and twist may be large. This is particularly true for thin walled open sections. Hence, we obtain:
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\mathbf {F} = R ^ {- 1} \left\{\lambda \mathbf {s} \mathbf {T} + \left[ - S ^ {\alpha} \epsilon_ {\alpha} ^ {\beta} \left(f b _ {\beta} + B w \frac {\partial \psi}{\partial S ^ {\beta}}\right) + \chi \psi \right] \mathbf {t} \mathbf {T} + \right.
f S ^ {\alpha} \epsilon_ {\alpha} ^ {\beta} (b - B) \mathbf {n} _ {\beta} \mathbf {T} + w \frac {\partial \psi}{\partial S ^ {\beta}} \mathbf {t} \mathbf {N} _ {\beta} + R f \mathbf {n} _ {\beta} \mathbf {N} _ {\beta} \Biggr \}.
We calculate the components of F in a corotational system with the approximation \mathbf { t } \cdot \mathbf { s } = 1 . This provides
F _ {1 1} = \mathbf {t} \cdot \mathbf {F} \cdot \mathbf {T} = R ^ {- 1} \left[ \lambda - S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \left(f b _ {\alpha} + B w \frac {\partial \psi}{\partial S ^ {\alpha}}\right) + \chi \psi \right],
{F _ {(\alpha + 1) 1}} {= \mathbf {n} _ {\alpha} \cdot \mathbf {F} \cdot \mathbf {T} = R ^ {- 1} (\lambda \gamma_ {\alpha} + \epsilon_ {\beta} ^ {\alpha} f S ^ {\beta} (b - B)),}
{F _ {1 (\beta + 1)}} {= \mathbf {t} \cdot \mathbf {F} \cdot \mathbf {N} _ {\beta} = R ^ {- 1} w \frac {d \psi}{d S ^ {\beta}},}
F _ {(\alpha + 1) (\beta + 1)} = \mathbf {n} _ {\alpha} \cdot \mathbf {F} \cdot \mathbf {N} _ {\beta} = f \delta_ {\alpha \beta}.
We again neglect all terms of order h ^ { n } / \ell ^ { n } for n \geq 2 , , except the term involving Bw. The equations then simplify to
{F _ {1 1}} {= \lambda - S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \left(f b _ {\alpha} + B w \frac {\partial \psi}{\partial S ^ {\alpha}} - \lambda B _ {\alpha}\right) + \chi \psi ,}
{F _ {(\alpha + 1) 1}} {= \lambda \gamma_ {\alpha} + \epsilon_ {\beta} ^ {\alpha} f S ^ {\beta} (b - B),}
{F _ {1 (\beta + 1)}} {= w \frac {d \psi}{d S ^ {\beta}},}
F _ {(\alpha + 1) (\beta + 1)} = f \delta_ {\alpha \beta}.
Consistent with traditional shell and beam theories, we slightly adapt the term involving the initial curvature B _ { \alpha } --instead of multiplying it with ¸, we multiply it with f . Such a change does not significantly increase the error in the curvature calculation, since we do not properly account for the initial curvature in the volume integration anyway. Hence, we find for F _ { 1 1 } :
F _ {1 1} = \lambda - S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \left(f (b _ {\alpha} - B _ {\alpha}) + B w \frac {\partial \psi}{\partial S ^ {\alpha}}\right) + \chi \psi .
We now make a multiplicative decomposition of F into a "stretch" part \mathbf { F } ^ { s } and a "distortion" part \mathbf { F } ^ { d } such that \mathbf { F } = \mathbf { F } ^ { d } \cdot \mathbf { F } ^ { s } .
