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Elements
\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 + \mathrm{d} W \lambda^ {- 1} \delta \overline {{\chi}} _ {\alpha})
F (\lambda^ {- 1} \mathrm{d} \delta \lambda + 2 \lambda^ {- 1} \delta \lambda f ^ {- 1} \mathrm{d} f + f ^ {2} \lambda^ {- 2} \left(\frac {I _ {p}}{A} + S _ {c} ^ {\alpha} S _ {c} ^ {\alpha}\right) \mathrm{d} b \delta b) +
F _ {\alpha} \mathrm{d} \delta \gamma_ {\alpha} + \overline {{M}} ^ {\alpha} \lambda^ {- 1} \mathrm{d} \delta b _ {\alpha} + \overline {{T}} \lambda^ {- 1} \mathrm{d} \delta b) d \ell .
We propose to make the "cross-section size" a function of the stretch ¸. For the thermal stretch of the section we use isotropic expansion \left( f _ { \theta } = \lambda _ { \theta } \right) ; and for the "mechanical" stretch of the section we assume an effective Poisson's ratio \nu \ ( f _ { m } = \lambda _ { m } ^ { - \nu } ) . The total cross-sectional stretch is
f = f _ {m} f _ {\theta} = \lambda_ {m} ^ {- \nu} \lambda_ {\theta} = \lambda^ {- \nu} \lambda_ {\theta} ^ {1 + \nu},
so that
\mathrm{d} f = - \nu \lambda^ {- 1 - \nu} \lambda_ {\theta} ^ {1 + \nu} \mathrm{d} \lambda .
Using this in the above expression for the rate of change of virtual work, we find
\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 + \mathrm{d} W \lambda^ {- 1} \delta \chi)
F (\lambda^ {- 1} \mathrm{d} \delta \lambda - 2 \nu \lambda^ {- 2} \mathrm{d} \lambda \delta \lambda + f ^ {2} \lambda^ {- 2} \left(\frac {I _ {p}}{A} + S _ {c} ^ {\alpha} S _ {c} ^ {\alpha}\right) \mathrm{d} b \delta b) +
F _ {\alpha} \mathrm{d} \delta \gamma_ {\alpha} + \overline {{M}} ^ {\alpha} \lambda^ {- 1} \mathrm{d} \delta b _ {\alpha} + \overline {{T}} \lambda^ {- 1} \mathrm{d} \delta b) d \ell .
This expression is symmetric if the material tensor [D] is symmetric.
Section integration
The formulation presented in the previous pages is valid for all possible beam types. However, different classes of beams will result in different final formulations. We consider three different classes of beams:
- Beams in which warping may be constrained. These beams generally have an open, thin walled section reinforced with some relatively solid parts or some relatively small closed cells and have a torsional stiffness that is considerably smaller than the polar moment of inertia. Hence, in the elastic range, the warping can be large, and warping prevention at the ends can contribute significantly to the torsional rigidity of the beam. In this case both the torsional shear stresses and the axial warping stresses can be of the same order of magnitude as the stresses due to axial forces and bending moments, and the complete theory must be used.
- Beams in which warping is unconstrained. These beams generally have a solid section or a closed, thin walled section and have a torsional stiffness that is of the same order of magnitude as the polar moment of inertia of the section. Hence, in the elastic range the warping is rather small, and it is assumed that warping prevention at the ends can be neglected. The axial warping stresses are
Elements
assumed to be negligible, but the torsional shear stresses are assumed to be of the same order of magnitude as the stresses due to axial forces and bending moments. In this case the warping is dependent on the twist and can be eliminated as an independent variable, which leads to a considerably simplified formulation.
- Beams in which warping constraints dominate the torsional rigidity. These beams generally have an open, thin walled section and have a torsional stiffness that is much smaller than the polar moment of inertia. In the elastic range the warping is likely to be large, and warping constraints are essential to provide torsional stiffness for the beam. In this case the axial stresses may be of the same order of magnitude as the stresses due to axial forces and bending moments, but the torsional shear stresses are relatively small. Hence, the warping can be coupled to the twist with a relatively stiff elastic constraint but cannot be eliminated because it must be possible to prevent warping at the nodes.
Examples of cross-sections for the last two classes will be derived after the general discussion of sections with and without warping prevention.
