1.2.2 Basic equations for a frame element

The governing differential equation for a truss element is

$$\frac{\mathrm{d}N_{x}}{\mathrm{d}x} = - q_{x}\left(x\right)$$ (1.41)

or

$$\frac{\mathrm{d}}{\mathrm{d}x}\left(EA\frac{\mathrm{d}u}{\mathrm{d}x}\right) = - q_{x}\left(x\right)$$ (1.42)

where $N_{x} = EA{\mathrm{d}u}/{\mathrm{d}x}$ and EA denotes the axial stiffness.

We now consider the basic equations of a frame element in symbolic matrix notation

$${\mathbf{Z_{x}}} = {\mathbf{U_{x}}}{\mathbf{Z_{A}}} + {\mathbf{\stackrel{\rm\circ}{Z}}}$$ (1.43)

where $\mathbf{Z_{x}}$, $\mathbf{Z_{A}}$ are the vectors of displacements and forces at the point with x-coordinate and at the beginning of the element, respectively,

$$\mathbf{Z_{x}} = \left[\begin{array}{c} u_{x} \\ w_{x} \\ \varp... ...{A} \\ \varphi_{A} \\ \ldots \\ N_{A} \\ Q_{A} \\ M_{A} \end{array}\right]$$ (1.44)

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

$$\mathbf{U_{x}} = {\large {\left[ \begin{array}{ccccccc} 1 & 0 & 0... ...\\ 0 & 0 & 0 & \vdots & 0 & - x & -1 \end{array} \right] }} \quad \normalfont$$ (1.45)

where $i_{0}=1$ is the scaling multiplier for the displacements ($i_{0}= EI/l$) and the loading vector (yzhqz.m, yzfzv.m, yzmyv.m) can be expressed as

$$\mathbf{\stackrel{\rm\circ}{Z}} = {\large { \left[ \begin{array}{... ...M_{y} \left( x - x_{a}\right)^{0}_{+} \end{array} \right] }} \quad \normalfont$$ (1.46)

Some other loading vectors are to be found in Tables C.1 and C.2 of Appendix C.