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Elements
This is a coarse approximation locally, but the intended applications concern analysis of pipelines, where you are mainly interested in inertia effects associated with locally low mode response. Table 3.9.1-1 shows a comparison of free vibration frequencies for a single 90° bend restrained against rigid body motion on its axis at one end and prevented from warping (but allowed to ovalize) at both ends. The agreement between the various models and the "standard shell element" is not very close, but we feel it is adequate for the intended use. More accurate modeling of inertia would incur a large penalty in computational cost.
| Dimensions: |
| R=0.9144 m (36 in) |
| r=0.2921 m (11.5 in) |
| t=0.0127m (0.5 in) |
| 90° bend |
| Material properties: | |
| Young's modulus: | 2.165 N/m2(31.4 lb/in2) |
| Poisson's ratio: | 0.3 |
| Density: | 7822.8 kg/m3(7.32 × 10-4lb sec2/in4) |
| Boundary conditions: | One end restrained on its axis against rigid body motion; both ends restrained against warping. |
Table 3.9.1-1. Free vibration frequencies for a 9 0 ^ { \circ } elbow.
| Model | Free vibration frequencies, Hertz | ||||||
| Mode 1 | Mode 2 | Mode 3 | Mode 4 | Mode 5 | |||
| Standard shell elements | 79.3 | 83.0 | 193 | 195 | 473 | ||
| Element type ELBOW31 B | 2 elements | P=4 | 73.1 | 78.6 | 163 | 166 | 506 |
| P=6 | 72.3 | 77.8 | 162 | 163 | 505 | ||
| 3 elements | P=4 | 75.4 | 81.4 | 162 | 163 | 176 | |
| P=6 | 74.4 | 80.6 | 160 | 161 | 174 | ||
| Element type ELBOW31 | 2 elements | P=4 | 79.1 | 84.5 | 167 | 169 | 424 |
| P=6 | 78.1 | 83.6 | 160 | 161 | 417 | ||
| 3 elements | P=4 | 82.2 | 88.3 | 179 | 180 | 453 | |
| P=6 | 81.0 | 87.3 | 172 | 174 | 410 | ||
Figure 3.9.1-1 Elbow geometry.
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Extrados t r° φ Crown Pipe axis Intrados R Axis of pipe bend
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(φ,S) a₂ r° a₃ a₁ -x
3.9.2 Pressure loadings on elbow elements
Elbow elements are often used to model pipelines in which the curvature of the pipe can change significantly while the pipe is subjected to uniform or hydrostatic pressure. Therefore, pressure loadings that include large geometry changes are developed for these elements, as described in this section.
The virtual work contribution of pressure on the lateral surface of the elbow is
\delta W _ {L} ^ {p} = \int_ {A} p \delta \mathbf {x} \cdot \mathbf {n} d A,
where
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p
is the pressure magnitude, \delta x is the variational displacement at a point on the midsurface of the lateral wall of the elbow,
n
is the normal to the lateral wall midsurface, and
A
is the area of space occupied by the lateral surface in the current configuration.
The product of the surface normal and the differential area can be rewritten in terms of material coordinates S0 along the pipe and Á around the pipe section:
\mathbf {n} d A = \frac {1}{r ^ {0}} \frac {\partial \mathbf {x}}{\partial \phi} \times \frac {\partial \mathbf {x}}{\partial S ^ {0}} r ^ {0} d \phi d S ^ {0},
where r ^ { 0 } is the initial pipe radius, so that
\delta W _ {L} ^ {p} = \int_ {A ^ {0}} p \frac {1}{r ^ {0}} \delta \mathbf {x} \cdot \frac {\partial \mathbf {x}}{\partial \phi} \times \frac {\partial \mathbf {x}}{\partial S ^ {0}} r ^ {0} d \phi d S ^ {0},
where A ^ { 0 } is the area of space occupied by the lateral surface in the reference configuration.
For hydrostatic pressure, the pressure magnitude is a function of position:
p = (z ^ {0} - \mathbf {x} \cdot \mathbf {k}) \frac {p ^ {1}}{(z ^ {0} - z ^ {1})},
where
p^1 is the reference pressure magnitude, z^0 is the zero pressure height, z^1 is the reference height, and \mathbf{k} is (0,0,1) , a unit vector in the vertical direction.