In ABAQUS we neglect the effect of shear stresses due to transverse shear forces at individual material points. We will consequently always assume elastic behavior of the section in transverse shear, leading to the relations
F _ {\alpha} = f _ {p} G A \overline {{\gamma}} _ {\alpha}, \qquad \mathrm{d} F _ {\alpha} = f _ {p} G A \mathrm{d} \overline {{\gamma}} _ {\alpha},
where F _ { \alpha } are the transverse shear forces, working at the shear center, and f _ { p } is the "slenderness compensation factor" used to prevent the shear stiffness from becoming too big in slender beams. The slenderness compensation factor is defined as
f _ {p} = \left(1 + 0. 2 5 \frac {\ell^ {2} A}{1 2 I}\right) ^ {- 1} \quad \mathrm{forfirst-orderTimoshenkobeamelementsand}
f _ {p} = \left(1 + 0. 2 5 \times 1 0 ^ {- 4} \frac {\ell^ {2} A}{1 2 I}\right) ^ {- 1} \mathrm{forallotherbeamelements},
where ` is the length of the element and I is the larger of the moments of inertia I _ { 1 1 } and I _ { 2 2 } . Hence, the transverse shear terms do not need to be considered in any further detail.
The fact that the transverse shear forces are considered separately allows us to write
\gamma_ {(\alpha + 1) 1} = \gamma_ {(\alpha + 1) 1} ^ {F} + \gamma_ {(\alpha + 1) 1} ^ {M},
\tau_ {(\alpha + 1) 1} = \tau_ {(\alpha + 1) 1} ^ {F} + \tau_ {(\alpha + 1) 1} ^ {M},
where \gamma _ { ( \alpha + 1 ) 1 } ^ { F } and \tau _ { ( \alpha + 1 ) 1 } ^ { F } are the strains and stresses due to transverse shear forces and \gamma _ { ( \alpha + 1 ) 1 } ^ { M } and ¿ M(®+1)1 are the strains and stresses due to a twisting around the shear center. Substitution in the \tau _ { ( \alpha + 1 ) 1 } ^ { M } expressions for the twisting and warping moments yields
Elements
\begin{array}{l} M _ {t} = \int_ {A} f \gamma_ {\alpha} ^ {t} (\tau_ {(\alpha + 1) 1} ^ {F} + \tau_ {(\alpha + 1) 1} ^ {M}) d A = \int_ {A} f \gamma_ {\alpha} ^ {t} \tau_ {(\alpha + 1) 1} ^ {M} d A, \\ M _ {w} = \int_ {A} f \left(\epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) - \gamma_ {\alpha} ^ {t}\right) (\tau_ {(\alpha + 1) 1} ^ {F} + \tau_ {(\alpha + 1) 1} ^ {M}) d A \\ = \epsilon_ {\beta} ^ {\alpha} (S _ {s} ^ {\beta} - S _ {c} ^ {\beta}) F _ {\alpha} + \int_ {A} f \left(\epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) - \gamma_ {\alpha} ^ {t}\right) \tau_ {(\alpha + 1) 1} ^ {M} d A, \\ \end{array}
where we used the fact that \gamma _ { \alpha } ^ { t } was calculated based on application of a twisting moment around the shear center and, hence, does not do any work on the shear stresses due to transverse shear forces.
The warping function − is assumed to be determined based on isotropic, homogeneous elastic behavior of the section in shear. For this case the elastic energy due to twist is
U _ {t} = \frac {1}{2} G \int_ {A} \gamma_ {(\alpha + 1) 1} ^ {M} \gamma_ {(\alpha + 1) 1} ^ {M} d A.
For twist without warping constraints w _ { p } vanishes and the energy per unit length of the beam is
U _ {t} = \frac {1}{2} f ^ {2} \lambda^ {- 2} (b - B) ^ {2} G \int_ {A} \gamma_ {\alpha} ^ {t} \gamma_ {\alpha} ^ {t} d A = \frac {1}{2} f ^ {2} \lambda^ {- 2} (b - B) ^ {2} G J,
where we have introduced the torsion integral J given by
J = \int_ {A} \gamma_ {\alpha} ^ {t} \gamma_ {\alpha} ^ {t} d A.
