C.1 Transfer matrices of first-order analysis

Transfer matrices of first-order structural analysis are treated in [Bor79a], [Krä91a], and [PW94]. The transfer equation for a frame element is

$$\mathbf{Z_{p}} = \mathbf{U\cdot Z_{v}} + \mathbf{\stackrel{\rm\circ}{Z}}$$ (C.1)

where

$$\mathbf{Z_{p}} = \left[ \begin{array}{c} u \\ w \\ \varphi_{y} ... ... w \\ \varphi_{y} \\ N_{x} \\ Q_{z} \\ M_{y} \end{array} \right]_{v} \quad$$ (C.2)

and $\mathbf{U_{x}}$ is the transfer matrix (Sign Convention 2):

$$\mathbf{U} = \left[ \begin{array}{cccccc} 1 & 0 & 0 & - \frac{\le... ...\ 0 & 0 & 0 & 0 & - \left( x_{p} - x_{v}\right) & -1 \end{array} \right] \quad$$ (C.3)

$$\mathbf{\stackrel{\rm\circ}{Z}} = \left[ \begin{array}{c} \int_{x... ...}^{x_{p}} q_{z} dx dx + \int_{x_{v}}^{x_{p}} m_{y} dx \end{array} \right] \quad$$ (C.4)

Table C.1: Loading vectors (to be continued in Table C.2)

The loading vector $\mathbf{\stackrel{\rm\circ}{Z}_{q}}$ for a uniformly distributed load $q_{z}$:

$$\mathbf{\stackrel{\rm\circ}{Z}_{q}} = \left[ \begin{array}{c} 0 \... ...ight) \\ - q_{z}\cdot\left( x_{p} - x_{v}\right)^{2} \end{array} \right] \quad$$ (C.5)

The loading vector $\mathbf{\stackrel{\rm\circ}{Z}_{F}}$ for a point load $F_{z}$:

$$\mathbf{\stackrel{\rm\circ}{Z}_{F}} = \left[ \begin{array}{c} 0 \... ... - F_{z} \\ - F_{z}\cdot\left( x_{p} - x_{a}\right) \end{array} \right] \quad$$ (C.6)

Table C.2: Loading vectors (continued from Table C.1)

The transfer matrix $\mathbf{U}$ of the first-order analysis for a frame (Sign Convention 2) can be computed using the GNU Octave function ylfhlin.m ; for a sparse transfer matrix U, the function ysplfhlin.m is to be used.

To compute the enlarged sparse transfer matrix $\mathbf{\widehat{IU}}$ of the first-order analysis

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for a beam (Sign Convention 2), the GNU Octave function yspTlvfmhvI.m is used. This function in its turn makes use of the function yspTlfhlin.m ;

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for a three-hinged frame (Sign Convention 2), the GNU Octave function yspSlvfmhvI.m , which comprises the function yspSlfhlin.m , is used;

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for a Gerber beam (Sign Convention 2), the GNU Octave function yspSTlvfmhvI.m comprising the function yspSTlfhlin.m is used.

The loading vectors in Tables C.1 and C.2 are comparable with those in [Krä91b] and [Bor79b].

The loading vectors $\mathbf{\stackrel{\rm\circ}{Z}_{q}}$, $\mathbf{\stackrel{\rm\circ}{Z}_{F}}$ can be computed using the GNU Octave functions yzhqz.m and yzfzv.m .

For a continuous beam, the loading vector $\mathbf{\stackrel{\rm\circ}{Z}}$ can be computed using the GNU Octave function ESTtalaKrmus.m ; to compute the vectors $\mathbf{\stackrel{\rm\circ}{Z}_{q}}$, $\mathbf{\stackrel{\rm\circ}{Z}_{F}}$, the functions yzThqz.m and yzTfzv.m are used.

For a three-hinged frame, the loading vector $\mathbf{\stackrel{\rm\circ}{Z}}$ can be computed using the GNU Octave function ESTSKrmus.m ; to compute the vectors $\mathbf{\stackrel{\rm\circ}{Z}_{q}}$, $\mathbf{\stackrel{\rm\circ}{Z}_{F}}$, the functions yzShqz.m and yzSfzv.m are used.

For a Gerber beam, the loading vector $\mathbf{\stackrel{\rm\circ}{Z}}$ can be computed using the GNU Octave function ESTSTKrmus.m ; to compute the vectors $\mathbf{\stackrel{\rm\circ}{Z}_{q}}$, $\mathbf{\stackrel{\rm\circ}{Z}_{F}}$, the functions yzSThqz.m and yzSTfzv.m are used.