In the elbow elements position on the lateral surface, x, is interpolated as
\mathbf {x} = \bar {\mathbf {x}} + (r ^ {0} + u ^ {r}) \mathbf {r} + u ^ {t} \mathbf {t} + y ^ {3} \mathbf {a} _ {3},
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where
\bar {\mathbf {x}} (S ^ {0})
is the position of a point on the pipe axis,
u ^ {r} (\phi , S ^ {0})
is the radial displacement,
u ^ {t} (\phi , S ^ {0})
is the tangential displacement,
r
is a1 cos \phi + \mathbf { a } _ { 2 } sin Á, and
t
is ¡a1 sin \phi + \mathbf { a } _ { 2 } cos Á; with { \bf a } _ { 1 } ( S ^ { 0 } ) and { \bf a } _ { 2 } ( S ^ { 0 } ) being the cross-sectional basis vectors.
The first variation of the position can now be expressed as
\delta \mathbf {x} = \delta \bar {\mathbf {x}} + \delta u ^ {r} \mathbf {r} + \delta u ^ {t} \mathbf {t} + \delta y ^ {3} \mathbf {a} _ {3} + \delta \pmb {\omega} \times [ (r ^ {0} + u ^ {r}) \mathbf {r} + u ^ {t} \mathbf {t} + y ^ {3} \mathbf {a} _ {3} ],
and the derivatives of the position with respect to the parametrization are
\frac {\partial \mathbf {x}}{\partial \phi} = (\frac {\partial u ^ {r}}{\partial \phi} - u ^ {t}) \mathbf {r} + (r ^ {0} + u ^ {r} + \frac {\partial u ^ {t}}{\partial \phi}) \mathbf {t} + \frac {\partial y ^ {3}}{\partial \phi} \mathbf {a} _ {3}
and
\begin{array}{l} \frac {\partial \mathbf {x}}{\partial S ^ {0}} = \left[ \frac {d \bar {\mathbf {x}}}{d S ^ {0}} \cdot \mathbf {r} + \frac {\partial u ^ {r}}{\partial S ^ {0}} + u ^ {t} \left(\frac {\partial \mathbf {t}}{\partial S ^ {0}} \cdot \mathbf {r}\right) + y ^ {3} \left(\frac {\partial \mathbf {a} _ {3}}{\partial S ^ {0}} \cdot \mathbf {r}\right) \right] \mathbf {r} \\ + \left[ \frac {d \bar {\mathbf {x}}}{d S ^ {0}} \cdot \mathbf {t} + \frac {\partial u ^ {t}}{\partial S ^ {0}} + (r ^ {0} + u ^ {r}) \frac {\partial \mathbf {r}}{\partial S ^ {0}} \cdot \mathbf {t} + y ^ {3} \frac {\partial \mathbf {a} _ {3}}{\partial S ^ {0}} \cdot \mathbf {t} \right] \mathbf {t} \\ + \left[ \frac {d \bar {\bf x}}{d S ^ {0}} \cdot {\bf a} _ {3} + \frac {\partial y ^ {3}}{\partial S ^ {0}} \cdot {\bf a} _ {3} + (r ^ {0} + u ^ {r}) \frac {\partial {\bf r}}{\partial S ^ {0}} \cdot {\bf a} _ {3} + u ^ {t} \frac {\partial {\bf t}}{\partial S ^ {0}} \cdot {\bf a} _ {3} \right] {\bf a} _ {3}. \\ \end{array}
Assuming that (1) terms in ( { \partial \mathbf { r } } / { \partial { S } ^ { 0 } } ) ¢ t and ( { \partial \mathbf { t } } / { \partial { S } ^ { 0 } } ) ¢ r can be ignored due to negligible twist in the pipe, (2) terms in y ^ { 3 } and its derivatives can be ignored due to negligible warping in the pipe, (3)
{ \bf a } _ { 3 } ( d S / d S ^ { 0 } ) = d \bar { \bf x } / d S ^ { 0 } , (4) the stretch d S / d S ^ { 0 } is unity, and (5) ur and u ^ { t } are small compared to r ^ { 0 } , we arrive at the following expression for the integrand of \delta W _ { L } ^ { p } :