For \mathbf { F } ^ { s } we choose
F _ {1 1} ^ {s} = \lambda , \quad F _ {(\alpha + 1) 1} ^ {s} = F _ {1 (\beta + 1)} ^ {s} = 0, \quad \mathrm{and} F _ {(\alpha + 1) (\beta + 1)} ^ {s} = f \delta_ {\alpha \beta};
and, hence, \mathbf { F } ^ { d } is
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F _ {1 1} ^ {d} = \lambda^ {- 1} F _ {1 1} = 1 - S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \lambda^ {- 1} \left(f (b _ {\alpha} - B _ {\alpha}) + B w \frac {\partial \psi}{\partial S ^ {\alpha}}\right) + \lambda^ {- 1} \chi \psi ,
{F _ {(\alpha + 1) 1} ^ {d}} {= \lambda^ {- 1} F _ {(\alpha + 1) 1} = \gamma_ {\alpha} + \epsilon_ {\beta} ^ {\alpha} f \lambda^ {- 1} S ^ {\beta} (b - B),}
{F _ {1 (\beta + 1)} ^ {d}} {= f ^ {- 1} F _ {1 (\beta + 1)} = f ^ {- 1} w \frac {\partial \psi}{\partial S ^ {\beta}},}
F _ {(\alpha + 1) (\beta + 1)} ^ {d} = f ^ {- 1} F _ {(\alpha + 1) (\beta + 1)} = \delta_ {\alpha \beta}.
Since \mathbf { F } ^ { s } is a diagonal tensor, the logarithmic "stretch" strains are immediately available as
e _ {1 1} ^ {s} = \ln \lambda , \quad \gamma_ {(\alpha + 1) 1} ^ {s} = 0, \quad \gamma_ {1 (\alpha + 1)} ^ {s} = 0, \quad e _ {(\alpha + 1) (\beta + 1)} ^ {s} = \delta_ {\alpha \beta} \ln (f).
Since the "distortional" strains are small, we obtain them from \mathbf { F } ^ { d } with the Green-Lagrange formula:
\mathbf {E} ^ {d} = \frac {1}{2} \left(\mathbf {F} ^ {d ^ {T}} \cdot \mathbf {F} ^ {d} - \mathbf {I}\right).
For the components this yields
\begin{array}{l} E _ {1 1} ^ {d} = S ^ {\beta} \epsilon_ {\alpha} ^ {\beta} \lambda^ {- 1} \left(f (b _ {\alpha} - B _ {\alpha}) + B w \frac {\partial \psi}{\partial S ^ {\alpha}}\right) + \lambda^ {- 1} \chi \psi \\ - S ^ {\beta} \epsilon_ {\alpha} ^ {\beta} \left(b _ {\alpha} - B _ {\alpha} + B w \frac {\partial \psi}{\partial S ^ {\alpha}}\right) \lambda^ {- 1} \chi \psi + \gamma_ {\alpha} \epsilon_ {\alpha} ^ {\beta} f \lambda^ {- 1} S ^ {\beta} (b - B) \\ + \frac {1}{2} S ^ {\gamma} \epsilon_ {\gamma} ^ {\alpha} S ^ {\beta} \epsilon_ {\beta} ^ {\delta} \left(b _ {\alpha} - B _ {\alpha} + B w \frac {\partial \psi}{\partial S ^ {\alpha}}\right) \left(b _ {\delta} - B _ {\delta} + B w \frac {\partial \psi}{\partial S ^ {\delta}}\right) \\ + \frac {1}{2} \lambda^ {- 2} \chi^ {2} \psi^ {2} + \frac {1}{2} \gamma_ {\alpha} \gamma_ {\alpha} + \frac {1}{2} f ^ {2} \lambda^ {- 2} S ^ {\alpha} S ^ {\alpha} (b - B) ^ {2}, \\ \end{array}
\begin{array}{l} E _ {(\alpha + 1) 1} = \frac {1}{2} \left[ f ^ {- 1} w \frac {\partial \psi}{\partial S ^ {\alpha}} + \gamma_ {\alpha} + \epsilon_ {\alpha} ^ {\beta} f \lambda^ {- 1} S ^ {\beta} (b - B) \right. \\ \left. - \lambda^ {- 1} w \frac {\partial \psi}{\partial S ^ {\alpha}} \epsilon_ {\beta} ^ {\gamma} \left(b _ {\gamma} - B _ {\gamma} + B w \frac {\partial \psi}{\partial S ^ {\gamma}}\right) + f ^ {- 1} w \frac {\partial \psi}{\partial S ^ {\alpha}} \lambda^ {- 1} \chi \psi \right], \\ \end{array}
E _ {(\alpha + 1) (\beta + 1)} = \frac {1}{2} f ^ {- 2} w ^ {2} \frac {\partial \psi}{\partial S ^ {\alpha}} \frac {\partial \psi}{\partial S ^ {\beta}}.
