2.7.10.9 Steifigkeitsmatrix des Materials

Steifigkeitsmatrix des Materials

$D_{11} = \frac{I_{0,{\varphi }_1} · E}{1 - {\nu }^2} = \frac{3.322 · {10}^{-4} · 11 · {10}^9}{1 - 0.{0328}^2} = 3656.74 \text{kNm}$

$D_{12} = D_{2.1} = 0.0328 · \sqrt{\left(3.656 · {10}^6 · 7.344 · {10}^6\right)} = 170.58 \text{kNm}$

$D_{33} =\frac{1 - \nu }{2} · \sqrt{D_{1.1}· D_{2.2}} = \frac{1 - 0.0328}{2} · \sqrt{\left(3.656 · {10}^6 ·7.344 · {10}^6\right)} = 2505.84 \mathrm{kNm}$

$D_{44} = D_{55} = \frac{5}{6} ·G · h = \frac{5}{6} · 11.8 · {10}^9 · 0.2 = 1 966 670 \text{kN/m}$

$D_{66} = \frac{E · A_{{\varphi }_1}}{1 - {\nu }^2} = \frac{11· {10}^9 · 0.096}{1 - 0.{0328}^2} = 1 055 590 \text{kN/m}$

$D_{67} = D_{76} · \nu · \sqrt{\left(D_{66} · D_{77}\right)} = 0.0328 · \sqrt{\left(1 055.59 · {10}^6 · 2505.84 ·{10}^6\right)}= 50 210.6 \text{kN/m}$

$D_{88} = G · h = 11.8 · {10}^9 · 0.2 = 2 360 000 \text{kN/m}$

$D_{16} = D_{61} = D_{61} · e_{{\varphi }_1} = 1 055.590 · {10}^9 · 0.0393 = 41 499.2 \text{kNm/m}$

$D_{27} = D_{72} = D_{77} · e_{{\varphi }_1} = 0$

$$\begin{gathered} D_{17} = D_{71} = \frac{\nu }{2} · \left(e_{{\varphi }_1} + e_{{\varphi }_2}\right) ·\sqrt{D_{66} · D_{77}} = \\ = \frac{0.0328}{2} · \left(0.393 + 0\right) · \sqrt{1 055.59 ·{10}^6 ·2505.84 ·{10}^6} = 987.0 \mathrm{kNm}/\mathrm{m} \end{gathered}$$

$D_{38} = D_{83} = \frac{1}{2} · G · h · \left(e_{{\varphi }_1} + e_{{\varphi }_2}\right) = \frac{1}{2} · 11 · {10}^9 · 0.2 ·\left(0.393 + 0\right) = 46 390.2 \mathrm{kNm}/\mathrm{m}$