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\begin{array}{l} \frac {p}{r ^ {0}} \delta \mathbf {x} \cdot \frac {\partial \mathbf {x}}{\partial \phi} \times \frac {\partial \mathbf {x}}{\partial S ^ {0}} = \frac {p}{r ^ {0}} \delta \bar {\mathbf {x}} \cdot (1 + r ^ {0} \frac {\partial \mathbf {r}}{\partial S ^ {0}} \cdot \mathbf {a} _ {3}) [ (r ^ {0} + u ^ {r} + \frac {\partial u ^ {t}}{\partial \phi}) \mathbf {r} + (u ^ {t} - \frac {\partial u ^ {r}}{\partial \phi}) \mathbf {t} ] \\ + \frac {p \delta \pmb {\omega}}{r ^ {0}} \cdot (1 + r ^ {0} \frac {\partial \mathbf {r}}{\partial S ^ {0}} \cdot \mathbf {a} _ {3}) [ (r ^ {0} + u ^ {r}) (u ^ {t} - \frac {\partial u ^ {r}}{\partial \phi}) \mathbf {a} _ {3} - u ^ {t} (r ^ {0} + u ^ {r} + \frac {\partial u ^ {t}}{\partial \phi}) \mathbf {a} _ {3} ] \\ + \frac {p \delta u ^ {r}}{r ^ {0}} (1 + r ^ {0} \frac {\partial \mathbf {r}}{\partial S ^ {0}} \cdot \mathbf {a} _ {3}) (r ^ {0} + u ^ {r} + \frac {\partial u ^ {t}}{\partial \phi}) \\ + \frac {p \delta u ^ {t}}{r ^ {0}} (1 + r ^ {0} \frac {\partial \mathbf {r}}{\partial S ^ {0}} \cdot \mathbf {a} _ {3}) (u ^ {t} - \frac {\partial u ^ {r}}{\partial \phi}). \\ \end{array}
For closed-end loading the virtual work contribution of pressure on the end-caps of the elbow is
\delta W _ {E} ^ {p} = \int_ {E ^ {0}} \frac {p}{r} \delta \mathbf {y} \cdot \frac {\partial \mathbf {y}}{\partial r} \times \frac {\partial \mathbf {y}}{\partial \phi} r d \phi d r,
where r and \phi are the material coordinates in a two-dimensional cylindrical coordinate system of points on the end-caps of the elbow element, y represents position on the end-caps, and E ^ { 0 } is the area of space occupied by the end-caps of the elbow element. We assume the following deformation for the end-caps:
\mathbf {y} (r, \phi ; S ^ {0}) = \bar {\mathbf {x}} (S ^ {0}) + \frac {r}{r ^ {0}} [ \mathbf {x} (\phi ; S ^ {0}) - \bar {\mathbf {x}} (S ^ {0}) ],
where S ^ { 0 } is the parameter that identifies the end-cap being considered. The assumed deformation arises naturally on considering a deformation of the end-caps in which radial rays of the reference end-cap configuration remain straight lines under deformation. It can be shown easily that the assumed deformation of the end-caps is differentiable as long as the deformations of the circumferential curves of the end-caps are differentiable. For the end-cap boundary shapes that arise in applications (primarily ovalized modes), the assumed deformation will be locally invertible so that integration of functions over the deformed surface is not likely to be a problem.
Ignoring the terms due to warping in the expression for position on the lateral surface, the first variation of position on an end-cap is
\delta \mathbf {y} = \delta \bar {\mathbf {x}} + \frac {r}{r ^ {0}} \big [ (\delta u ^ {r} - u ^ {t} \delta \pmb {\omega} \cdot \mathbf {a} _ {3}) \mathbf {r} + (\delta u ^ {t} + (r ^ {0} + u ^ {r}) \delta \pmb {\omega} \cdot \mathbf {a} _ {3}) \mathbf {t} + (u ^ {t} \delta \pmb {\omega} \cdot \mathbf {r} - (r ^ {0} + u ^ {r}) \delta \pmb {\omega} \cdot \mathbf {t}) \mathbf {a} _ {3} \big ],
and the derivatives of \mathbf { y } with respect to r and \phi are
\frac {\partial \mathbf {y}}{\partial r} = \frac {1}{r ^ {0}} [ (r ^ {0} + u ^ {r}) \mathbf {r} + u ^ {t} \mathbf {t} ]
and
\frac {\partial \mathbf {y}}{\partial \phi} = \frac {r}{r ^ {0}} [ (\frac {\partial u ^ {r}}{\partial \phi} - u ^ {t}) \mathbf {r} + (r ^ {0} + u ^ {r} + \frac {\partial u ^ {t}}{\partial \phi}) \mathbf {t} ].