With complete warping prevention the w _ { p } = f ^ { 2 } \lambda ^ { - 1 } ( b - B ) and the energy per unit length of the beam is
U _ {t} = \frac {1}{2} f ^ {2} \lambda^ {- 2} (b - B) ^ {2} G \int_ {A} \epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\gamma} ^ {\alpha} (S ^ {\gamma} - S _ {c} ^ {\gamma}) d A = \frac {1}{2} f ^ {2} \lambda^ {- 2} (b - B) ^ {2} G I _ {p},
where we have introduced the polar moment of inertia
I _ {p} = \int_ {A} (S ^ {\beta} - S _ {c} ^ {\beta}) (S ^ {\beta} - S _ {c} ^ {\beta}) d A = \int_ {A} \epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\gamma} ^ {\alpha} (S ^ {\gamma} - S _ {c} ^ {\gamma}) d A.
For unconstrained warping \tau _ { ( \alpha + 1 ) 1 } = G \gamma _ { \alpha } ^ { t } . Since the twisting moment must be equal to
M _ {t} = \int_ {A} f (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\beta} ^ {\alpha} \tau_ {(\alpha + 1) 1} d A = G \int_ {A} f (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\beta} ^ {\alpha} \gamma_ {\alpha} ^ {t} d A,
and since
M _ {t} = \int_ {A} f \gamma_ {\alpha} ^ {t} \tau_ {(\alpha + 1) 1} d A = G \int_ {A} f \gamma_ {\alpha} ^ {t} \gamma_ {\alpha} ^ {t} d A,
it follows that
\int_ {A} (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\beta} ^ {\alpha} \gamma_ {\alpha} ^ {t} d A = \int_ {A} \gamma_ {\alpha} ^ {t} \gamma_ {\alpha} ^ {t} d A = J.
Beams with unconstrained warping
For this beam type, warping prevention is not taken into consideration. Hence we assume w _ { p } = 0 . In addition we assume that axial strains due to warping can be neglected: \chi \Omega \approx 0 .
For the strains at a material point this yields
\begin{array}{l} {e _ {1 1}} {= \overline {{e}} - (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\beta} ^ {\alpha} (f \lambda^ {- 1} b _ {\alpha} - B _ {\alpha}),} \\ {\gamma_ {(\alpha + 1) 1} ^ {M}} {= f \lambda^ {- 1} \gamma_ {\alpha} ^ {t} (b - B),} \\ \end{array}
and for the variations in the strains
\begin{array}{l} \delta e _ {1 1} \quad = \delta \overline {{e}} - f \lambda^ {- 1} (S ^ {\beta} - S _ {c} ^ {\beta}) \epsilon_ {\beta} ^ {\alpha} \delta b _ {\alpha}, \\ \delta \gamma_ {(\alpha + 1) 1} ^ {M} = f \lambda^ {- 1} \gamma_ {\alpha} ^ {t} \delta b. \\ \end{array}
Substitution in the virtual work statement yields
\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) d \ell ,
where the expressions derived earlier for F , F _ { \alpha } , M ^ { \alpha } , and M _ { t } apply.
Although there is no warping prevention in the section, the warping moment M _ { w } does not vanish. From the expression obtained earlier follows
M _ {w} = \epsilon_ {\beta} ^ {\alpha} (S _ {s} ^ {\beta} - S _ {c} ^ {\beta}) F _ {\alpha}.
This yields for the torque around the centroid
T = \epsilon_ {\beta} ^ {\alpha} (S _ {s} ^ {\beta} - S _ {c} ^ {\beta}) F _ {\alpha} + M _ {t}
and for the torque around the origin
\overline {{T}} = \epsilon_ {\beta} ^ {\alpha} S _ {s} ^ {\beta} F _ {\alpha} + M _ {t}.
For the rate of change of virtual work we obtain
\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 +
F (\lambda^ {- 1} \mathrm{d} \delta \lambda - 2 \nu \lambda^ {- 2} \mathrm{d} \lambda \delta \lambda + f ^ {2} \lambda^ {- 2} \left(\frac {I _ {p}}{A} + S _ {c} ^ {\alpha} S _ {c} ^ {\alpha}\right) \mathrm{d} b \delta b) +
F _ {\alpha} \mathrm{d} \delta \gamma_ {\alpha} + \overline {{M}} ^ {\alpha} \lambda^ {- 1} \mathrm{d} \delta b _ {\alpha} + \overline {{T}} \lambda^ {- 1} \mathrm{d} \delta b) d \ell .
Beams with elastic torsion and constrained warping
Consider the case that the shear stresses are defined from the shear strains by linear elastic response, with a constant shear modulus G:
\tau_ {(\alpha + 1) 1} ^ {M} = G \gamma_ {(\alpha + 1) 1} ^ {M} = f \lambda^ {- 1} G \gamma_ {\alpha} ^ {t} (b - B) + f ^ {- 1} G \left(\epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) - \gamma_ {\alpha} ^ {t}\right).