Note that these strains are small. Since the various terms have different dependence on the cross-sectional coordinates, this leads to the conditions
\begin{array}{l} S ^ {\beta} \epsilon_ {\alpha} ^ {\beta} f \lambda^ {- 1} (b _ {\alpha} - B _ {\alpha}) \ll 1, \\ \lambda^ {- 1} \chi \psi \ll 1, \\ \gamma_ {\alpha} \ll 1, \\ \end{array}
f ^ {- 1} w \frac {\partial \psi}{\partial S ^ {\alpha}} + \epsilon_ {\alpha} ^ {\beta} f \lambda^ {- 1} S ^ {\beta} (b - B) \ll 1.
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The last condition will generally require that both w and b - B are small, since \partial \psi / \partial S ^ { \alpha } will not be proportional to \epsilon _ { \alpha } ^ { \beta } S ^ { \beta } . However, for thin walled open section beams the proportionality is approximately satisfied and, therefore, we obtain in that case
w \frac {\partial \psi}{\partial S ^ {\alpha}} \approx - f ^ {2} \lambda^ {- 1} \epsilon_ {\alpha} ^ {\beta} S ^ {\beta} (b - B).
Note that within the desired accuracy this equation even holds for other sections: in that case both the right-hand and left-hand side are very small. Substitution in the expression for the Green-Lagrange strain yields the (small) distortional strains:
e _ {1 1} ^ {d} = - f \lambda^ {- 1} S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} (b _ {\alpha} - B _ {\alpha}) + \lambda^ {- 1} \chi \psi + \frac {1}{2} f ^ {2} \lambda^ {- 2} S ^ {\alpha} S ^ {\alpha} (b ^ {2} - B ^ {2}),
\gamma_ {(\alpha + 1) 1} ^ {d} = \gamma_ {\alpha} - \epsilon_ {\beta} ^ {\alpha} f \lambda^ {- 1} S ^ {\beta} (b - B) + f ^ {- 1} w \frac {\partial \psi}{\partial S ^ {\alpha}},
{\gamma_ {1 (\alpha + 1)} ^ {d}} {= \gamma_ {(\alpha + 1) 1} ^ {d},}
e _ {(\alpha + 1) (\beta + 1)} ^ {d} = \frac {1}{2} f ^ {2} \lambda^ {- 2} S ^ {\alpha} S ^ {\alpha} (b ^ {2} - B ^ {2}).
The total strains are obtained simply by addition as
e _ {1 1} = \ln \lambda - f \lambda^ {- 1} S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} (b _ {\alpha} - B _ {\alpha}) + \lambda^ {- 1} \chi \psi + \frac {1}{2} f ^ {2} \lambda^ {- 2} S ^ {\alpha} S ^ {\alpha} (b ^ {2} - B ^ {2}),
\gamma_ {(\alpha + 1) 1} = \gamma_ {\alpha} + \epsilon_ {\beta} ^ {\alpha} f \lambda^ {- 1} S ^ {\beta} (b - B) + f ^ {- 1} w \frac {\partial \psi}{\partial S ^ {\alpha}},
\gamma_ {1 (\alpha + 1)} = \gamma_ {(\alpha + 1) 1},
e _ {(\alpha + 1) (\beta + 1)} = \delta_ {\alpha \beta} \ln (f) + \frac {1}{2} f ^ {2} \lambda^ {- 2} S ^ {\alpha} S ^ {\alpha} (b ^ {2} - B ^ {2}).
We assume that there are no stresses in the ( \alpha + 1 ) ( \beta + 1 ) directions. Hence, the strains in these directions do not contribute to virtual work and do not need to be considered any further.
It is useful to split the total warping w in two parts: a part due to "free" warping w _ { f } minus a part due to warping prevention w _ { p } \colon w = w _ { f } - w _ { p } . We assume that the warping function \psi is chosen such that the free warping is related to the twist with the relation
w _ {f} = f ^ {2} \lambda^ {- 1} (b - B) \qquad \text {and, hence,} \quad w _ {p} = f ^ {2} \lambda^ {- 1} (b - B) - w.
This makes it possible to write the expression for \gamma _ { ( \alpha + 1 ) 1 } as
\gamma_ {(\alpha + 1) 1} = \gamma_ {\alpha} + f \lambda^ {- 1} \left(\epsilon_ {\beta} ^ {\alpha} S ^ {\beta} + \frac {\partial \psi}{\partial S ^ {\alpha}}\right) (b - B) - f ^ {- 1} w _ {p} \frac {\partial \psi}{\partial S ^ {\alpha}}.