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The integrand of the expression for the virtual work of pressure on the end-caps can now be expressed as
\begin{array}{l} \frac {p}{r} \delta \mathbf {y} \cdot \frac {\partial \mathbf {y}}{\partial r} \times \frac {\partial \mathbf {y}}{\partial \phi} = p \{r ^ {0 2} + 2 r ^ {0} u ^ {r} + (r ^ {0} + u ^ {r}) \frac {\partial u ^ {t}}{\partial \phi} \\ - u ^ {t} \frac {\partial u ^ {r}}{\partial \phi} \} [ \frac {1}{r ^ {0 ^ {2}}} \delta \bar {\bf x} \cdot {\bf n} + \frac {r}{r ^ {0 ^ {3}}} \delta {\pmb \omega} \cdot \{u ^ {t} {\bf r} - (r ^ {0} + u ^ {r}) {\bf t} \} {\bf a} _ {3} \cdot {\bf n} ], \\ \end{array}
where n is { \bf a } _ { 3 } if the center of the end-cap is node 1 of the element and { \bf - a _ { 3 } } if the center is node 2 or 3 of the element.
The load stiffness for the pressure loading, which by definition is the first variation of the virtual work of the pressure load, is given by d ( \delta W _ { L } ^ { p } + \delta W _ { E } ^ { p } ) . The following expressions are required for its calculation:
\begin{array}{l} d (\frac {p}{r ^ {0}} \delta \mathbf {x} \cdot \frac {\partial \mathbf {x}}{\partial \phi} \times \frac {\partial \mathbf {x}}{\partial S ^ {0}}) = \frac {d p}{r ^ {0}} \delta \mathbf {x} \cdot \frac {\partial \mathbf {x}}{\partial \phi} \times \frac {\partial \mathbf {x}}{\partial S ^ {0}} \\ + \left\{\frac {p \delta \mathbf {x} \cdot \frac {\partial \mathbf {x}}{\partial \phi} \times \frac {\partial \mathbf {x}}{\partial S ^ {0}}}{(1 + r ^ {0} \frac {\partial \mathbf {r}}{\partial S ^ {0}} \cdot \mathbf {a} _ {3})} \right\} \left[ \left(\frac {\partial d \pmb {\omega}}{\partial S ^ {0}} \times \mathbf {r} + d \pmb {\omega} \times \frac {\partial \mathbf {r}}{\partial S ^ {0}}\right) \cdot \mathbf {a} _ {3} + \frac {\partial \mathbf {r}}{\partial S ^ {0}} \cdot d \pmb {\omega} \times \mathbf {a} _ {3} \right] \\ \end{array}
+ \frac {p}{r ^ {0}} (1 + r ^ {0} \frac {\partial \mathbf {r}}{\partial S ^ {0}} \cdot \mathbf {a} _ {3}) \delta \bar {\mathbf {x}} \cdot [ (d u ^ {r} + \frac {\partial d u ^ {t}}{\partial \phi}) \mathbf {r} + (r ^ {0} + u ^ {r} + \frac {\partial u ^ {t}}{\partial \phi}) d \boldsymbol {\omega} \times \mathbf {r}
+ (d u ^ {t} - \frac {\partial d u ^ {r}}{\partial \phi}) \mathbf {t} + (u ^ {t} - \frac {\partial u ^ {r}}{\partial \phi}) d \pmb {\omega} \times \mathbf {t} ]
+ \frac {p}{r ^ {0}} (1 + r ^ {0} \frac {\partial \mathbf {r}}{\partial S ^ {0}} \cdot \mathbf {a} _ {3}) \delta \pmb {\omega} \cdot \big [ \{d u ^ {r} (u ^ {t} - \frac {\partial u ^ {r}}{\partial \phi}) + (r ^ {0} + u ^ {r}) (d u ^ {t} - \frac {\partial d u ^ {r}}{\partial \phi}) \} \big ]
- d u ^ {t} (r ^ {0} + u ^ {r} + \frac {\partial u ^ {t}}{\partial \phi}) - u ^ {t} (d u ^ {r} + \frac {\partial d u ^ {t}}{\partial \phi}) \} \mathbf {a} _ {3}