This allows us to write for the twisting and warping moments:
\begin{array}{l} M _ {t} = f ^ {2} \lambda^ {- 1} G (b - B) \int_ {A} \gamma_ {\alpha} ^ {t} \gamma_ {\alpha} ^ {t} d A + G w _ {p} \int_ {A} \gamma_ {\alpha} ^ {t} \left(\epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) - \gamma_ {\alpha} ^ {t}\right) d A \\ = f ^ {2} \lambda^ {- 1} G J (b - B) \\ \end{array}
M _ {w} = f ^ {2} \lambda^ {- 1} G (b - B) \int_ {A} \gamma_ {\alpha} ^ {t} \left(\epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) - \gamma_ {\alpha} ^ {t}\right) d A
+ G w _ {p} \int_ {A} \left(\epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {c} ^ {\beta}) - \gamma_ {\alpha} ^ {t}\right) \left(\epsilon_ {\gamma} ^ {\alpha} (S ^ {\gamma} - S _ {c} ^ {\gamma}) - \gamma_ {\alpha} ^ {t}\right) d A
= G (I _ {p} - J) w _ {p}.
Substitution of these expressions in the part of the virtual work equation related to torsion yields
\delta \Pi_ {t} = \int_ {\ell} (\lambda^ {- 1} M _ {t} \delta b + f ^ {- 2} M _ {w} \delta w _ {p}) d \ell = \int_ {\ell} \left(f ^ {2} \lambda^ {- 2} G J (b - B) \delta b + f ^ {- 2} G (I _ {p} - J) w _ {p} \delta w _ {p}\right) d \ell .
Note that the torque T around the centroid is
T = M _ {t} + M _ {w} = G J f ^ {2} \lambda^ {- 1} (b - B) + G (I _ {p} - J) w _ {p}.
Hence, we can write virtual work in terms of the primary variables b and w:
\delta \Pi_ {t} = \int_ {\ell} \left(\lambda^ {- 1} T \delta b - f ^ {- 2} M _ {w} \delta w\right) d A
= \int_ {\ell} \left[ \left(f ^ {2} \lambda^ {- 2} G J (b - B) + \lambda^ {- 1} G (I _ {p} - J) w _ {p}\right) \delta b - f ^ {- 2} G (I _ {p} - J) w _ {p} \delta w \right] d \ell .
The complete virtual work equation has the form
Elements
\delta \Pi = \int_ {\ell} (F \delta \overline {{{e}}} + F _ {\alpha} \delta \overline {{{\gamma}}} _ {\alpha} + \lambda^ {- 1} M ^ {\alpha} \delta b _ {\alpha} + f ^ {2} \lambda^ {- 2} G J b \delta b + f ^ {- 2} G (I _ {p} - J) w _ {p} \delta w _ {p} + \lambda^ {- 1} W \delta \chi) d \ell ,
where F , F _ { \alpha } , M ^ { \alpha } and W are defined as before.
For the rate of change of virtual work we obtain similarly
\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} + f ^ {2} \lambda^ {- 2} G J \mathrm{d} b \delta b + f ^ {- 2} G (I _ {p} - J) \mathrm{d} w _ {p} \delta w _ {p} +
\mathrm{d} W \lambda^ {- 1} \delta \chi + F (\lambda^ {- 1} \mathrm{d} \delta \lambda - 2 \nu \lambda^ {- 2} \mathrm{d} \lambda \delta \lambda + f ^ {2} \lambda^ {- 2} \left(\frac {I _ {p}}{A} + S _ {c} ^ {\alpha} S _ {c} ^ {\alpha}\right) \mathrm{d} b \delta b) +
F _ {\alpha} \mathrm{d} \delta \gamma_ {\alpha} + \overline {{M}} ^ {\alpha} \lambda^ {- 1} \mathrm{d} \delta b _ {\alpha} + \overline {{T}} \lambda^ {- 1} \mathrm{d} \delta b) d \ell
with
\overline {{T}} = T + f \epsilon_ {\beta} ^ {\alpha} S _ {c} ^ {\beta} F _ {\alpha}.
We now discuss some specific section types incorporated in ABAQUS.
