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F2.1 Tvärsnittskonstanter


F2.1.1 Ytstorheter

Statiska momentet $S_\lambda$ och yttröghetsmomentet $I_\lambda$ m.a.p. axeln $\mathrm{\lambda}$ definieras som:

$$ S_{\lambda}=\sum_{i} A_{i}a_{i}=\int_A a \mathrm{d}A $$
(F2–1)

$$ I_{\lambda}=\sum_{i} A_{i}a^{2}_{i}=\int_A a^{2} \mathrm{d}A $$
(F2–2)

Speciellt gäller statiska moment och yttröghetsmoment m.a.p. $y$ och $z$-axlarna:

$$ S_{y}=\int_A z \, {\rm d} A \;, \quad S_{z}=\int_A y \, {\rm d} A \;, \quad I_{y}=\int_A z^2 \, {\rm d} A \;, \quad I_{z}=\int_A y^2 \, {\rm d} A $$
(F2–3)

Steiners sats:

$$ I_{\lambda}=\overline{I}_{\lambda}+A \,d^{2} $$
(F2–4)

där $\bar{I}_{\lambda}$ är yttröghetsmomentet m.a.p. den med $\lambda$ parallella axeln genom kroppens ytcentrum(YC).

Koordinater för ytcentrum (YC):

$$ \bar{y}=\frac{S_z}{A} \;, \quad \bar{z}=\frac{S_y}{A} $$
(F2–5)

F2.1.2 Tvärsnittsdata

$I_{y}=\frac{ba^{3}}{3} \quad I_{z}=\frac{ab^{3}}{3}$

$\bar{I}_y = \frac{b a^3}{12} \quad \bar{I}_z = \frac{ab^3}{12}$

$\bar{I}_y = \bar{I}_z = \frac{\pi}{4}(b^4-a^4)$

$K_{\mathrm{v}}=\frac{\pi}{2}(b^4-a^4)$

$t \ll d$

$A \approx \pi d t$

$\bar{I}_y = \bar{I}_z \approx \frac{\pi d^3 t}{8}$

$K_{\mathrm{v}} \approx \frac{\pi d^3 t}{4}$

$A=\frac{ \pi a b }{4}$

$\bar{I}_{y}= \frac{\pi b a^3}{64}$

$\bar{I}_{z}=\frac{\pi a b^3}{64}$

$I_{y}=\frac{bh^{3}}{12}$

$\overline{I}_{y}=\frac{bh^{3}}{36}$

$\bar{y}=\frac{a+b}{3} \quad \bar{z}=\frac{h}{3}$

$t_1, t_2 \ll h, b$

$\overline{I}_{y} \approx \frac{t_2 h^3}{12} + \frac{t_1 b h^2}{2}$

$\overline{I}_{z} \approx \frac{t_1 b^3}{6}$

$I_{y}=\frac{bh^{3}}{12}$

$\overline{I}_{y}=\frac{bh^{3}}{36}$

$\bar{y}=\frac{a+b}{3} \quad \bar{z}=\frac{h}{3}$