It is desirable to choose the cross-sectional resultants such that they are completely uncoupled. In addition, we assume that the axial strain variation across the section due to the second-order term in the twist is not significant. Therefore, we consider only the average axial strain due to the
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second-order twist term. Hence, we introduce the average axial strain
\overline {{e}} = \ln \lambda - f \lambda^ {- 1} S _ {c} ^ {\beta} \epsilon_ {\beta} ^ {\alpha} (b _ {\alpha} - B _ {\alpha}) + \lambda^ {- 1} \chi \overline {{\psi}} + \frac {1}{2} \left(\frac {I _ {p}}{A} + S _ {c} ^ {\alpha} S _ {c} ^ {\alpha}\right) f ^ {2} \lambda^ {- 2} (b ^ {2} - B ^ {2}),
where S _ { c } ^ { \beta } are the coordinates of the centroid, I _ { p } is the polar moment of inertia, and \overline { { \psi } } is the average value of the warping function:
\overline {{\psi}} = \frac {1}{A} \int_ {A} \psi d A.
Similarly, we introduce the average shear strain
\overline {{\gamma}} _ {\alpha} = \gamma_ {\alpha} + f \lambda^ {- 1} \left(\epsilon_ {\beta} ^ {\alpha} S _ {c} ^ {\beta} + \frac {1}{A} \int_ {A} \frac {\partial \psi}{\partial S ^ {\alpha}} d A\right) (b - B) - f ^ {- 1} w _ {p} \frac {1}{A} \int_ {A} \frac {\partial \psi}{\partial S ^ {\alpha}} d A.
This last expression can be simplified by the introduction of the shear center coordinates S _ { s } ^ { \beta } , which are related to the warping function by
\epsilon_ {\beta} ^ {\alpha} S _ {s} ^ {\beta} = \epsilon_ {\beta} ^ {\alpha} S _ {c} ^ {\beta} + \frac {1}{A} \int_ {A} \frac {\partial \psi}{\partial S ^ {\alpha}} d A, \quad \mathrm{or} \quad S _ {s} ^ {\beta} = S _ {c} ^ {\beta} + \frac {1}{A} \int_ {A} \epsilon_ {\beta} ^ {\alpha} \frac {\partial \psi}{\partial S ^ {\alpha}} d A.
This yields
\overline {{\gamma}} _ {\alpha} = \gamma_ {\alpha} + f \lambda^ {- 1} \epsilon_ {\beta} ^ {\alpha} S _ {s} ^ {\beta} (b - B) + f ^ {- 1} w _ {p} \epsilon_ {\beta} ^ {\alpha} (S _ {c} ^ {\beta} - S _ {s} ^ {\beta}).
Note that the average value is in fact the value at the shear center if warping prevention is absent ( w _ { p } = 0 ) . However, for full warping prevention ( w _ { p } = f ^ { 2 } \lambda ^ { - 1 } ( b - B ) ) the average value corresponds to the value obtained at the centroid.
Instead of the original warping function \psi ( S ^ { \alpha } ) we now introduce a modified warping function \Omega ( S ^ { \alpha } ) related to \psi by
\Omega (S ^ {\alpha}) \stackrel {\mathrm{def}} {=} \psi (S ^ {\alpha}) - \overline {{\psi}}.
This function in fact represents the classical definition of a warping function with an area weighted average of zero. The average value \overline { { \psi } } can be obtained from the classical warping function with the condition that \psi ( 0 ) = 0 :
\overline {{\psi}} = \psi (S ^ {\alpha}) - \Omega (S ^ {\alpha}) = \psi (0) - \Omega (0) = - \Omega (0) = - \Omega_ {\circ}.
The expression for the average axial strain then becomes
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\overline {{e}} = \ln \lambda - f \lambda^ {- 1} S _ {c} ^ {\beta} \epsilon_ {\beta} ^ {\alpha} (b _ {\alpha} - B _ {\alpha}) - \lambda^ {- 1} \chi \Omega_ {\circ} + \frac {1}{2} \left(\frac {I _ {p}}{A} + S _ {c} ^ {\alpha} S _ {c} ^ {\alpha}\right) f ^ {2} \lambda^ {- 2} (b ^ {2} - B ^ {2}),
and the location of the shear center is
S _ {s} ^ {\beta} = S _ {c} ^ {\beta} + \frac {1}{A} \int_ {A} \epsilon_ {\beta} ^ {\alpha} \frac {\partial \Omega}{\partial S ^ {\alpha}} d A.
