C.2 Transfer matrices of second-order analysis

The transfer equations with axial and transverse shear forces are of the form

$$\mathbf{{\tilde{Z}}_{p}} - \mathbf{\tilde{U}}\mathbf{{\tilde{Z}}_{v}} = \mathbf{\stackrel{\rm\circ}{\tilde{Z}}}$$ (C.7)

where

$$\mathbf{{\tilde{Z}}_{p}} = \left[ \begin{array}{c} u \\ w \\ \v... ... w \\ \varphi_{y} \\ S_{x} \\ H_{z} \\ M_{y} \end{array} \right]_{v} \quad$$ (C.8)

where
$S$ - axial force,
$H$ - transverse shear force, see Fig. 5.1.

Numerical difficulties may occur when the transfer matrix manipulation involves differences of large numbers (forces and displacements can differ ca $10^{3}$ times), leading to inaccuracies in computations [PW94], [PL63]. Round-off errors can be reduced by scaling. We will scale (multiply) the displacements and rotations by $i_{0}= EI_{basic}/l_{basic}$ (the scaling multiplier for the displacements). After solving a system of linear equations of a boundary value problem, we divide each of the displacements and rotations found by $i_{0}= EI_{basic}/l_{basic}$. Displacements and forces at the beginning of members determine unscaled initial parameter vectors for the structure members.

The transfer matrix $\mathbf{U}$ from Eqs. (C.7), (C.9), and (C.10) of a second-order analysis for a frame (Sign Convention 2) can be computed using the GNU Octave function ylfmhvII.m (p. ).
The transfer matrix $\mathbf{U}$ for a frame in compression, Sign Convention 2:

$$\mathbf{\tilde {U}} = \left[\begin{array}{cccccc} 1 & 0 & 0 & - i... ...\frac{- \sin\frac{\nu x}{l}}{\nu}{x} & - \cos\frac{\nu x}{l} \end{array}\right]$$ (C.9)

Table C.3: Loading vectors of compression (axial force $S$ and transverse shear force $H$)

The transfer matrix $\mathbf{U}$ for a frame in tension, Sign Convention 2:

$$\mathbf{\tilde {U}_{t}} = \left[\begin{array}{cccccc} 1 & 0 & 0 &... ...m{sh}\frac{\nu x}{l}}{\nu}{x} & - \mathrm{ch}\frac{\nu x}{l} \end{array}\right]$$ (C.10)

Table C.4: Loading vectors of tension (axial force $S$ and transverse shear force $H$)

Table C.5: Loading vectors of compression (normal force $N$ and shear force $Q$)

In the case of a uniformly distributed load $q_{z}$, the loading vector $\mathbf{\stackrel{\rm\circ}{Z}}$ (see Eq. (C.7)) of a second-order analysis for a frame member under axial compression $S_{x}$ is

$$\mathbf{\stackrel{\rm\circ}{\tilde {Z}}} = \left[\begin{array}{c}... ...\left( \nu \frac{x}{l} \right) \right]\frac{ql^{2}}{\nu^{2}} \end{array}\right]$$ (C.11)

and for a frame member under axial tension $S_{x}$,

$$\mathbf{\stackrel{\rm\circ}{\tilde {Z}}_{t}} = \left[\begin{array... ...\left( \nu \frac{x}{l} \right) \right]\frac{ql^{2}}{\nu^{2}} \end{array}\right]$$ (C.12)

Table C.6: Loading vectors of tension (normal force $N$ and shear force $Q$)

The loading vectors in Tables C.3, C.4, C.5, and C.6 are comparable with those in [Krä91b] and [Bor79b].

The loading vectors of Eqs. (C.11) and (C.12) can be computed using the GNU Octave function ylqvII.m .

In the case of a point load $F_{z}$, the loading vector $\mathbf{\stackrel{\rm\circ}{Z}}$ (see Eq. (C.7)) of a second-order analysis for a frame member under axial compression $S_{x}$ is

$$\mathbf{\stackrel{\rm\circ}{\tilde {Z}}} = \left[\begin{array}{c}... ...frac{\left(x - a\right)_{+}}{l}\right) \right]\frac{Fl}{\nu} \end{array}\right]$$ (C.13)

and for a frame member under axial tension $S_{x}$,

$$\mathbf{\stackrel{\rm\circ}{\tilde {Z}}_{t}} = \left[\begin{array... ...frac{\left(x - a\right)_{+}}{l}\right) \right]\frac{Fl}{\nu} \end{array}\right]$$ (C.14)

where $\left(x -a \right)_{+}$ is the Heaviside function:

$$\left(x -a \right)_{+} = \left\{ \begin{array}{ccc} 0, & if & {\l... ...< 0} \\ {x - a} , & if & {\left( x -a \right) \geq 0} \end{array}\right. \quad$$ (C.15)

The loading vectors of Eqs. (C.13) and (C.14) can be computed using the GNU Octave function ylfhvzII.m .

The basic system of equations or the frame element of Eq. (C.7) can be rewritten in the form

$$\mathbf{\widehat{IU}}\cdot\mathbf{\widehat{Z}} = \mathbf{\stackrel{\rm\circ}{Z}}$$ (C.16)

where the matrix $\mathbf{\widehat{IU}}$ can be computed using the GNU Octave function ysplvfmhvII.m , in which the GNU Octave function ysplfmhvII.m (p. ) is used. These functions differentiate between pressure and tensile axial loading.

The loading vectors of a uniformly distributed load and a point load for a frame can be computed using the GNU Octave functions ylqvII.m (p. ) and ylfhvzII.m (p. ). These functions differentiate between pressure and tensile axial loading.

The transfer equations with normal and shear forces ($N$ - normal force, $Q$ - shear force shown in Fig. 5.1):

$$\mathbf{Z_{p}} = \mathbf{U_{x}\cdot Z_{v}} + \mathbf{\stackrel{\rm\circ}{Z}}$$ (C.17)

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.18)

and the transfer matrix $\mathbf{U_{x}}\equiv\mathbf{U^{\left(N-2\right)}}$ for a compressive normal force (Sign Conven-tion 2)

$$\mathbf{U^{\left(N-2\right)}} = \left[ \begin{array}{cccccc} 1 & ... ... & -\cos\left( \frac{\nu x}{l}\right) \end{array}\right] \quad % see oli enne$$ (C.19)

while $\mathbf{U_{x}}\equiv\mathbf{U^{\left(N+2\right)}}$ for a tensile normal force (Sign Convention 2)

$$\mathbf{U^{\left(N+2\right)}} = \left[ \begin{array}{cccccc} 1 & ... ...mathrm{ch}\left(\nu\frac{x}{l}\right) \end{array}\right] \quad % see oli enne$$ (C.20)

Transfer matrices (C.19) and (C.20) can be computed with the GNU Octave function ylfmII.m (p. ).