Circular section
For this type of section, warping is absent. Hence,
\Omega = \frac {\partial \Omega}{\partial S ^ {\alpha}} = 0.
Solid noncircular sections
Solid sections such as rectangles or trapezoids are included in this category. The warping function à is a harmonic function and is subject to the condition that no shear stress component can act normal to the boundary of the cross-section. Although it is possible to determine the warping function à in this manner, we choose to work in terms of the Saint-Venant's stress function because of its simplicity. Following standard procedures we normalize this function so that the (elastic) shear strains can be derived directly from it. We introduce the function \varphi ( S ^ { \beta } ) , which is differentiable in the cross-section and has the property that
\gamma_ {\alpha} ^ {t} = \epsilon_ {\alpha} ^ {\beta} \frac {\partial \varphi}{\partial S ^ {\beta}}.
The stress function is determined by solving the differential equation of the form
\nabla^ {2} \varphi = - 2 \quad \text {in} A, \text {with} \varphi = 0 \quad \text {on} S,
where S represents the boundary of the section. This boundary condition ensures that no shear stress component can act normal to the boundary.
Elements
For the solid noncircular sections this differential equation is solved numerically using a second-order isoparametric finite element. The torsional constant of the bar is then equal to twice the volume under the normalized stress function surface.
Closed, thin walled cross-sections
In this case we assume that the shear strain perpendicular to the section must vanish so that
(\epsilon_ {\beta} ^ {\alpha} S ^ {\beta} + \frac {\partial \Omega}{\partial S ^ {\alpha}}) n _ {\alpha} = 0.
Since t _ { \beta } = - \epsilon _ { \beta } ^ { \alpha } n _ { \alpha } , this yields
\frac {\partial \Omega}{\partial n} = \frac {\partial \Omega}{\partial S ^ {\alpha}} n _ {\alpha} = t _ {\beta} S ^ {\beta},
which can be identically satisfied anywhere. The normalized shear strain along the section is
\overline {{\gamma}} = (\epsilon_ {\beta} ^ {\alpha} S ^ {\beta} + \frac {\partial \Omega}{\partial S ^ {\alpha}}) t _ {\alpha} = n _ {\beta} S ^ {\beta} + \frac {\partial \Omega}{\partial s},
where s is the distance along the section. Integrating this around the circumference yields
\oint \overline {{\gamma}} d s = \oint n _ {\alpha} S ^ {\alpha} d s + \oint \frac {\partial \Omega}{\partial s} d s = 2 A _ {C},
where A _ { C } is the area enclosed by the section. With the assumption that the shear modulus is constant along the section, the total torsional elastic strain energy is
U _ {T} = \oint \frac {1}{2} G \overline {{\gamma}} ^ {2} t d s,
where t is the wall thickness. Minimization of the energy with the constraint enforced with a Lagrange multiplier \mu yields
\delta P = \oint (G \overline {{\gamma}} h - \mu) \delta \overline {{\gamma}} d s - (\oint \overline {{\gamma}} d s - 2 A _ {C}) \delta \mu = 0.
Hence, \overline { { { \gamma } } } = \frac { \mu } { G h } and \frac { \mu } { G } \oint \frac { 1 } { h } d s - A _ { C } = 0 ; which combine to define
\overline {{\gamma}} = \frac {2 A _ {C}}{h \oint \frac {1}{h} d s}.
This allows calculation of \overline { \gamma } at any point in the section based on the section geometry.
Thin walled open sections
Elements
The most important sections that exhibit substantial warping are the thin walled open sections. For a single branch section we can conveniently express − as a function of the coordinate s along the section and the coordinate z perpendicular to the section. A suitable approximation for − is
\Omega (s, z) = \Omega_ {c} (s) + z \eta (s),
where \Omega _ { c } ( s ) is the value at the centerline of the wall. Minimization of the torsional elastic energy yields
\int_ {s _ {1}} ^ {s _ {2}} \int_ {- h / 2} ^ {h / 2} \left[ \left(t _ {\alpha} \epsilon_ {\beta} ^ {\alpha} (S ^ {\beta} - S _ {s} ^ {\beta}) + \frac {d \Omega_ {c}}{d s} + z \frac {d \eta}{d s}\right) \left(\frac {d \delta \Omega_ {c}}{d s} + z \frac {d \delta \eta}{d s}\right) + (n _ {\alpha} \epsilon_ {\beta} ^ {\alpha} S ^ {\beta} + \eta) \delta \eta \right] d z d s = 0.