+ \left\{(r ^ {0} + u ^ {r}) (u ^ {t} - \frac {\partial u ^ {r}}{\partial \phi}) - u ^ {t} (r ^ {0} + u ^ {r} + \frac {\partial u ^ {t}}{\partial \phi}) \right\} d \pmb {\omega} \times \mathbf {a} _ {3} ]
+ \frac {p}{r ^ {0}} (1 + r ^ {0} \frac {\partial \mathbf {r}}{\partial S ^ {0}} \cdot \mathbf {a} _ {3}) \delta u ^ {r} (d u ^ {r} + \frac {\partial d u ^ {t}}{\partial \phi})
+ \frac {p}{r ^ {0}} (1 + r ^ {0} \frac {\partial \mathbf {r}}{\partial S ^ {0}} \cdot \mathbf {a} _ {3}) \delta u ^ {t} (d u ^ {t} - \frac {\partial d u ^ {r}}{\partial \phi})
and
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\begin{array}{l} d (\frac {p}{r} \delta \mathbf {y} \cdot \frac {\partial \mathbf {y}}{\partial r} \times \frac {\partial \mathbf {y}}{\partial \phi}) = \frac {d p}{r} \delta \mathbf {y} \cdot \frac {\partial \mathbf {y}}{\partial r} \times \frac {\partial \mathbf {y}}{\partial \phi} \\ + p [ 2 r ^ {0} d u ^ {r} + d u ^ {r} \frac {\partial u ^ {t}}{\partial \phi} + (r ^ {0} + u ^ {r}) \frac {\partial d u ^ {t}}{\partial \phi} - d u ^ {t} \frac {\partial u ^ {r}}{\partial \phi} - u ^ {t} \frac {\partial d u ^ {r}}{\partial \phi} ] \\ \big [ \frac {1}{r ^ {0 ^ {2}}} \delta \bar {\bf x} \cdot {\bf n} + \frac {r}{r ^ {0 ^ {3}}} ({\bf a} _ {3} \cdot {\bf n}) \delta {\pmb \omega} \cdot \{u ^ {t} {\bf r} - (r ^ {0} + u ^ {r}) {\bf t} \} \big ] \\ + p [ r ^ {0 ^ {2}} + 2 r ^ {0} u ^ {r} + (r ^ {0} + u ^ {r}) \frac {\partial u ^ {t}}{\partial \phi} - u ^ {t} \frac {\partial u ^ {r}}{\partial \phi} ] \\ \big [ \frac {r}{r ^ {0 ^ {3}}} (\mathbf {a} _ {3} \cdot \mathbf {n}) \delta \pmb {\omega} \cdot \{d u ^ {t} \mathbf {r} + u ^ {t} d \pmb {\omega} \times \mathbf {r} - d u ^ {r} \mathbf {t} - (r ^ {0} + u ^ {r}) d \pmb {\omega} \times \mathbf {t} \} \\ + \left. \frac {1}{r ^ {0 ^ {2}}} \delta \bar {\mathbf {x}} \cdot d \boldsymbol {\omega} \times \mathbf {n} \right]. \\ \end{array}
In the above dp is nonzero only in the case of hydrostatic pressure, when it is given in the first case by
d p = - \frac {p ^ {1}}{(z ^ {0} - z ^ {1})} \mathbf {k} \cdot d \mathbf {x}
and in the second case by
d p = - \frac {p ^ {1}}{(z ^ {0} - z ^ {1})} \mathbf {k} \cdot d \mathbf {y}.
3.9.3 Frame elements with lumped plasticity
Frame elements are designed for the analysis of initially straight, slender beams. The elements are implemented for large displacements and large rotations but small strains. The elastic response of the frame elements follows Euler-Bernoulli beam theory. Plasticity is included in the element's response through a lumped plasticity model with kinematic hardening, which permits yielding only at the ends of the beam. The hardening data are given as a relationship between the generalized force and generalized displacement. Hence, the plastic response of the element is length dependent. The elastic-plastic frame elements are designed to represent plastic hinge formation in frame-like structures, where a single frame element can be used as a member between the structure's nodes.