The strains can, thus, be written in the form
{e _ {1 1}} {= \overline {{{e}}} - f \lambda^ {- 1} (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\beta} ^ {\alpha} (b _ {\alpha} - B _ {\alpha}) + \lambda^ {- 1} \chi \Omega ,}
\gamma_ {(\alpha + 1) 1} = \overline {{\gamma}} _ {\alpha} + f \lambda^ {- 1} \left(\epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {s} ^ {\beta}) + \frac {\partial \Omega}{\partial S ^ {\alpha}}\right) (b - B) + f ^ {- 1} w _ {p} \left(\epsilon_ {\beta} ^ {\alpha} (S _ {s} ^ {\beta} - S _ {c} ^ {\beta}) - \frac {\partial \Omega}{\partial S ^ {\alpha}}\right).
The second term in the expression for \gamma _ { ( \alpha + 1 ) 1 } is proportional to the shear strain field for pure elastic torsion:
\gamma_ {\alpha} ^ {t} (S ^ {\gamma}) \stackrel {\mathrm{def}} {=} \epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {s} ^ {\beta}) + \frac {\partial \Omega}{\partial S ^ {\alpha}}.
We use this definition to eliminate the gradient of − from the expression for the shear strain, which yields as final expression for the strains
e _ {1 1} = \overline {{e}} - f \lambda^ {- 1} (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\beta} ^ {\alpha} (b _ {\alpha} - B _ {\alpha}) + \lambda^ {- 1} \chi \Omega ,
\gamma_ {(\alpha + 1) 1} = \overline {{\gamma}} _ {\alpha} + f \lambda^ {- 1} \gamma_ {\alpha} ^ {t} (b - B) + f ^ {- 1} w _ {p} \left(\epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) - \gamma_ {\alpha} ^ {t}\right).
Virtual work
Since it was assumed that there are no stresses in the ( \alpha + 1 ) ( \beta + 1 ) directions, the virtual work contribution is
\begin{array}{l} \delta \Pi = \int_ {V} \left(\sigma_ {1 1} \delta e _ {1 1} + \tau_ {(\alpha + 1) 1} \delta \gamma_ {(\alpha + 1) 1}\right) d V \\ = \int_ {\ell} d \ell \int_ {A} (\sigma_ {1 1} \delta e _ {1 1} + \tau_ {(\alpha + 1) 1} \delta \gamma_ {(\alpha + 1) 1}) d A. \\ \end{array}
The strain variations are obtained by linearization of the expressions for the strains
\delta e _ {1 1} \qquad = \delta \overline {{e}} - f \lambda^ {- 1} (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\beta} ^ {\alpha} \delta b _ {\alpha} + \lambda^ {- 1} \Omega \delta \chi ,
\delta \gamma_ {(\alpha + 1) 1} = \delta \overline {{\gamma}} _ {\alpha} + f \lambda^ {- 1} \gamma_ {\alpha} ^ {t} \delta b + f ^ {- 1} \left(\epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) - \gamma_ {\alpha} ^ {t}\right) \delta w _ {p},
where all terms of the order of the "distortional" strains have been neglected. From the expressions for the average axial strain and average shear strain, we obtain the average axial and shear strain variations as
Elements
\delta \overline {{e}} = \lambda^ {- 1} \delta \lambda - \lambda^ {- 1} f S _ {c} ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \delta b _ {\alpha} - \lambda^ {- 1} \Omega_ {\circ} \delta \chi + f ^ {2} \lambda^ {- 2} \left(\frac {I _ {p}}{A} + S _ {c} ^ {\alpha} S _ {c} ^ {\alpha}\right) b \delta b,
\delta \overline {{\gamma}} _ {\alpha} = \delta \gamma_ {\alpha} + \lambda^ {- 1} f S _ {s} ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \delta b + f ^ {- 1} \epsilon_ {\beta} ^ {\alpha} (S _ {c} ^ {\beta} - S _ {s} ^ {\beta}) \delta w _ {p},
where \delta w _ { p } = f ^ { 2 } \lambda ^ { - 1 } \delta b - \delta w and again all terms of the order of the "distortional" strains have been neglected.