We now introduce S _ { 0 } ^ { \beta } , which is the position of the middle of the wall. With t _ { \alpha } \epsilon _ { \beta } ^ { \alpha } = n _ { \beta } and
S ^ { \beta } = S _ { 0 } ^ { \beta } + z n _ { \beta } , and after carrying out the integration over z, the minimization condition simplifies to
\int_ {s _ {1}} ^ {s _ {2}} \left[ h \left(n _ {\beta} (S _ {0} ^ {\beta} - S _ {s} ^ {\beta}) + \frac {d \Omega_ {c}}{d s}\right) \frac {d \delta \Omega_ {c}}{d s} + h \left(- t _ {\beta} (S _ {0} ^ {\beta} - S _ {s} ^ {\beta}) + \eta\right) \delta \eta + \frac {1}{1 2} h ^ {3} \left(1 + \frac {d \eta}{d s}\right) \frac {d \delta \eta}{d s} \right] d s = 0.
Clearly, the dominant terms are of order h. This yields the equations
\frac {d \Omega_ {c}}{d s} = - n _ {\beta} (S _ {0} ^ {\beta} - S _ {s} ^ {\beta}), \qquad \eta = t _ {\beta} (S _ {0} ^ {\beta} - S _ {s} ^ {\beta}),
so that \Omega ( s , z ) is
\Omega (s, z) = \int_ {0} ^ {s} - n _ {\beta} (S _ {0} ^ {\beta} - S _ {s} ^ {\beta}) d s + t _ {\beta} (S _ {0} ^ {\beta} - S _ {s} ^ {\beta}) z + \Omega_ {s},
where \Omega _ { s } is the value of − at the start of the integration over the section. Observe that
\Omega_ {c} (s) = \int_ {0} ^ {s} - n _ {\beta} (S _ {0} ^ {\beta} - S _ {s} ^ {\beta}) d s + \Omega_ {s}
represents the sectorial area between two points on the midsection relative to the shear center.
Therefore, it readily follows that
\int_ {0} ^ {s _ {e n d}} h \Omega_ {c} S ^ {\alpha} d s = 0,
so there is no coupling between twist and bending in the section for linear elastic material behavior. As was discussed before, \Omega _ { s } must be chosen such that
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\overline {{\Omega}} = \int_ {0} ^ {s _ {e n d}} h \Omega_ {c} d s / \int_ {0} ^ {s _ {e n d}} h d s = 0,
which eliminates the coupling between twist and axial extension in the section for linear elasticity.
Note that coupling terms still exist but that they are incorporated in the generalized strain displacement relations. The coupling between twist and extension is governed by \Omega _ { 0 } , the value of the warping function at the origin of the cross-sectional coordinate system. If the origin is on the section, this value can be evaluated properly. If the origin is not on the section (which means that the node is not connected to the section), we assume that \Omega _ { 0 } = 0 .
The torsion integral J is readily obtained as
\begin{array}{l} J = \int_ {s _ {1}} ^ {s _ {2}} \int_ {- h / 2} ^ {h / 2} \left[ \left(n _ {\beta} (S ^ {\beta} - S _ {s} ^ {\beta}) + \frac {d \Omega}{d s}\right) ^ {2} + \left(- t _ {\beta} (S ^ {\beta} - S _ {s} ^ {\beta}) + \frac {d \Omega}{d z}\right) ^ {2} \right] d z d s \\ = \int_ {s _ {1}} ^ {s _ {2}} \int_ {- h / 2} ^ {h / 2} (2 z) ^ {2} d z d s = \int_ {s _ {1}} ^ {s _ {2}} \frac {1}{3} h ^ {3} d s, \\ \end{array}
and the polar moment of inertia is given by the expression
I _ {p} = \int_ {s _ {1}} ^ {s _ {2}} h (S ^ {\beta} - S _ {c} ^ {\beta}) (S ^ {\beta} - S _ {c} ^ {\beta}) d s.
The above derivations cover single branch open sections. Multibranch open sections can be transformed into single branch open sections by connecting the end of one branch with the beginning of the next branch with a section that has thickness h = 0 . Such dummy sections do not yield any contribution to the area, the moments of inertia, or the torsion integral and, hence, have no influence on the results.
3.5.3 Euler-Bernoulli beam elements
In these elements it is assumed that the internal virtual work rate is associated with axial strain and torsional shear only. Further, it is assumed that the cross-section does not deform in its plane or warp out of its plane, and that this cross-sectional plane remains normal to the beam axis. These are the classical assumptions of the Euler-Bernoulli beam theory, which provides satisfactory results for slender beams.