Degrees of freedom on the element
Frame elements are formulated in terms of the solution variables at user-defined end nodes and extra internal degrees of freedom associated with an internal node. The three-dimensional version of the element is discussed here. The two-dimensional version is found by appropriate reduction of the three-dimensional degrees of freedom. The element has three nodes (two user-defined and one internal), 12 external degrees of freedom, and three internal degrees of freedom. Each of the two end nodes has six external degrees of freedom: three displacements and three rotations. An internal node (at the center of the element) has three displacement degrees of freedom only, as shown in Figure 3.9.3-1.
Figure 3.9.3-1 Frame element in space.
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M_{x1} \rightarrow \boxed{1} \rightarrow N_{x1} \rightarrow N_{x3} \rightarrow N_{x2} \rightarrow N_{x2} \rightarrow M_{x2} \rightarrow x \n M_{y1} \rightarrow \boxed{2} \rightarrow N_{y1} \rightarrow N_{y3} \rightarrow N_{y2} \rightarrow N_{y1} \rightarrow M_{x1} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow x \n M_{x1} \rightarrow M_{y1} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} = -\frac{L}{2} \rightarrow x = \frac{L}{2} \rightarrow M_{x1} \rightarrow M_{y1} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M {x1} \rightarrow M{y1} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y1} \rightarrow M_{x1} \rightarrow M_{y1} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} \rightarrow M_{y2} \rightarrow M_{x2} = -\frac{L}{2} \rightarrow x = \underline{L} / 2
The element is formulated in a local system with the x-direction representing the axial direction and the y- and z-directions representing the directions transverse to the frame element axis. In this local coordinate system the element's degrees of freedom can be written
\mathbf {q} _ {1} = \left\{u _ {x 1}, u _ {y 1}, u _ {z 1}, \phi_ {x 1}, \phi_ {y 1}, \phi_ {z 1} \right\} ^ {T},
\mathbf {q} _ {2} = \left\{u _ {x 2}, u _ {y 2}, u _ {z 2}, \phi_ {x 2}, \phi_ {y 2}, \phi_ {z 2} \right\} ^ {T},
\mathbf {q} _ {3} = \left\{u _ {x 3}, u _ {y 3}, u _ {z 3} \right\} ^ {T}, \quad \mathrm{and}
\mathbf {q} = \left\{\mathbf {q} _ {1} ^ {T}, \mathbf {q} _ {2} ^ {T}, \mathbf {q} _ {3} ^ {T} \right\} ^ {T}.
The elastic formulation
The elastic response of the element is governed by Euler-Bernoulli beam theory. The displacement interpolation for the deflections transverse to the frame element axis (the y- and z-directions) uses fourth-order polynomials, allowing for quadratic variation of the curvature along the beam axis. Let \xi \in [ - 1 , 1 ] be the isoparametric coordinate along the length of the beam. Then,
u _ {y} = a _ {0} + a _ {1} \xi + a _ {2} \xi^ {2} + a _ {3} \xi^ {3} + a _ {4} \xi^ {4},
u _ {z} = b _ {0} + b _ {1} \xi + b _ {2} \xi^ {2} + b _ {3} \xi^ {3} + b _ {4} \xi^ {4}.
The transverse displacement interpolations incorporate exact solutions to force and moment loading at the ends and constant distributed loads along the beam axis (such as gravity loading). The
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displacement interpolation function along the frame element axis (the x-direction) is a second-order polynomial, allowing for linear variation of the axial strain along the frame element axis:
u _ {x} = c _ {0} + c _ {1} \xi + c _ {2} \xi^ {2}.
The twist rotation degree of freedom interpolation along the beam axis (rotation about the x-axis) is linear, allowing for constant twist strain:
\phi_ {x} = d _ {0} + d _ {1} \xi .