We now introduce the generalized section forces as defined below:
| Axial force | $F = \int_{A} \sigma_{11} dA$ |
| Shear forces | $F_{\alpha} = \int_{A} \tau_{(\alpha+1)1} dA$ |
| Bending moments | $M^{\alpha} = \int_{A} -f(S^{\beta} - S^{\beta}_{c}) \epsilon_{\beta}^{\alpha} \sigma_{11} dA$ |
| Twisting moment | $M_{t} = \int_{A} f \gamma_{\alpha}^{t} \tau_{(\alpha+1)1} dA$ |
| Warping moment | $M_{w} = \int_{A} f \left( \epsilon_{\beta}^{\alpha} (S^{\beta} - S^{\beta}_{c}) - \gamma_{\alpha}^{t} \right) \tau_{(\alpha+1)1} dA$ |
| Bimoment | $W = \int_{A} \Omega \sigma_{11} dA$ |
which transforms the virtual work contribution into
\delta \Pi = \int_ {\ell} (F \delta \overline {{{e}}} + F _ {\alpha} \delta \overline {{{\gamma}}} _ {\alpha} + M ^ {\alpha} \lambda^ {- 1} \delta b _ {\alpha} + M _ {t} \lambda^ {- 1} \delta b + M _ {w} f ^ {- 2} \delta w _ {p} + W \lambda^ {- 1} \delta \chi) d \ell .
Observe that the total torque T relative to the centroid of the section is the sum of the twisting moment and the warping moment:
T = \int_ {A} f \epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) \tau_ {(\alpha + 1) 1} d A = M _ {t} + M _ {w}.
The rate of change of virtual work
To obtain the rate of change of virtual work, we first transform the integrations in the virtual work equation to the original volume such that
\delta \Pi = \int_ {\ell^ {\circ}} d \ell^ {\circ} \int_ {A ^ {\circ}} \big (\sigma_ {1 1} f ^ {2} \lambda \delta e _ {1 1} + \tau_ {(\alpha + 1) 1} f ^ {2} \lambda \delta \gamma_ {(\alpha + 1) 1} \big) d A ^ {\circ}.
The strain variations relative to the original state are then
\begin{array}{l} f ^ {2} \lambda \delta e _ {1 1} = f ^ {2} \lambda \delta \overline {{e}} - f ^ {3} (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\beta} ^ {\alpha} \delta b _ {\alpha} + f ^ {2} \Omega \delta \chi , \\ {f ^ {2} \lambda \delta \gamma_ {(\alpha + 1) 1}} {= f ^ {2} \lambda \delta \overline {{\gamma}} _ {\alpha} + f ^ {3} \gamma_ {\alpha} ^ {t} \delta b + f \lambda \left(\epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) - \gamma_ {\alpha} ^ {t}\right) \delta w _ {p}.} \\ \end{array}
The strain variations relative to the original state are
Elements
{f ^ {2} \lambda \delta \overline {{e}}} {= f ^ {2} \delta \lambda - f ^ {3} S _ {c} ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \delta b _ {\alpha} - f ^ {2} \Omega_ {\circ} \delta \chi + f ^ {4} \lambda^ {- 1} \left(\frac {I _ {p}}{A} + S _ {c} ^ {\alpha} S _ {c} ^ {\alpha}\right) b \delta b,}
f ^ {2} \lambda \delta \overline {{\gamma}} _ {\alpha} = f ^ {2} \lambda \delta \gamma_ {\alpha} + f ^ {3} S _ {s} ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \delta b + f \lambda \epsilon_ {\beta} ^ {\alpha} (S _ {c} ^ {\beta} - S _ {s} ^ {\beta}) \delta w _ {p}.
The rate of change of virtual work is then
\begin{array}{l} \mathrm{d} \delta \Pi = \int_ {\ell^ {\circ}} d \ell^ {\circ} \int_ {A ^ {\circ}} \left[ f ^ {2} \lambda \left(\mathrm{d} \sigma_ {1 1} \delta e _ {1 1} + \mathrm{d} \tau_ {(\alpha + 1) 1} \delta \gamma_ {(\alpha + 1) 1} \right. \right. \\ + \sigma_ {1 1} (2 f \mathrm{d} f \delta \lambda - 3 f ^ {2} S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \mathrm{d} f \delta b _ {\alpha} + 2 f \psi \mathrm{d} f \delta \chi + f ^ {4} \lambda^ {- 1} \left(\frac {I _ {p}}{A} + S _ {c} ^ {\alpha} S _ {c} ^ {\alpha}\right) \mathrm{d} b \delta b \\ + f ^ {2} \mathrm{d} \delta \lambda - f ^ {3} S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \mathrm{d} \delta b _ {\alpha} + f ^ {2} \psi \mathrm{d} \delta \chi) \\ + \tau_ {(\alpha + 1) 1} \left(f ^ {2} \mathrm{d} \lambda \delta \gamma_ {\alpha} + 2 \lambda f \mathrm{d} f \delta \gamma_ {\alpha} + 3 f ^ {2} S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \delta b \mathrm{d} f + f \frac {\partial \psi}{\partial S ^ {\alpha}} \mathrm{d} \lambda \delta w \right. \\ \left. + \lambda \frac {\partial \psi}{\partial S ^ {\alpha}} \mathrm{d} f \delta w + \lambda f ^ {2} \mathrm{d} \delta \gamma_ {\alpha} + f ^ {3} S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \mathrm{d} \delta b + \lambda f \frac {\partial \psi}{\partial S ^ {\alpha}} \mathrm{d} \delta w) \right] d A ^ {\circ}, \\ \end{array}
where we have neglected terms of order b. The changes in stress follow from the constitutive law
\left\{ \begin{array}{l} \mathrm{d} \sigma_ {1 1} \\ \mathrm{d} \tau_ {2 1} \\ \mathrm{d} \tau_ {3 1} \end{array} \right\} = \left[ \begin{array}{c c c} D _ {1 1} & D _ {1 2} & D _ {1 3} \\ D _ {2 1} & D _ {2 2} & D _ {2 3} \\ D _ {3 1} & D _ {3 2} & D _ {3 3} \end{array} \right] \left\{ \begin{array}{l} \mathrm{d} e _ {1 1} \\ \mathrm{d} \gamma_ {2 1} \\ \mathrm{d} \gamma_ {3 1} \end{array} \right\}.
We approximate \mathrm { d } e _ { 1 1 } and \mathbf { d } \gamma _ { 1 1 } with the same relations as used for virtual work:
\mathrm{d} e _ {1 1} = \mathrm{d} \overline {{e}} - f \lambda^ {- 1} (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\beta} ^ {\alpha} \mathrm{d} b _ {\alpha} + \lambda^ {- 1} \Omega \mathrm{d} \chi ,
\mathrm{d} \gamma_ {(\alpha + 1) 1} = \mathrm{d} \overline {{\gamma}} _ {\alpha} + f \lambda^ {- 1} \gamma_ {\alpha} ^ {t} \mathrm{d} b + f ^ {- 1} \left(\epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) - \gamma_ {\alpha} ^ {t}\right) \mathrm{d} w _ {p},
and for the average strain rates
{\mathrm{d} \overline {{e}}} {= \lambda^ {- 1} \mathrm{d} \lambda - \lambda^ {- 1} f S _ {c} ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \mathrm{d} b _ {\alpha} - \lambda^ {- 1} \Omega_ {\circ} \mathrm{d} \chi + f ^ {2} \lambda^ {- 2} \left(\frac {I _ {p}}{A} + S _ {c} ^ {\alpha} S _ {c} ^ {\alpha}\right) b \mathrm{d} b,}
\mathrm{d} \overline {{\gamma}} _ {\alpha} = \mathrm{d} \gamma_ {\alpha} + \lambda^ {- 1} f S _ {s} ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \mathrm{d} b + f ^ {- 1} \epsilon_ {\beta} ^ {\alpha} (S _ {c} ^ {\beta} - S _ {s} ^ {\beta}) \mathrm{d} w _ {p}.
We now neglect all terms that involve the product of a stress tensor; a variation in curvature, twist, or warping; and a change in axial strain of the cross-section. Using the previously obtained expressions for the second variations in \lambda , b _ { \alpha } , b , \gamma _ { \alpha } , and \chi and transforming the results into the current state, this provides
Elements
\mathrm{d} \delta \Pi = \int_ {\ell} d \ell \int_ {A} \left[ \mathrm{d} \sigma_ {1 1} \delta e _ {1 1} + \mathrm{d} \tau_ {(\alpha + 1) 1} \delta \gamma_ {(\alpha + 1) 1} \right.
+ \sigma_ {1 1} (2 \lambda^ {- 1} \delta \lambda f ^ {- 1} \mathrm{d} f + \lambda^ {- 1} \mathrm{d} \delta \lambda - \lambda^ {- 1} f S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \mathrm{d} \delta b _ {\alpha} + f ^ {2} \lambda^ {- 2} \left(\frac {I _ {p}}{A} + S _ {c} ^ {\alpha} S _ {c} ^ {\alpha}\right) \mathrm{d} b \delta b)
\left. + \tau_ {(\alpha + 1) 1} \left(\mathrm{d} \delta \gamma_ {\alpha} + \lambda^ {- 1} f S ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \mathrm{d} \delta b + f ^ {- 1} \frac {\partial \psi}{\partial S ^ {\alpha}} \mathrm{d} \delta w\right) \right] d A.
The incremental moments, forces, etc. are defined as
\mathrm{d} F = \int_ {A} \mathrm{d} \sigma_ {1 1} d A,
\mathrm{d} F _ {\alpha} = \int_ {A} \mathrm{d} \tau_ {(\alpha + 1) 1} d A,
\mathrm{d} M ^ {\alpha} = \int_ {A} - f (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\beta} ^ {\alpha} \mathrm{d} \sigma_ {1 1} d A,
\mathrm{d} M _ {t} = \int_ {A} f \gamma_ {\alpha} ^ {t} \mathrm{d} \tau_ {(\alpha + 1) 1} d A,
\mathrm{d} M _ {w} = \int_ {A} f \left(\epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) - \gamma_ {\alpha} ^ {t}\right) \mathrm{d} \tau_ {(\alpha + 1) 1} d A,
{\mathrm{d} W} {= \int_ {A} \Omega \mathrm{d} \sigma_ {1 1} d A.}
For the determination of the initial stress stiffness, we assume that the second variations of the warping function and its derivative vanish: \mathrm { d } \delta w = \mathrm { d } \delta \chi = 0 . Consequently,
\mathrm{d} \delta w _ {p} = f ^ {2} \lambda^ {- 1} \mathrm{d} \delta b,
and, therefore, only the torque relative to the centroid plays a role in the initial stress contribution to the rate of change of virtual work:
\begin{array}{l} \mathrm{d} \delta \Pi = \int_ {\ell} (\mathrm{d} F \delta \overline {{e}} + \mathrm{d} F _ {\alpha} \delta \overline {{\gamma}} _ {\alpha} + \mathrm{d} M ^ {\alpha} \lambda^ {- 1} \delta b _ {\alpha} + \mathrm{d} M _ {t} \lambda^ {- 1} \delta b + \mathrm{d} M _ {w} f ^ {- 2} \delta w _ {p} + \mathrm{d} W \lambda^ {- 1} \delta \chi \\ + F \mathrm{d} \delta \overline {{e}} + F _ {\alpha} \mathrm{d} \delta \overline {{\gamma}} _ {\alpha} + M ^ {\alpha} \lambda^ {- 1} \mathrm{d} \delta b _ {\alpha} + T \lambda^ {- 1} \mathrm{d} \delta b) d \ell . \\ \end{array}
The second variations of \overline { { e } } and \overline { { \gamma } } _ { \alpha } contain contributions of the second variation in curvature and twist. These can be separated out by defining the bending moments and the torque relative to the origin of the cross-sectional coordinate system:
\overline {{M}} ^ {\alpha} = M ^ {\alpha} + \int_ {A} f S _ {c} ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \sigma_ {1 1} d A = M ^ {\alpha} + f S _ {c} ^ {\beta} \epsilon_ {\beta} ^ {\alpha} F,
{\overline {{T}}} {= T + \int_ {A} f S _ {c} ^ {\beta} \epsilon_ {\beta} ^ {\alpha} \tau_ {(\alpha + 1) 1} d A = T + f S _ {c} ^ {\beta} \epsilon_ {\beta} ^ {\alpha} F _ {\alpha}.}
The expression for the rate of change of virtual work then takes the form