Let ( S , g , h ) be material coordinates such that S locates points on the beam axis and ( g , h ) measures distance in the cross-section. In addition, let \mathbf { n } _ { 1 } , \ \mathbf { n } _ { 2 } be unit vectors normal to the beam axis in the current configuration: { \mathbf n } _ { 1 } = { \mathbf n } _ { 1 } ( S ) ; { \mathbf { n } } _ { 2 } = { \mathbf { n } } _ { 2 } ( S ) . Then the position of a point of the beam in the current configuration is
\mathbf {x} ^ {f} = \mathbf {x} + g \mathbf {n} _ {1} + h \mathbf {n} _ {2},
where \mathbf { x } = \mathbf { x } ( S ) is the point on the beam axis of the cross-section containing \mathbf { x } ^ { f } . Then
Elements
\frac {d \mathbf {x} ^ {f}}{d S} = \frac {d \mathbf {x}}{d S} + g \frac {d \mathbf {n} _ {1}}{d S} + h \frac {d \mathbf {n} _ {2}}{d S},
and so length on the fiber at ( S , g , h ) is measured in the current configuration as
\begin{array}{l} (d l ^ {f}) ^ {2} = \frac {d \mathbf {x} ^ {f}}{d S} \cdot \frac {d \mathbf {x} ^ {f}}{d S} (d S) ^ {2} \\ = \left(\frac {d \mathbf {x}}{d S} + g \frac {d \mathbf {n} _ {1}}{d S} + h \frac {d \mathbf {n} _ {2}}{d S}\right) \cdot \left(\frac {d \mathbf {x}}{d S} + g \frac {d \mathbf {n} _ {1}}{d S} + h \frac {d \mathbf {n} _ {2}}{d S}\right) (d S) ^ {2}. \\ \end{array}
Now since the beam is slender, we will neglect terms of second-order in g and h, the distance measuring material coordinates in the cross-section. Thus,
Equation 3.5.3-1
(d l ^ {f}) ^ {2} = \left(\frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {x}}{d S} + 2 g \frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {n} _ {1}}{d S} + 2 h \frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {n} _ {2}}{d S}\right) (d S) ^ {2}.
Strain measures
The internal virtual work rate associated with axial stress is
\delta W _ {1} ^ {I} = \int_ {L ^ {f}} \int_ {A} \sigma^ {f} \delta \varepsilon^ {f} d A d L ^ {f},
where \sigma ^ { f } and \delta \varepsilon ^ { f } are any material stress and strain measures associated with axial deformation at the point ( S , g , h ) of the beam, since strains are assumed to be small. For this purpose we will use Green's strain so that
\varepsilon^ {f} = \frac {1}{2} [ (\lambda^ {f}) ^ {2} - 1 ],
where ( \lambda ^ { f } ) ^ { 2 } = ( d l ^ { f } ) ^ { 2 } / ( d L ^ { f } ) ^ { 2 } , the square of the ratio of current configuration length to reference configuration length in the axial direction on the fiber. From Equation 3.5.3-1 and its equivalent in the reference configuration, we have
\begin{array}{l} \varepsilon^ {f} d L ^ {f} = \frac {1}{2} \left[ \left(\frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {x}}{d S} + 2 g \frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {n} _ {1}}{d S} + 2 h \frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {n} _ {2}}{d S}\right) \right. \\ \left. \cdot \left(\frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {x}}{d S} + 2 g \frac {d \mathbf {X}}{d S} \cdot \frac {d \mathbf {N} _ {1}}{d S} + 2 h \frac {d \mathbf {x}}{d S} \cdot \frac {d \mathbf {N} _ {2}}{d S}\right) ^ {- 1} - 1 \right] \\ \cdot \left(\frac {d \mathbf {X}}{d S} \cdot \frac {d \mathbf {X}}{d S} + 2 g \frac {d \mathbf {X}}{d S} \cdot \frac {d \mathbf {N} _ {1}}{d S} + 2 h \frac {d \mathbf {X}}{d S} \cdot \frac {d \mathbf {N} _ {2}}{d S}\right) ^ {\frac {1}{2}} d S. \\ \end{array}
Again, neglecting all but first-order terms in g and h because of the slenderness assumption, this becomes