The generalized strains, following Euler-Bernoulli beam theory, are
\epsilon_ {x} = \frac {2}{L} \frac {\mathrm{d} u _ {x}}{\mathrm{d} \xi}, \quad \kappa_ {z} = \frac {4}{L ^ {2}} \frac {\mathrm{d} ^ {2} u _ {y}}{\mathrm{d} \xi^ {2}}, \quad \kappa_ {y} = - \frac {4}{L ^ {2}} \frac {\mathrm{d} ^ {2} u _ {z}}{\mathrm{d} \xi^ {2}}, \quad \epsilon_ {t} = \frac {2}{L} \frac {\mathrm{d} \phi_ {x}}{\mathrm{d} \xi},
where \epsilon _ { x } is the axial strain, \kappa _ { z } and \kappa _ { y } are the beam curvatures, and \epsilon _ { t } is the twist strain. The 15 undetermined constants in the interpolation equations for the displacements are determined by introducing the nodal degrees of freedom; that is,
u _ {y} (\xi = - 1) \stackrel {\mathrm{def}} {=} u _ {y 1}, \quad \frac {\partial u _ {z}}{\partial \xi} (\xi = 1) \stackrel {\mathrm{def}} {=} - \frac {L}{2} \phi_ {y 2}, \quad u _ {x} (\xi = 0) \stackrel {\mathrm{def}} {=} u _ {x 3}, \quad \mathrm{etc.}
The interpolations in terms of the nodal degrees of freedom are described below in the section discussing the large-displacement formulation.
The strain-displacement relationship is written in matrix form as
\varepsilon = \mathbf {B q},
where B is a 4 15 matrix and
\pmb {\varepsilon} = \left\{\epsilon_ {x}, \kappa_ {z}, \kappa_ {y}, \epsilon_ {t} \right\} ^ {T}.
The elastic stiffness matrix is integrated numerically and used to calculate 15 nodal forces and moments--12 forces/moments (also called generalized forces) associated with the two end nodes,
\mathbf {F} _ {1} = \left\{N _ {x 1}, N _ {y 1}, N _ {z 1}, M _ {x 1}, M _ {y 1}, M _ {z 1} \right\} ^ {T},
\mathbf {F} _ {2} = \left\{N _ {x 2}, N _ {y 2}, N _ {z 2}, M _ {x 2}, M _ {y 2}, M _ {z 2} \right\} ^ {T},
and three forces associated with the internal node,
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\mathbf {F} _ {3} = \left\{N _ {x 3}, N _ {y 3}, N _ {z 3} \right\} ^ {T}.
The vector of forces and moments for the frame element can be written as
\mathbf {F} = \left\{\mathbf {F} _ {1} ^ {\mathbf {T}}, \mathbf {F} _ {2} ^ {\mathbf {T}}, \mathbf {F} _ {3} ^ {\mathbf {T}} \right\} ^ {T}.
The elastic stiffness is, therefore, a 15 £ 15 matrix relating the force vector, F, and the nodal displacement vector, q:
\mathbf {F} = \mathbf {K} ^ {\mathbf {e}} \cdot \mathbf {q}.
Material properties of frame elements can, in general, be temperature dependent. Let us define the elastic strain vector as
\varepsilon^ {e l a s t i c} = \varepsilon - \varepsilon^ {t h} + \varepsilon^ {i n i t i a l},
where " denotes the total strain and \varepsilon ^ { t h } denotes the thermal expansion strain, where only the axial strain is nonzero and is given by
\epsilon_ {x} ^ {t h} = \alpha (\theta) (\theta - \theta^ {0}) - \alpha (\theta_ {I}) (\theta_ {I} - \theta^ {0}),
where
\alpha (\theta) is the thermal expansion coefficient, \theta is the current temperature at the frame element section, \theta^0 is the reference temperature for \alpha , and \theta_I
is the initial temperature at this point defined on the *INITIAL CONDITIONS option (``Initial conditions,'' Section 19.2.1 of the ABAQUS/Standard User's Manual).
The temperature field is defined by the user at the element's ends and is assumed to be linear along the element axis but constant within the element cross-section. If the thermal expansion coefficient is temperature dependent, it is evaluated at the nodes. Thermal strains are calculated at the element's end nodes, and thermal strains at the integration points are interpolated from the nodal points using appropriate interpolation schemes: axial strains are interpolated linearly, curvatures are interpolated quadratically, and twist strain is constant along the frame element axis.
Initial generalized strains, "initial , are calculated from the initial generalized forces given by the user on the *INITIAL CONDITIONS, TYPE=STRESS option, using the following relationship: