Równanie wektorowe powłoki walca
$\overrightarrow{r} = a\left\lbrack \cos\left( u^{2} \right)\overrightarrow{i} + sin\left( u^{2} \right)\overrightarrow{j} \right\rbrack + u^{1}\left\lbrack \cos\left( u^{2} + \alpha \right)\overrightarrow{i} + sin\left( u^{2} + \alpha \right)\overrightarrow{j} \right\rbrack cos\beta + u^{1}\text{sinβ}\overrightarrow{k}$
gdzie:
u1, u2− współrzędne krzywoliniowe
α − parametr kątowy; kąt zawarty miedzy rzutem tworzącej na płaszczyznę podstawy XOY a promieniem podstawy
β − parametr kątowy, określa nachylenie tworzącej do płaszczyzny XOY
a− promień podstawy
Ostateczne równanie powłoki walca:
$$\overrightarrow{r} = a\left\lbrack \cos\left( u^{2} \right)\overrightarrow{i} + sin\left( u^{2} \right)\overrightarrow{j} \right\rbrack + u^{1} \bullet \overrightarrow{k}$$
Kowariantne wektory bazy
$$r_{i} = \frac{\partial\overrightarrow{r}}{\partial u^{i}}$$
$$r_{1} = \frac{\partial\overrightarrow{r}}{\partial u^{1}} = \left\lbrack \cos\left( u^{2} + \alpha \right)\overrightarrow{i} + sin\left( u^{2} + \alpha \right)\overrightarrow{j} \right\rbrack cos\beta + sin\beta\overrightarrow{k} = \overrightarrow{k}$$
$$r_{2} = \frac{\partial\overrightarrow{r}}{\partial u^{2}} = a\left\lbrack \operatorname{-sin}\left( u^{2} \right)\overrightarrow{i} + cos\left( u^{2} \right)\overrightarrow{j} \right\rbrack$$
Korzystając z iloczynu skalarnego $\left| \overrightarrow{i} \right| \bullet \left| \overrightarrow{j} \right|$ otrzymano:
$$g_{11} = \overrightarrow{r_{1}} \bullet \overrightarrow{r_{1}} = \overrightarrow{k} \bullet \overrightarrow{k} = 1$$
$$g_{12} = g_{21} = \overrightarrow{r_{1}} \bullet \overrightarrow{r_{2}} = \overrightarrow{k} \bullet a\left\lbrack \operatorname{-sin}\left( u^{2} \right)\overrightarrow{i} + cos\left( u^{2} \right)\overrightarrow{j} \right\rbrack = 0$$
$$g_{22} = \overrightarrow{r_{2}} \bullet \overrightarrow{r_{2}} = \left( a\left\lbrack \operatorname{-sin}\left( u^{2} \right)\overrightarrow{i} + cos\left( u^{2} \right)\overrightarrow{j} \right\rbrack \right)^{2} = a^{2}\left\lbrack \operatorname{}\left( u^{2} \right) + \cos^{2}\left( u^{2} \right) \right\rbrack = a^{2}$$
$$g = \left| g_{\text{ij}} \right| = \left| \begin{matrix}
1 & 0 \\
0 & a^{2} \\
\end{matrix} \right| = a^{2}$$
Kowariantny tensor metryczny
$$g^{\text{ij}} = {\overrightarrow{r}}^{i} \bullet {\overrightarrow{r}}^{j}$$
$$g^{\text{ij}} = \frac{G^{\text{ij}}}{g}$$
Gij−dopełnienie algebraiczne elementu gij
$$g^{11} = \frac{G^{11}}{g} = \frac{a^{2}}{a^{2}} = 1$$
$$g^{12} = g^{21} = \frac{G^{12}}{g} = \frac{G^{21}}{g} = \frac{0}{a^{2}} = 0$$
$$g^{22} = \frac{G^{22}}{g} = \frac{1}{a^{2}}$$
$$g^{\text{ij}} = \left| \begin{matrix}
1 & 0 \\
0 & \frac{1}{a^{2}} \\
\end{matrix} \right| = \frac{1}{a^{2}}$$
Wektor jednostkowy m
$$\sqrt{g} \bullet \overrightarrow{m} = \overrightarrow{r_{1}}\text{\ x\ }\overrightarrow{r_{2}} = \left| \begin{matrix}
\overrightarrow{i} & \overrightarrow{j} & \overrightarrow{k} \\
0 & 0 & 1 \\
\operatorname{-asin}\left( u^{2} \right) & \text{acos}\left( u^{2} \right) & 0 \\
\end{matrix} \right| = \operatorname{-asin}u^{2}\overrightarrow{j} - acos\text{\ u}^{2}\overrightarrow{i}$$
$$\sqrt{g} = a$$
$$\overrightarrow{m} = \operatorname{-sin}u^{2}\overrightarrow{j} - cos\text{\ u}^{2}\overrightarrow{i}$$
$$\sqrt{a^{2}} \bullet \overrightarrow{m} = \overrightarrow{r_{1}}\text{\ x\ }\overrightarrow{r_{2}} = \operatorname{-asin}u^{2}\overrightarrow{j} - acos\text{\ u}^{2}\overrightarrow{i}$$
$$\overrightarrow{m_{i}} = \frac{\partial\overrightarrow{m}}{\partial u^{i}}$$
$$\overrightarrow{m_{1}} = \frac{\partial\overrightarrow{m}}{\partial u^{1}} = 0$$
$$\overrightarrow{m_{2}} = \frac{\partial\overrightarrow{m}}{\partial u^{2}} = \sin u^{2}\overrightarrow{i} - \cos u^{2}\overrightarrow{j}$$
Współczynniki drugiej formy różniczkowej (kowariantnej):
$$b_{\text{ij}} = \overrightarrow{r_{i}} \bullet \overrightarrow{m_{j}}$$
$$b_{11} = \overrightarrow{r_{1}} \bullet \overrightarrow{m_{1}} = \overrightarrow{k} \bullet 0 = 0$$
$$b_{12} = \overrightarrow{r_{1}} \bullet \overrightarrow{m_{2}} = \overrightarrow{k} \bullet (\sin u^{2}\overrightarrow{i} - \cos u^{2}\overrightarrow{j}) = 0$$
$$b_{21} = \overrightarrow{r_{2}} \bullet \overrightarrow{m_{1}} = \lbrack a\left\lbrack \operatorname{-sin}\left( u^{2} \right)\overrightarrow{i} + cos\left( u^{2} \right)\overrightarrow{j} \right\rbrack\rbrack \bullet 0 = 0$$
$$b_{22} = \overrightarrow{r_{2}} \bullet \overrightarrow{m_{2}} = a\left\lbrack \operatorname{-sin}\left( u^{2} \right)\overrightarrow{i} + cos\left( u^{2} \right)\overrightarrow{j} \right\rbrack\rbrack \bullet (\sin u^{2}\overrightarrow{i} - \cos u^{2}\overrightarrow{j}) = a$$
Wyznacznik drugiej formy różniczkowej
$$b = \left| b_{\text{ij}} \right| = \left| \begin{matrix}
0 & 0 \\
0 & a \\
\end{matrix} \right| = 0$$
Kowariantny tensor drugiej formy różniczkowej
bkl = bijgikgjl = b11g1kg1l + b12g1kg2l + b21g2kg1l + b22g2kg2l
b11 = b11g11g11 + b12g11g21 + b21g21g11 + b22g21g21 = 0
b12 = b11g11g12 + b12g11g22 + b21g21g12 + b22g21g22 = 0
b21 = b11g12g11 + b12g12g21 + b21g22g11 + b22g22g21 = 0
$$b^{22} = b_{11}g^{12}g^{12} + b_{12}g^{12}g^{22} + b_{21}g^{22}g^{12} + b_{22}g^{22}g^{22} = a \bullet \frac{1}{a^{2}} \bullet \frac{1}{a^{2}} = \frac{1}{a^{3}}$$
Mieszany tensor drugiej formy różniczkowej
bjk = bij • gik = b1j • g1k + b2j • g2k
b11 = b11 • g11 + b21 • g21 = 0
b12 = b11 • g12 + b21 • g22 = 0
b21 = b12 • g11 + b22 • g21 = 0
$$b_{2}^{2} = b_{12} \bullet g^{12} + b_{22} \bullet g^{22} = a \bullet \frac{1}{a^{2}} = \frac{1}{a}$$
Współczynniki trzeciej formy różniczkowej
$$c_{\text{ij}} = \overrightarrow{m_{i}} \bullet \overrightarrow{m_{j}}$$
$$c_{11} = \overrightarrow{m_{1}} \bullet \overrightarrow{m_{1}} = 0 \bullet 0 = 0$$
$$c_{12} = \overrightarrow{m_{1}} \bullet \overrightarrow{m_{2}} = 0 \bullet \operatorname{(sin}u^{2}\overrightarrow{i} - \cos u^{2}\overrightarrow{j}) = 0$$
$$c_{21} = \overrightarrow{m_{2}} \bullet \overrightarrow{m_{1}} = \operatorname{(sin}u^{2}\overrightarrow{i} - \cos u^{2}\overrightarrow{j}) \bullet 0 = 0$$
$$c_{22} = \overrightarrow{m_{2}} \bullet \overrightarrow{m_{2}} = (\sin u^{2}\overrightarrow{i} - \cos u^{2}\overrightarrow{j})^{2} = 1$$
Krzywizna Gaussa (zapewnia opis powierzchni środkowej)
$$K = \frac{b}{g} = \frac{0}{a^{2}} = 0$$
Krzywizna średnia
$$H = \frac{1}{2}g^{\text{ij}} \bullet b_{\text{ij}}$$
$$H = \frac{1}{2}\left( b_{11} \bullet g^{11} + b_{12} \bullet g^{12} + b_{21} \bullet g^{21} + b_{22} \bullet g^{22} \right) = \frac{1}{2} \bullet \frac{1}{a^{2}} \bullet a = \frac{1}{2a}$$
Symbole Christoffela drugiego rodzaju:
$$\Gamma_{\text{ij}}^{k} = \frac{1}{2}g^{\text{km}}\left\lbrack g_{jm,k} + g_{mi,j} = g_{ij,m} \right\rbrack$$
$$\Gamma_{\text{ij}}^{k} = \frac{1}{2}g^{k1}\left\lbrack g_{j1,k} + g_{1i,j} - g_{ij,1} \right\rbrack + \frac{1}{2}g^{k2}\left\lbrack g_{j2,k} + g_{2i,j} - g_{ij,2} \right\rbrack$$
Ponieważ wszystkie współczynniki pierwszej formy różniczkowej są stałymi zatem ich pochodne są równe 0. Stąd symbole Christoffela drugiego rodzaju są równe 0.
Γijk = 0
Po podstawieniu znanych wartości otrzymano:
Dane:
a = R = 32, 4 m
Współczynniki pierwszej formy różniczkowej
g11 = 1
g12 = g21 = 0
g22 = a2 = 32, 42 = 1049, 76 m
Wyznacznik pierwszej formy różniczkowej
g = a2 = R2 = 32, 42 = 1049, 76 m2
Kowariantny tensor metryczny
g11 = 1
g12 = g21 = 0
$$g^{22} = \frac{1}{a^{2}} = \frac{1}{{32,4}^{2}} = 9,53 \bullet 10^{- 4}$$
Współczynniki drugiej formy różniczkowej
b11 = 0
b12 = b21 = 0
b22 = a = R = 32, 4 m
Wyznacznik drugiej formy różniczkowej
b = |bij| = 0
Kowariantny tensor drugiej formy różniczkowej
b11 = 0
b12 = b21 = 0
$$b^{22} = \frac{1}{a^{3}} = \frac{1}{{32,4}^{3}} = 2,94 \bullet 10^{- 5}$$
Mieszany tensor drugiej formy różniczkowej
b11 = 0
b12 = b21 = 0
$$b_{2}^{2} = \frac{1}{a} = \frac{1}{32,4} = 0,0309$$
Współczynniki trzeciej formy różniczkowej
c11 = 0
c12 = c21 = 0
c22 = 1
Krzywizna Gaussa
K = 0
Krzywizna średnia
$$H = \frac{1}{2a} = \frac{1}{2 \bullet 32,4} = 0,015$$
Symbole Christoffela drugiego rodzaju:
Γijk = 0
Równania równowagi
Nij|i − Qibij + Pj = 0
Nijbij + Qj|j + P3 = 0
Mij|i − Qj = 0
Wykorzystując umowę o sumowaniu otrzymano:
N11|1 + N21|2 − Q1b11 − Q2b21 + P1 = 0
N12|1 + N22|2 − Q1b12 − Q2b22 + P2 = 0
N11b11 + N12b12 + N21b21 + N22b22 + P2 = 0
M11|1 + M21|2 − Q1 = 0
M12|1 + M22|2 − Q2 = 0
Wykorzystując wzory na pochodne kowariantne otrzymano:
Nij|k = Nij,k + Γkmi • Nmj + Γkmj • Nim
Qj|k = Qj,k + Γkij • Qi
Mij|k = Mij,k + Γkmi • Mmj + Γkmj • Mim
Symbole Christoffela drugiego rodzaju:
Nij|k = Nij,k
Qj|k = Qj,k
Mij|k = Mij,k
Stan błonowy
Równania równowagi
Nij|k + Pj = 0
Nijbij + P3 = 0
Stosując umowę o sumowaniu otrzymano:
$\left. \ {\overset{\overline{}}{N}}^{11} \right|_{1} + \left. \ {\overset{\overline{}}{N}}^{21} \right|_{2} + P^{1} = 0$ (1)
$\left. \ {\overset{\overline{}}{N}}^{12} \right|_{1} + \left. \ {\overset{\overline{}}{N}}^{22} \right|_{2} + P^{2} = 0$ (2)
${\overset{\overline{}}{N}}^{11}b_{11} + N^{12}b_{12} + N^{21}b_{21} + N^{22}b_{22} + P^{3} = 0$ (3)
Wykorzystując wzory na pochodne kowariantne i podstawiając dane wartości bij otrzymano
${{\overset{\overline{}}{N}}^{11}}_{,1} + {{\overset{\overline{}}{N}}^{21}}_{,2} + P^{1} = 0$ (1)
${{\overset{\overline{}}{N}}^{12}}_{,1} + {{\overset{\overline{}}{N}}^{22}}_{,2} + P^{2} = 0$ (2)
N22a + P3 = 0 (3)
Składowe wektora obciążeń
Wektor obciążenia
$$\overrightarrow{P} = P^{i}{\overrightarrow{r}}_{i} + P^{3}\overrightarrow{m}$$
$$\overrightarrow{P} = P^{1}{\overrightarrow{r}}_{1} + P^{2}{\overrightarrow{r}}_{2} + P^{3}\overrightarrow{m}$$
Ciężar własny
q = 2hγz
P1 = −2hγz
P2 = 0
P3 = 0
Wyznaczenie sił wewnętrznych z równania
(3) ${\overset{\overline{}}{N}}^{22} = \frac{1}{a}P^{3} = \frac{1}{a}\gamma_{c}(L - U^{1})$
(2) ${\overset{\overline{}}{N}}^{12} = \int_{}^{}{( - {\overset{\overline{}}{N}}_{12}^{22} - P^{2}})du^{1} = \int_{}^{}0du^{1} = 0$
Warunek brzegowy
$${\overset{\overline{}}{N}}^{12}\left( U^{'} = L \right) = 0 \rightarrow C_{1} = 0$$
$${\overset{\overline{}}{N}}^{12} = 0$$
(1) ${\overset{\overline{}}{N}}^{11} = \int_{}^{}{( - {\overset{\overline{}}{N}}_{12}^{21} - P^{1}})du^{1} = \int_{}^{}{2h\gamma_{z}}du^{1} = 2h\lambda_{z}U^{1} + C_{1}$
$${\overset{\overline{}}{N}}^{11}\left( U^{1} = L \right) = 0$$
2hγz = L + C2 = 0 → C2 = −2hγzL
$${\overset{\overline{}}{N}}^{11} = 2h\gamma_{z}(U^{1} - L)$$
Przemieszczenia i odkształcenia
w− przemieszczenia
γ− odkształcenia
Związek fizyczny
$$\gamma^{\text{ij}} = \frac{1}{2Eh}\left\lbrack \left( 1 + \nu \right){\overset{\overline{}}{N}}^{\text{ij}} - \nu\overset{\overline{}}{N}g^{\text{ij}} \right\rbrack$$
gdzie:
$$\overset{\overline{}}{N} = g^{\text{ij}}{\overset{\overline{}}{N}}^{\text{ij}}$$
$$\overset{\overline{}}{N} = {\overset{\overline{}}{N}}^{11} + R^{2}{\overset{\overline{}}{N}}^{22}$$
Składowe tensora odkształcenia
$$\gamma^{11} = \frac{1}{2Eh}\left\lbrack \left( 1 + \nu \right){\overset{\overline{}}{N}}^{11} - \nu({\overset{\overline{}}{N}}^{11} + a^{2}{\overset{\overline{}}{N}}^{22})g^{11} \right\rbrack = \frac{1}{2Eh}\left\lbrack \left( 1 + \nu \right){\overset{\overline{}}{N}}^{11} - \nu({\overset{\overline{}}{N}}^{11} + a^{2}{\overset{\overline{}}{N}}^{22}) \bullet 1 \right\rbrack = \frac{1}{2Eh}\left\lbrack {\overset{\overline{}}{N}}^{11} - \nu a^{2}{\overset{\overline{}}{N}}^{22} \right\rbrack$$
$$\gamma^{12} = \gamma^{21} = \frac{1}{2Eh}\left\lbrack \left( 1 + \nu \right){\overset{\overline{}}{N}}^{12} - \nu({\overset{\overline{}}{N}}^{11} + a^{2}{\overset{\overline{}}{N}}^{22})g^{12} \right\rbrack = \frac{1}{2Eh}\left\lbrack \left( 1 + \nu \right){\overset{\overline{}}{N}}^{12} - \nu({\overset{\overline{}}{N}}^{11} + a^{2}{\overset{\overline{}}{N}}^{22}) \bullet 0 \right\rbrack = \frac{1}{2Eh}\left\lbrack \left( 1 + \nu \right){\overset{\overline{}}{N}}^{12} \right\rbrack$$
$$\gamma^{22} = \frac{1}{2Eh}\left\lbrack \left( 1 + \nu \right){\overset{\overline{}}{N}}^{22} - \nu({\overset{\overline{}}{N}}^{11} + a^{2}{\overset{\overline{}}{N}}^{22})g^{22} \right\rbrack = \frac{1}{2Eh}\left\lbrack \left( 1 + \nu \right){\overset{\overline{}}{N}}^{22} - \nu({\overset{\overline{}}{N}}^{11} + a^{2}{\overset{\overline{}}{N}}^{22}) \bullet \frac{1}{a^{2}} \right\rbrack = \frac{1}{2Eh}\left\lbrack {\overset{\overline{}}{N}}^{22} - \nu\frac{1}{a^{2}}{\overset{\overline{}}{N}}^{11} \right\rbrack$$
Kowariantny tensor odkształcenia
γkl = γijgikgjl = γ11g1kg1l + γ12g1kg2l + γ21g2kg1l + γ22g2kg2l
γ11 = γ11g11g11 + γ12g11g21 + γ21g21g11 + γ22g21g21
γ12 = γ11g11g12 + γ12g11g22 + γ21g21g12 + γ22g21g22
γ21 = γ11g12g11 + γ12g12g21 + γ21g22g11 + γ22g22g21
γ22 = γ11g12g12 + γ12g12g22 + γ21g22g12 + γ22g22g22
Podstawiając wartości gij otrzymano
γ11 = γ11
γ12 = γ12 • a2 = γ12 • R2
γ21 = γ21 • a2 = γ21 • R2
γ22 = γ22 • a2 • a2 = γ22 • R4
Związek geometryczny
$$2\gamma_{y} = w\frac{k}{i}g_{\text{jk}} + w\frac{k}{j}g_{\text{ik}} + 2b_{\text{ij}}w^{3}$$
$$2\gamma_{11} = w\frac{1}{1}g_{11} + w\frac{2}{1}g_{12} + w\frac{1}{1}g_{11} + w\frac{2}{1}g_{12} + 2b_{11}w^{3}$$
$$2\gamma_{12} = w\frac{1}{1}g_{21} + w\frac{2}{1}g_{22} + w\frac{1}{2}g_{11} + w\frac{2}{2}g_{12} + 2b_{12}w^{3}$$
$$2\gamma_{22} = w\frac{1}{2}g_{21} + w\frac{2}{2}g_{22} + w\frac{1}{1}g_{21} + w\frac{2}{2}g_{22} + 2b_{22}w^{3}$$
Podstawiając znane wartości gij i bij otrzymano:
γ11 = wI11
$$\gamma_{12} = \frac{1}{2}\left( a^{2}w_{I1}^{2} + w_{I2}^{1} \right)$$
γ22 = a2wI22 − aw3
Podstawiając za składowe tensora odkształcenia zależności z punktu 2.3 otrzymano:
$$w_{I1}^{1} = \frac{1}{2Eh}\left( {\overset{\overline{}}{N}}^{11} - \nu R^{2}{\overset{\overline{}}{N}}^{22} \right)$$
$$a^{2}w_{I1}^{2} + w_{I2}^{1} = 2a^{2}\frac{1}{2Eh}\left( 1 + \nu \right){\overset{\overline{}}{N}}^{12}$$
$$a^{2}w_{I2}^{2} - aw^{3} = a^{4}\frac{1}{2Eh}\left( {\overset{\overline{}}{N}}^{22} - \nu\frac{1}{a^{2}}{\overset{\overline{}}{N}}^{11} \right)$$
Wyznaczenie składowych wektora przemieszczenia z równania:
składowa pionowa w1
$$w^{'} = \int_{}^{}{\frac{1}{2Eh}\left\lbrack 2h\gamma_{z}\left( u^{'} - L \right) - \nu a^{2}\frac{1}{a}\gamma_{c}\left( u^{'} - L \right)du' \right\rbrack}$$
$$w^{1} = \int_{}^{}{\frac{1}{2Eh}\left\lbrack 2h\gamma_{z}\left( u^{1} - L \right) - \nu a^{2}\frac{1}{a}\gamma_{c}\left( u^{1} - L \right)du^{1} \right\rbrack} = \frac{1}{2Eh}\left\lbrack 2h\gamma_{z}\frac{\left( u^{1} - L \right)^{2}}{2} + \nu a\gamma_{c}\frac{\left( u^{1} - L \right)^{2}}{2} + S_{1} \right\rbrack$$
Warunek brzegowy
w1(u1=0) = 0
$$\frac{1}{2Eh}\left\lbrack 2h\gamma_{z} - \frac{L^{2}}{2} + \nu a\gamma_{c}\frac{L^{2}}{2} + S_{1} \right\rbrack = 0$$
$$S_{1} = \frac{L^{2}}{2}\left( 2h\gamma_{z} + \text{νa}\gamma_{c} \right)$$
$$w^{1} = \frac{1}{2Eh}\left\lbrack 2h\gamma_{z}\frac{\left( u^{1} - L \right)^{2}}{2} + \nu a\gamma_{c}\frac{\left( u^{1} - L \right)^{2}}{2} - \frac{L^{2}}{2}\left( 2h\gamma_{z} + \text{νa}\gamma_{c} \right) \right\rbrack$$
$$w^{1} = \frac{1}{2Eh}\left\lbrack \left( 2h\gamma_{z} + \nu a\gamma_{c} \right)\frac{\left( u^{1} \right)^{2}}{2} - Lu^{1} \right\rbrack$$
składowa styczna do obwodu
$${\overset{\overline{}}{N}}^{12} = 0 \land w_{I2}^{1} = 0\ zatem\ w^{2} = 0$$
składowa pozioma w3
$$w_{I2}^{2} = Rw^{3} = a^{4}\frac{1}{2Eh}\left( {\overset{\overline{}}{N}}^{22} - \nu\frac{1}{R^{2}}{\overset{\overline{}}{N}}^{11} \right)$$
$$w^{3} = R^{3}\frac{1}{2Eh}\left\lbrack \frac{1}{R}\gamma_{c}\left( L - u^{1} \right) - \nu\frac{1}{R}2h\gamma_{z}\left( u^{1} - L \right) \right\rbrack$$
$$w^{3} = R^{3}\frac{1}{2Eh}\left\lbrack \frac{1}{R}\gamma_{c}\left( u^{1} - L \right) + \nu\frac{1}{R^{2}}2h\gamma_{z}\left( u^{1} - L \right) \right\rbrack$$
$$w^{3} = \frac{R^{2}\gamma_{c} + \nu R2h\gamma_{z}}{2Eh}\left( u^{1} - L \right)$$
Obliczenie wartości sił wewnetrznych
w stanie błonowym
Dane:
2h = (0, 014; 0, 012; 0, 01; 0, 01; 0, 01; 0, 008; 0, 008) m
γz = 25 kN/m3
L = 30, 322 m
$${\overset{\overline{}}{N}}^{11} = 2h\gamma_{z}\left( u^{1} - L \right)$$
Obliczenia wykonanoza pomocą programu Excel.
$$u^{1} = 0 \longmapsto {\overset{\overline{}}{N}}^{11} = 0,014 \bullet 25\left( 0 - 30,322 \right) = - 10,613\ kN/m$$
$$u^{1} = 4,332 \longmapsto {\overset{\overline{}}{N}}^{11} = - 9,097\ kN/m$$
$$u^{1} = 8,664 \longmapsto {\overset{\overline{}}{N}}^{11} = - 6,497\ kN/m$$
$$u^{1} = 12,996 \longmapsto {\overset{\overline{}}{N}}^{11} = - 4,332\ kN/m$$
$$u^{1} = 17,328 \longmapsto {\overset{\overline{}}{N}}^{11} = - 3,249\ kN/m$$
$$u^{1} = 21,660 \longmapsto {\overset{\overline{}}{N}}^{11} = - 2,166\ kN/m$$
$$u^{1} = 25,992 \longmapsto {\overset{\overline{}}{N}}^{11} = - 0,866\ kN/m$$
$$u^{1} = 30,322 \longmapsto {\overset{\overline{}}{N}}^{11} = 0\ kN/m$$
$${\overset{\overline{}}{N}}^{12} = 0$$
$${\overset{\overline{}}{N}}^{22} = \frac{1}{R}\gamma_{c}\left( L - u^{1} \right)$$
R = 32, 4 m
γc = 13, 0 kN/m3
$$u^{1} = 0 \longmapsto {\overset{\overline{}}{N}}^{22} = \frac{1}{32,4} \bullet 13,0 \bullet \left( 30,322 - 0 \right) = 12,166\ kN/m$$
$$u^{1} = 4,332 \longmapsto {\overset{\overline{}}{N}}^{22} = 10,428\ kN/m$$
$$u^{1} = 8,664 \longmapsto {\overset{\overline{}}{N}}^{22} = 8,690\ kN/m$$
$$u^{1} = 12,996 \longmapsto {\overset{\overline{}}{N}}^{22} = 6,952\ kN/m$$
$$u^{1} = 17,328 \longmapsto {\overset{\overline{}}{N}}^{22} = 5,214\ kN/m$$
$$u^{1} = 21,660 \longmapsto {\overset{\overline{}}{N}}^{22} = 3,475\ kN/m$$
$$u^{1} = 25,992 \longmapsto {\overset{\overline{}}{N}}^{22} = 1,737\ kN/m$$
$$u^{1} = 30,322 \longmapsto {\overset{\overline{}}{N}}^{22} = 0\ kN/m$$
Obliczanie wartości przemieszczeń
2h = (0, 014; 0, 012; 0, 01; 0, 01; 0, 01; 0, 008; 0, 008) m
γz = 25 kN/m3
L = 30, 322 m
R = 32, 4 m
γc = 13, 0 kN/m3
ν = 0, 2
$$\frac{C37}{45} \rightarrow E = 34\ GPa = 34 \bullet 10^{6}\ kN/m^{2}$$
$${\overset{\overline{}}{w}}^{1} = \frac{1}{2Eh}\left\lbrack \left( 2h\gamma_{z} + \nu R\gamma_{c} \right)\frac{\left( u^{1} \right)^{2}}{2} - Lu^{1} \right\rbrack$$
$$u^{1} = 0 \longmapsto {\overset{\overline{}}{w}}^{1} = \frac{1}{0,014 \bullet 34 \bullet 10^{6}}\left\lbrack \left( 0,014 \bullet 25 + 0,2 \bullet 32,4 \bullet 13 \right)\frac{0^{2}}{2} - 30,322 \bullet 0 \right\rbrack = 0$$
$$u^{1} = 4,332 \longmapsto {\overset{\overline{}}{w}}^{1} = 1,39 \bullet 10^{- 3}\text{\ m}$$
$$u^{1} = 8,664 \longmapsto {\overset{\overline{}}{w}}^{1} = 7,13 \bullet 10^{- 3}\ m$$
$$u^{1} = 12,996 \longmapsto {\overset{\overline{}}{w}}^{1} = 1,98 \bullet 10^{- 2}\ m$$
$$u^{1} = 17,328 \longmapsto {\overset{\overline{}}{w}}^{1} = 3,58 \bullet 10^{- 2}\ m$$
$$u^{1} = 21,660 \longmapsto {\overset{\overline{}}{w}}^{1} = 5,64 \bullet 10^{- 2}\ m$$
$$u^{1} = 25,992 \longmapsto {\overset{\overline{}}{w}}^{1} = 1,02 \bullet 10^{- 1}\ m$$
$$u^{1} = 30,322 \longmapsto {\overset{\overline{}}{w}}^{1} = 1,39 \bullet 10^{- 1}\ m$$
${\overset{\overline{}}{w}}^{2} = 0 -$ składowa styczna do obwodu
$${\overset{\overline{}}{w}}^{3} = \frac{R^{2}\gamma_{c} + \nu R2h\gamma_{z}}{2Eh}\left( u^{1} - L \right)$$
$$u^{1} = 0 \longmapsto {\overset{\overline{}}{w}}^{3} = \frac{{32,4}^{2} \bullet 13 + 0,2 \bullet 32,4 \bullet 0,014 \bullet 25}{0,014 \bullet 34 \bullet 10^{6}}\left( 0 - 30,322 \right) = - 0,869\ m$$
$$u^{1} = 4,332 \longmapsto {\overset{\overline{}}{w}}^{3} = - 0,745\ m$$
$$u^{1} = 8,664 \longmapsto {\overset{\overline{}}{w}}^{3} = - 0,725\ m$$
$$u^{1} = 12,996 \longmapsto {\overset{\overline{}}{w}}^{3} = - 0,696\ m$$
$$u^{1} = 17,328 \longmapsto {\overset{\overline{}}{w}}^{3} = - 0,522\ m$$
$$u^{1} = 21,660 \longmapsto {\overset{\overline{}}{w}}^{3} = - 0,348\ m$$
$$u^{1} = 25,992 \longmapsto {\overset{\overline{}}{w}}^{3} = - 0,217\ m$$
$$u^{1} = 30,322 \longmapsto {\overset{\overline{}}{w}}^{3} = 0\ m$$
Stan zgięciowy
Siły przekrojowe
$${\hat{N}}^{\text{ij}} = \left\lbrack d^{y}C_{\text{kl}}^{'} - b^{y}C_{i} \right\rbrack \bullet S^{\text{kl}}$$
$${\hat{Q}}^{j} = \frac{\varepsilon}{\omega} \bullet g \bullet g^{\text{ij}} \bullet \left\lbrack n_{i} - C_{12}^{1} + m_{i} \bullet C_{m} \right\rbrack \bullet S^{\text{kl}}$$
$${\hat{M}}^{\text{ij}} = - \frac{1}{\omega^{2}}\left( 1 - \nu \right) \bullet \alpha^{\text{ij}} \bullet C_{\text{kl}}^{2} + \left\lbrack \left( 1 - \gamma \right) \bullet \beta^{\text{ij}} - g\varepsilon g^{\text{ij}}C_{\text{kl}}^{1} \right\rbrack \bullet S^{\text{kl}}$$
Przemieszczenia
$${\hat{w}}_{s} = \frac{g}{2Eh} \bullet C_{\text{kl}}^{2} \bullet S^{d}$$
$${\hat{w}}^{d} = \frac{1}{2Eh\varepsilon} \bullet \left\lbrack \frac{2H}{\varepsilon} - \left( 1 - \nu \right) \right\rbrack \bullet \hat{Q}$$
gdzie:
$$\omega^{4} = \frac{3\left( 1 - \nu^{2} \right)}{h^{2}}$$
$$\varepsilon = 1 \pm \sqrt{H^{2} - k}$$
$$\alpha^{\text{ij}} = \left\{ \begin{matrix}
\alpha^{11} = \left( n_{2} \right)^{2} - \left( m_{2} \right)^{2} \\
\alpha^{12} = n_{1}n_{2} + m_{1}m_{2} \\
\alpha^{13} = \left( n_{1} \right)^{2} - \left( m_{1} \right)^{2} \\
\end{matrix} \right.\ $$
$\beta^{\text{ij}} = \left\{ \begin{matrix} \beta^{11} = 2n_{2}m_{2} \\ \beta^{12} = - \left( n_{1}m_{2} + n_{2}m_{1} \right) \\ \beta^{13} = {2n}_{1}m_{1} \\ \end{matrix} \right.\ $
$$S^{\text{kl}} = \left\{ \begin{matrix}
\begin{matrix}
S^{11} = e^{\omega m_{1}u^{i}} \bullet \sin{\omega n_{1}u^{1} \bullet \sin{\omega n_{2}}}u^{2} \\
S^{12} = e^{\omega m_{1}u^{i}} \bullet \sin{\omega n_{1}u^{1} \bullet \cos{\omega n_{2}u^{2}}} \\
S^{21} = e^{\omega m_{1}u^{i}} \bullet \operatorname{cos\omega}{\bullet \sin{\omega n_{2}}}u^{2} \\
\end{matrix} \\
S^{22} = e^{\omega m_{1}u^{i}} \bullet \cos{\omega n_{1}u^{1}} \bullet \cos{\omega n_{2}u^{2}} \\
\end{matrix} \right.\ $$
m2 = n2 = 0
$$m_{1} = n_{1} = - \sqrt{\frac{1}{2a}}$$
$$\varepsilon = H \pm \sqrt{H^{2} - k}\ ;\ \ \ \ \ \ \ \ \ \ \ H = \frac{1}{2a}$$
$$\varepsilon = 0\ \ \ \vee \ \ \ \varepsilon = \frac{1}{a}$$
po podstawieniu znanych wartości:
$$m_{1} = n_{1} = \varepsilon_{k} \bullet \sqrt{\frac{\text{εg}}{2gz}}$$
otrzymano:
$$\alpha^{\text{ij}} = \left\{ \begin{matrix}
\alpha^{11} = 0 \\
\alpha^{12} = 0 \\
\alpha^{13} = 0 \\
\end{matrix} \right.\ $$
$$\beta^{\text{ij}} = \left\{ \begin{matrix}
\beta^{11} = 0 \\
\beta^{12} = 0 \\
\beta^{13} = {2n}_{1}n_{2} = \frac{1}{a} \\
\end{matrix} \right.\ $$
$$S^{\text{kl}} = \left\{ \begin{matrix}
\begin{matrix}
S^{11} = 0 \\
S^{12} = e^{\omega m_{1}u^{i}} \bullet \sin{\omega n_{1}u^{1}} \\
S^{21} = 0 \\
\end{matrix} \\
S^{22} = e^{\omega m_{1}u^{i}} \bullet \cos{\omega n_{1}u^{1}} \\
\end{matrix} \right.\ $$
Siły przekrojowe
$${\hat{N}}^{11} = {\hat{N}}^{12} = {\hat{N}}^{21} = C$$
$${\hat{N}}^{22} = - \beta^{22} \bullet \left\lbrack C_{12}^{2} \bullet S^{12} + C_{22}^{2} \bullet S^{22} \right\rbrack$$
$${\hat{Q}}^{1} = \frac{\varepsilon}{\omega} \bullet g \bullet g^{1} \bullet n_{1}\left\lbrack \left( C_{12}^{1} + C_{12}^{2} \right) \bullet S^{12} + \left( C_{22}^{1} + C_{22}^{2} \right) \bullet S^{22} \right\rbrack$$
$${\hat{Q}}^{2} = 0$$
$${\hat{M}}^{11} = - \frac{\text{gε}g^{11}}{\omega^{2}} + C_{12}^{1} \bullet S^{12} + C_{22}^{1} \bullet S^{22}$$
$${\hat{M}}^{12} = {\hat{M}}^{21} = 0$$
$${\hat{M}}^{22} = - \frac{1}{\omega^{2}}\left\lbrack \left( 1 - \nu \right) \bullet \beta^{22} - g\varepsilon g^{22} \right\rbrack \bullet \left\lbrack C_{12}^{1} \bullet S^{12} + C_{22}^{1} \bullet S^{22} \right\rbrack$$
Przemieszczenia
$${\hat{w}}^{1} = \frac{1}{2Eh\varepsilon} \bullet \left\lbrack \frac{2H}{\varepsilon} - \left( 1 - \nu \right) \right\rbrack \bullet {\hat{Q}}^{1}$$
$${\hat{w}}^{2} = 0$$
$${\hat{w}}^{3} = \frac{g}{2Eh} \bullet \left\lbrack C_{12}^{2} \bullet S^{12} + C_{22}^{2} \bullet S^{22} \right\rbrack$$
Warunki brzegowe (dla dolnego brzegu u1 = 0
podparcie płaszcza powłoki
1) $w^{3} = {\overset{\overline{}}{w}}^{3} + {\hat{w}}^{3} = 0$
$$d^{1} = {\overset{\overline{}}{d}}^{1} + {\hat{d}}^{1} = 0$$
2)$\ w_{11}^{3} = {\overset{\overline{}}{w}}_{11}^{3} + {\hat{w}}_{11}^{3} = 0$
$$d^{1} = - g^{4}\left( w^{k}b_{\text{kl}} + w^{\frac{3}{1}} \right)$$
Składowe wektora przemieszczenia w stanie błonowym
$${\overset{\overline{}}{w}}^{3} = \frac{u^{'} - L}{2Eh}\left\lbrack 2ha\nu\gamma_{z} + a^{2}\gamma_{c} \right\rbrack$$
$${\overset{\overline{}}{w}}_{11}^{3} = - \frac{1}{2Eh}\left\lbrack 2ha\nu\gamma_{z} + a^{2}\gamma_{c} \right\rbrack$$
$${\overset{\overline{}}{w}}^{3}\left( u^{'} = 0 \right) = - \frac{L}{2Eh}\left\lbrack 2ha\nu\gamma_{z} + a^{2}\gamma_{c} \right\rbrack$$
$${\overset{\overline{}}{w}}_{11}^{3}(u^{'} = 0) = \frac{1}{2Eh}\left\lbrack 2ha\nu\gamma_{z} + a^{2}\gamma_{c} \right\rbrack$$
Składowa przemieszczenia w stanie zgięciowym
$${\hat{w}}^{3} = \frac{g}{2Eh} \bullet \left\lbrack C_{12}^{2} \bullet S^{12} + C_{22}^{2} \bullet S^{22} \right\rbrack$$
$${\hat{w}}_{11}^{3} = \frac{g}{2Eh} \bullet \left\lbrack C_{12}^{2} \bullet S_{11}^{12} + C_{22}^{2} \bullet S_{11}^{22} \right\rbrack$$
$$\left( \frac{S_{11}^{12} = \omega_{m},e^{\omega_{m},u^{'}} \bullet \sin{\omega_{n},u^{'}} + \omega_{n},e^{\omega_{m},u^{'}} \bullet \cos{\omega_{n},u^{'}}}{\begin{matrix}
S_{11}^{22} = \omega_{m},e^{\omega_{m}u^{'}} \bullet \cos{\omega_{n},u^{'}} + \omega_{n},e^{\omega_{m},u^{'}} \bullet \sin{\omega_{n},u^{'}} \\
S^{12}\left( u^{'} = 0 \right) = 0;\ S^{22}\left( u^{'} = 0 \right) = e^{\omega_{m},u^{'}} \bullet \cos{\omega_{n},u^{'}} = 1 \\
S_{11}^{12}\left( u^{'} = 0 \right) = \omega_{n};S_{11}^{22}\left( u^{'} = 0 \right) = \omega_{m} \\
\end{matrix}} \right)$$
$${\hat{w}}^{3}\left( u^{'} = 0 \right) = \frac{g}{2Eh} \bullet C_{22}^{2}$$
$${\hat{w}}_{11}^{3}\left( u^{'} = 0 \right) = \frac{g}{2Eh} \bullet \omega_{n} \bullet \left\lbrack C_{12}^{2} + C_{22}^{2} \right\rbrack$$
Podstawiając do warunków brzegowych otrzymano:
$$w^{3} = - \frac{L}{2Eh}\left\lbrack 2ha\nu\gamma_{z} + a^{2}\gamma_{c} \right\rbrack + \frac{g}{2Eh} \bullet C_{22}^{2} = 0$$
$$w_{11}^{3} = \frac{1}{2Eh}\left\lbrack 2ha\nu\gamma_{z} + a^{2}\gamma_{c} \right\rbrack + \frac{g}{2Eh} \bullet \omega_{n} \bullet \left\lbrack C_{12}^{2} + C_{22}^{2} \right\rbrack = 0$$
$$C_{22}^{2} = \frac{L}{g}\left\lbrack 2ha\nu\gamma_{z} + a^{2}\gamma_{c} \right\rbrack$$
$$C_{12}^{2} = - \left( \frac{1}{\omega_{n}} - \frac{L}{g} \right) \bullet \left\lbrack 2ha\nu\gamma_{z} + a^{2}\gamma_{c} \right\rbrack$$
w3 = 0
w113 = 0
Stan zgięciowy
$$M_{i}^{\text{ij}} - {\hat{Q}}^{j} = 0$$
$$M_{1}^{11} + \hat{M} - {\hat{Q}}^{1} = 0$$
$$M_{1}^{11} = {\hat{Q}}^{1}$$
S1112 = (S22+S12) • ωn
S1112 = (S22−S12) • ωn
$${\hat{M}}_{11}^{11} = \frac{\varepsilon}{\omega} \bullet gg^{11} \bullet n_{1} \bullet \left\lbrack \left( C_{12}^{1} - C_{22}^{1} \right)S^{12} + \left( C_{12}^{1} - C_{22}^{1} \right)S^{22} \right\rbrack$$
$${\hat{Q}}^{1} = \frac{\varepsilon}{\omega} \bullet gg^{11} \bullet n_{1} \bullet \left\lbrack \left( C_{12}^{1} + C_{12}^{2} \right)S^{12} + \left( C_{22}^{1} + C_{22}^{2} \right)S^{22} \right\rbrack$$
$${\overset{\overline{}}{M}}_{11}^{11} = {\hat{Q}}^{1}$$
C221 = −C122
C121 = C222
$$C_{12}^{2} = - \left( \frac{1}{\omega_{n} \bullet g} - \frac{L}{g} \right) \bullet \left\lbrack 2ha\nu\gamma_{z} + a^{2}\gamma_{c} \right\rbrack$$
Obliczenie wartości S
S22 = eωm, u′cosωn, u′
S12 = eωm, u′sinωn, u′
gdzie:
$$\omega^{4} = \frac{3\left( 1 - \nu^{2} \right)}{h^{2}}$$
h = (0, 007; 0, 006; 0, 005; 0, 005; 0, 005; 0, 004; 0, 004) m
R = 32, 4 m
υ = 0, 2
$$\omega^{4} = \frac{3\left( 1 - {0,2}^{2} \right)}{{0,007}^{2}} = 58775,5 \rightarrow \ \omega = 15,57$$
$$m_{1} = n_{1} = - \sqrt{\frac{1}{2R}} = - \sqrt{\frac{1}{2 \bullet 32,4}} = - 0,124$$
u′ = 0 → S22 = 1
S12 = 0
S11 = 0
u1 = 4, 332 ↦ S22 = e15, 57 • (−0,124) • 4, 332 • cos(15,57•(−0,124)•4,332)= − 1, 14 • 10−4
S12 = e15, 57 • (−0,124) • 4, 332 • sin(15,57•(−0,124)•4,332)= − 2, 04 • 10−4
u1 = 8, 664 ↦ S22 = 1, 01 • 10−8
S12 = 1, 002 • 10−8
u1 = 12, 996 ↦ S22 = −1, 99 • 10−14
S12 = 1, 26 • 10−13
u1 = 17, 328 ↦ S22 = −1, 99 • 10−18
S12 = −6, 12 • 10−18
u1 = 21, 660 ↦ S22 = 2, 29 • 10−22
S12 = 2, 29 • 10−22
u1 = 25, 992 ↦ S22 = −1, 35 • 10−29
S12 = 5, 92 • 10−30
u1 = 30, 322 ↦ S22 = −1, 07 • 10−34
S12 = −2, 06 • 10−34
Obliczenie wartości S
$${\hat{N}}^{22} = - \beta^{22} \bullet \left\lbrack C_{12}^{2} \bullet S^{12} + C_{22}^{2} \bullet S^{22} \right\rbrack$$
Dane:
$$\beta^{22} = \frac{1}{R} = \frac{1}{32,4} = 0,031$$
$$C_{12}^{2} = - \left( \frac{1}{\omega_{n} \bullet n_{1} \bullet g} \pm \frac{L}{g} \right) \bullet \left\lbrack 2ha\nu\gamma_{z} + a^{2}\gamma_{c} \right\rbrack = - \left( \frac{1}{15,57 \bullet \left( - 0,124 \right) \bullet 1049,76} \pm \frac{30,322}{1049,76} \right) \bullet \left( 0,014 \bullet 32,4 \bullet 0,2 \bullet 25 + {32,4}^{2} \bullet 13 \right) = - 387,5/400,99$$
$$C_{22}^{2} = \frac{L}{g}\left\lbrack 2ha\nu\gamma_{z} + a^{2}\gamma_{c} \right\rbrack = \frac{30,322}{1049,76}\left\lbrack 0,014 \bullet 32,4 \bullet 0,2 \bullet 25 + {32,4}^{2} \bullet 13 \right\rbrack = 394,25$$
$${\hat{N}}^{22} = - 0,031 \bullet \left\lbrack 400,99 \bullet S^{12} + 394,25 \bullet S^{22} \right\rbrack$$
$$u^{1} = 0 \longmapsto {\hat{N}}^{22} = - 0,031 \bullet \left\lbrack 400,99 \bullet 0 + 394,25 \bullet 1 \right\rbrack = 394,25$$
$$u^{1} = 4,332 \longmapsto {\hat{N}}^{22} = - 0,031 \bullet \left\lbrack 400,99 \bullet \left( - 2,04 \bullet 10^{- 4} \right) + 394,25 \bullet \left( - 1,14 \bullet 10^{- 4} \right) \right\rbrack = - 0,04$$
$$u^{1} = 8,664 \longmapsto {\hat{N}}^{22} = 3,86 \bullet 10^{- 6}$$
$$u^{1} = 12,996 \longmapsto {\hat{N}}^{22} = - 9,40 \bullet 10^{- 12}$$
$$u^{1} = 17,328 \longmapsto {\hat{N}}^{22} = - 7,10 \bullet 10^{- 16}$$
$$u^{1} = 21,660 \longmapsto {\hat{N}}^{22} = 8,76 \bullet 10^{- 20}$$
$$u^{1} = 25,992 \longmapsto {\hat{N}}^{22} = - 5,39 \bullet 10^{- 27}$$
$$u^{1} = 30,322 \longmapsto {\hat{N}}^{22} = - 3,94 \bullet 10^{- 32}$$
$${\hat{Q}}^{1} = \frac{\varepsilon}{\omega} \bullet gg^{11} \bullet n_{1} \bullet \left\lbrack \left( C_{12}^{1} + C_{12}^{2} \right)S^{12} + \left( C_{22}^{1} + C_{22}^{2} \right)S^{22} \right\rbrack$$
Dane:
$$\varepsilon = \frac{1}{R} = \frac{1}{32,4} = 0,031$$
g = a2 = 32, 42 = 1049, 76
g11 = 1
$$n_{1} = - \sqrt{\frac{1}{2a}} = - 0,124$$
C221 = −C122 ⇒ C221 = 387, 5; C122 = −387, 5
C121 = C222 ⇒ C121 = 394, 25; C222 = 394, 25
$$u^{1} = 0 \longmapsto {\hat{Q}}^{1} = \frac{0,031}{15,57} \bullet 1049,76 \bullet 1 \bullet \left( - 0,124 \right) \bullet \left\lbrack \left( 394,25 + \left( - 387,5 \right) \right)0 + \left( 387,5 + 394,25 \right)1 \right\rbrack = - 1,75$$
$$u^{1} = 4,332 \longmapsto {\hat{Q}}^{1} = \frac{0,031}{15,57} \bullet 1049,76 \bullet 1 \bullet \left( - 0,124 \right) \bullet \left\lbrack \left( 394,25 + \left( - 387,5 \right) \right)\left( - 2,04 \bullet 10^{- 4} \right) + \left( 387,5 + 394,25 \right)\left( - 1,14 \bullet 10^{- 4} \right) \right\rbrack = 5,13 \bullet 10^{- 4}$$
$$u^{1} = 8,664 \longmapsto {\hat{Q}}^{1} = - 2,75 \bullet 10^{- 8}$$
$$u^{1} = 12,996 \longmapsto {\hat{Q}}^{1} = - 1,33 \bullet 10^{- 13}$$
$$u^{1} = 17,328 \longmapsto {\hat{Q}}^{1} = 1,01 \bullet 10^{- 17}$$
$$u^{1} = 21,660 \longmapsto {\hat{Q}}^{1} = - 5,11 \bullet 10^{- 22}$$
$$u^{1} = 25,992 \longmapsto {\hat{Q}}^{1} = 7,57 \bullet 10^{- 30}$$
$$u^{1} = 30,322 \longmapsto {\hat{Q}}^{1} = 4,81 \bullet 10^{- 32}$$
$${\hat{M}}^{11} = \frac{\text{gε}g^{11}}{\omega^{2}} \bullet C_{12}^{1} \bullet S^{12} + C_{22}^{1} \bullet S^{22}$$
$$u^{1} = 0 \longmapsto {\hat{M}}^{11} = \frac{1049,76 \bullet 0,031 \bullet 1}{{15,57}^{2}} \bullet \left\lbrack 394,25 \bullet 1 + 387,5 \bullet 0 \right\rbrack = 823,99$$
$$u^{1} = 4,332 \longmapsto {\hat{M}}^{11} = \frac{1049,76 \bullet 0,031 \bullet 1}{{15,57}^{2}} \bullet \left\lbrack 394,25 \bullet ( - 2,04 \bullet 10^{- 4}) + 388,03 \bullet ( - 1,13 \bullet 10^{- 4}) \right\rbrack = - 0,075$$
$$u^{1} = 8,664 \longmapsto {\hat{M}}^{11} = 5,04 \bullet 10^{- 8}$$
$$u^{1} = 12,996 \longmapsto {\hat{M}}^{11} = 1,015 \bullet 10^{- 10}$$
$$u^{1} = 17,328 \longmapsto {\hat{M}}^{11} = - 2,89 \bullet 10^{- 15}$$
$$u^{1} = 21,660 \longmapsto {\hat{M}}^{11} = 1,59 \bullet 10^{- 21}$$
$$u^{1} = 25,992 \longmapsto {\hat{M}}^{11} = 1,20 \bullet 10^{- 26}$$
$$u^{1} = 30,322 \longmapsto {\hat{M}}^{11} = - 1,94 \bullet 10^{- 31}$$
$${\hat{M}}^{22} = - \frac{1}{\omega^{2}}\left\lbrack \left( 1 - \nu \right) \bullet \beta^{22} - g\varepsilon g^{22} \right\rbrack \bullet \left\lbrack C_{12}^{1} \bullet S^{12} + C_{22}^{1} \bullet S^{22} \right\rbrack$$
$${g^{22} = \frac{1}{a^{2}} = \frac{1}{{32,4}^{2}} = 9,53 \bullet 10^{- 4}\backslash n}{u^{1} = 0 \longmapsto {\hat{M}}^{22} = - \frac{1}{{15,57}^{2}}\left\lbrack \left( 1 - 0,2 \right) \bullet 0,031 - 1049,76 \bullet 0,031 \bullet 9,53 \bullet 10^{- 4} \right\rbrack \bullet \left\lbrack 394,25 \bullet 1 + 387,5 \bullet 0 \right\rbrack = 1,01 \bullet 10^{- 2}}$$
$$u^{1} = 4,332 \longmapsto {\hat{M}}^{22} = - \frac{1}{{15,57}^{2}}\left\lbrack \left( 1 - 0,2 \right) \bullet 0,031 - 1049,76 \bullet 0,031 \bullet 9,53 \bullet 10^{- 4} \right\rbrack \bullet \left\lbrack 394,25 \bullet \left( - 2,04 \bullet 10^{- 4} \right) + 388,03 \bullet \left( - 1,13 \bullet 10^{- 4} \right) \right\rbrack = - 9,24 \bullet 10^{- 7}$$
$$u^{1} = 8,664 \longmapsto {\hat{M}}^{22} = 5,72 \bullet 10^{- 13}$$
$$u^{1} = 12,996 \longmapsto {\hat{M}}^{22} = 1,05 \bullet 10^{- 15}$$
$$u^{1} = 17,328 \longmapsto {\hat{M}}^{22} = - 2,99 \bullet 10^{- 20}$$
$$u^{1} = 21,660 \longmapsto {\hat{M}}^{22} = 1,65 \bullet 10^{- 26}$$
$$u^{1} = 25,992 \longmapsto {\hat{M}}^{22} = 1,11 \bullet 10^{- 31}$$
$$u^{1} = 30,322 \longmapsto {\hat{M}}^{22} = - 1,79 \bullet 10^{- 36}$$
$${\hat{w}}^{1} = \frac{1}{2Eh\varepsilon} \bullet \left\lbrack \frac{2H}{\varepsilon} - \left( 1 - \nu \right) \right\rbrack \bullet {\hat{Q}}^{1}$$
$$u^{1} = 8,664 \longmapsto {\hat{w}}^{1} = \frac{1}{0,014 \bullet 34 \bullet 10^{6} \bullet 0,031} \bullet \left\lbrack \frac{30,322}{0,031} - \left( 1 - 0,2 \right) \right\rbrack \bullet \left( - 2,75 \bullet 10^{- 8} \right) = - 2,12 \bullet 10^{- 9}$$
$$u^{1} = 12,996 \longmapsto {\hat{w}}^{1} = - 1,23 \bullet 10^{- 14}$$
$$u^{1} = 17,328 \longmapsto {\hat{w}}^{1} = 9,37 \bullet 10^{- 19}$$
$$u^{1} = 21,660 \longmapsto {\hat{w}}^{1} = - 4,74 \bullet 10^{- 23}$$
$$u^{1} = 25,992 \longmapsto {\hat{w}}^{1} = 8,77 \bullet 10^{- 31}$$
$$u^{1} = 30,322 \longmapsto {\hat{w}}^{1} = 5,57 \bullet 10^{- 33}$$
$${\hat{w}}^{3} = \frac{g}{2Eh} \bullet \left\lbrack C_{12}^{2} \bullet S^{12} + C_{22}^{2} \bullet S^{22} \right\rbrack$$
$$u^{1} = 8,664 \longmapsto {\hat{w}}^{3} = \frac{1049,76}{0,012 \bullet 34 \bullet 10^{6}} \bullet \left\lbrack \left( - 388,56 \right) \bullet 1,002 \bullet 10^{- 8} + 394,24 \bullet \left( - 1,01 \right) \bullet 10^{- 8} \right\rbrack = 2,27 \bullet 10^{- 10}$$
$$u^{1} = 12,996 \longmapsto {\hat{w}}^{3} = - 1,76 \bullet 10^{- 13}$$
$$u^{1} = 17,328 \longmapsto {\hat{w}}^{3} = 4,91 \bullet 10^{- 18}$$
$$u^{1} = 21,660 \longmapsto {\hat{w}}^{3} = - 4,40 \bullet 10^{- 24}$$
$$u^{1} = 25,992 \longmapsto {\hat{w}}^{3} = - 2,94 \bullet 10^{- 29}$$
$$u^{1} = 30,322 \longmapsto {\hat{w}}^{3} = 4,72 \bullet 10^{- 34}$$
Obliczenie fizyczne sił przekrojowych
$$N_{\overset{\overline{}}{\text{ij}}} = \sqrt{\frac{g_{\text{ij}}}{g^{\text{ii}}}\left( {\overset{\overline{}}{N}}_{\text{ij}} + {\hat{N}}_{\text{ij}} \right)}$$
$$Q_{\overset{\overline{}}{i}} = \frac{1}{\sqrt{g^{\text{ij}}}}{\hat{Q}}^{i}$$
$$M_{\overset{\overline{}}{i1}} = \sqrt{\frac{g \bullet g^{11}}{g^{\text{ii}}}}{\hat{M}}^{i2}$$
$$M_{\overset{\overline{}}{i2}} = \sqrt{\frac{g \bullet g^{22}}{g^{\text{ii}}}}{\hat{M}}^{i1}$$
Wartości fizyczne przemieszczeń:
$$w_{1}^{-} = \sqrt{g_{\text{ii}}}\left( {\overset{\overline{}}{w}}^{1} + {\hat{w}}^{1} - {\hat{w}}^{1} \bullet (u^{1} = 0 \right))$$
$$w_{3}^{-} = {\overset{\overline{}}{w}}^{3} + {\hat{w}}^{3}$$
Obliczenie fizyczne sił przekrojowych
$$N_{\overset{\overline{}}{11}} = \sqrt{\frac{g_{11}}{g^{11}}}\left( {\overset{\overline{}}{N}}^{11} + {\hat{N}}^{11} \right)$$
g11 = 1
g11 = 1
$${\hat{N}}^{11} = 0$$
$$N_{\overset{\overline{}}{11}} = {\overset{\overline{}}{N}}^{11}$$
$$N_{\overset{\overline{}}{22}} = \sqrt{\frac{g_{22}}{g^{22}}}\left( {\overset{\overline{}}{N}}^{22} + {\hat{N}}^{22} \right)$$
g22 = a2 = 1049, 76
$$g^{22} = \frac{1}{a^{2}} = \frac{1}{{32,4}^{2}} = 9,53 \bullet 10^{- 4}$$
$${N_{\overset{\overline{}}{22}} = 1049,53\left( {\overset{\overline{}}{N}}^{22} + {\hat{N}}^{22} \right)\backslash n}{M_{\overset{\overline{}}{21}} = - \sqrt{\frac{g \bullet g^{11}}{g^{22}}}{\hat{M}}^{22}}$$
g = a2 = 1049, 76
$$g^{22} = \frac{1}{a^{2}} = \frac{1}{{32,4}^{2}} = 9,53 \bullet 10^{- 4}$$
g11 = 1
$$M_{\overset{\overline{}}{21}} = - 1049,54\ {\hat{M}}^{22}$$
$$M_{\overset{\overline{}}{12}} = - \sqrt{\frac{g \bullet g^{22}}{g^{11}}}{\hat{M}}^{11}$$
g = a2 = 1049, 76
$$g^{22} = \frac{1}{a^{2}} = \frac{1}{{32,4}^{2}} = 9,53 \bullet 10^{- 4}$$
g11 = 1
$$M_{\overset{\overline{}}{12}} = {\hat{M}}^{11}$$
$$w_{1}^{-} = \sqrt{g_{\text{ii}}}\left( {\overset{\overline{}}{w}}^{1} + {\hat{w}}^{1} - {\hat{w}}^{1} \bullet (u^{1} = 0 \right))$$
g11 = 1
$$w_{1}^{-} = \left( {\overset{\overline{}}{w}}^{1} + {\hat{w}}^{1} - {\hat{w}}^{1} \bullet (u^{1} = 0 \right))$$
$$w_{3}^{-} = {\overset{\overline{}}{w}}^{3} + {\hat{w}}^{3}$$
ZESTAWIENIE WARTOŚCI FIZYCZNYCH SIŁ PRZEKROJOWYCH ORAZ PRZEMIESZCZEŃ
| Ui | N-11 | N-22 | M-11 | M-22 | Q-1 | w-1 | w-3 |
|---|---|---|---|---|---|---|---|
| 0 | -16,4497 | 6,083117 | -7,26157E-05 | 2,68534E-05 | 0 | 0 | -0,07041 |
| 0,1 | -16,3412 | 6,042994 | -7,21368E-05 | 2,66763E-05 | -6,11296E-07 | -4,4E-05 | -0,06995 |
| 0,2 | -16,2327 | 6,00287 | -7,16578E-05 | 2,64992E-05 | -1,21855E-06 | -8,7E-05 | -0,06948 |
| 0,3 | -16,1242 | 5,962747 | -7,11788E-05 | 2,6322E-05 | -1,82175E-06 | -0,00013 | -0,06902 |
| 0,4 | -16,0157 | 5,922623 | -7,06999E-05 | 2,61449E-05 | -2,42091E-06 | -0,00017 | -0,06855 |
| 0,5 | -15,9072 | 5,8825 | -7,02209E-05 | 2,59678E-05 | -3,01602E-06 | -0,00022 | -0,06809 |
| 0,6 | -15,7987 | 5,842377 | -6,97419E-05 | 2,57907E-05 | -3,60709E-06 | -0,00026 | -0,06762 |
| 0,7 | -15,6902 | 5,802253 | -6,9263E-05 | 2,56135E-05 | -4,19412E-06 | -0,0003 | -0,06716 |
| 0,8 | -15,5817 | 5,76213 | -6,8784E-05 | 2,54364E-05 | -4,7771E-06 | -0,00034 | -0,0667 |
| 0,9 | -15,4732 | 5,722006 | -6,83051E-05 | 2,52593E-05 | -5,35603E-06 | -0,00038 | -0,06623 |
| 1 | -15,3647 | 5,681883 | -6,78261E-05 | 2,50822E-05 | -5,93092E-06 | -0,00043 | -0,06577 |
| 1,1 | -15,2562 | 5,641759 | -6,73471E-05 | 2,49051E-05 | -6,50176E-06 | -0,00047 | -0,0653 |
| 1,2 | -15,1477 | 5,601636 | -6,68682E-05 | 2,47279E-05 | -7,06855E-06 | -0,00051 | -0,06484 |
| 1,3 | -15,0392 | 5,561512 | -6,63892E-05 | 2,45508E-05 | -7,63131E-06 | -0,00055 | -0,06437 |
| 1,4 | -14,9307 | 5,521389 | -6,59102E-05 | 2,43737E-05 | -8,19001E-06 | -0,00059 | -0,06391 |
| 1,5 | -14,8222 | 5,481265 | -6,54313E-05 | 2,41966E-05 | -8,74467E-06 | -0,00063 | -0,06344 |
| 1,6 | -14,7137 | 5,441142 | -6,49523E-05 | 2,40195E-05 | -9,29529E-06 | -0,00067 | -0,06298 |
| 1,7 | -14,6052 | 5,401019 | -6,44733E-05 | 2,38423E-05 | -9,84186E-06 | -0,00071 | -0,06252 |
| 1,8 | -14,4967 | 5,360895 | -6,39944E-05 | 2,36652E-05 | -1,03844E-05 | -0,00074 | -0,06205 |
| 1,9 | -14,3882 | 5,320772 | -6,35154E-05 | 2,34881E-05 | -1,09229E-05 | -0,00078 | -0,06159 |
| 2 | -14,2797 | 5,280648 | -6,30364E-05 | 2,3311E-05 | -1,14573E-05 | -0,00082 | -0,06112 |
| 2,1 | -14,1712 | 5,240525 | -6,25575E-05 | 2,31338E-05 | -1,19877E-05 | -0,00086 | -0,06066 |
| 2,2 | -14,0627 | 5,200401 | -6,20785E-05 | 2,29567E-05 | -1,2514E-05 | -0,00105 | -0,07022 |
| 2,3 | -13,9542 | 5,160278 | -6,15996E-05 | 2,27796E-05 | -1,30363E-05 | -0,00109 | -0,06968 |
| 2,4 | -13,8457 | 5,120154 | -6,11206E-05 | 2,26025E-05 | -1,35546E-05 | -0,00113 | -0,06914 |
| 2,5 | -13,7372 | 5,080031 | -6,06416E-05 | 2,24254E-05 | -1,40688E-05 | -0,00118 | -0,0686 |
| 2,6 | -13,6287 | 5,039907 | -6,01627E-05 | 2,22482E-05 | -1,45789E-05 | -0,00122 | -0,06805 |
| 2,7 | -13,5202 | 4,999784 | -5,96837E-05 | 2,20711E-05 | -1,50851E-05 | -0,00126 | -0,06751 |
| 2,8 | -13,4117 | 4,95966 | -5,92047E-05 | 2,1894E-05 | -1,55871E-05 | -0,0013 | -0,06697 |
| 2,9 | -13,3032 | 4,919537 | -5,87258E-05 | 2,17169E-05 | -1,60852E-05 | -0,00134 | -0,06643 |
| 3 | -13,1947 | 4,879414 | -5,82468E-05 | 2,15398E-05 | -1,65791E-05 | -0,00139 | -0,06589 |
| 3,1 | -13,0862 | 4,83929 | -5,77678E-05 | 2,13626E-05 | -1,70691E-05 | -0,00143 | -0,06534 |
| 3,2 | -12,9777 | 4,799167 | -5,72889E-05 | 2,11855E-05 | -1,7555E-05 | -0,00147 | -0,0648 |
| 3,3 | -12,8692 | 4,759043 | -5,68099E-05 | 2,10084E-05 | -1,80368E-05 | -0,00151 | -0,06426 |
| 3,4 | -12,7607 | 4,71892 | -5,6331E-05 | 2,08313E-05 | -1,85146E-05 | -0,00155 | -0,06372 |
| 3,5 | -12,6522 | 4,678796 | -5,5852E-05 | 2,06541E-05 | -1,89884E-05 | -0,00159 | -0,06318 |
| 3,6 | -12,5437 | 4,638673 | -5,5373E-05 | 2,0477E-05 | -1,94581E-05 | -0,00163 | -0,06264 |
| 3,7 | -12,4352 | 4,598549 | -5,48941E-05 | 2,02999E-05 | -1,99237E-05 | -0,00166 | -0,06209 |
| 3,8 | -12,3267 | 4,558426 | -5,44151E-05 | 2,01228E-05 | -2,03853E-05 | -0,0017 | -0,06155 |
| 3,9 | -12,2182 | 4,518302 | -5,39361E-05 | 1,99457E-05 | -2,08429E-05 | -0,00174 | -0,06101 |
| 4 | -10,3797 | 4,478179 | -3,3664E-05 | 1,45238E-05 | -1,56464E-05 | -0,00178 | -0,06047 |
| 4,1 | -10,2867 | 4,438056 | -3,33624E-05 | 1,43937E-05 | -1,59766E-05 | -0,00182 | -0,05993 |
| 4,2 | -10,1937 | 4,397932 | -3,30607E-05 | 1,42636E-05 | -1,63039E-05 | -0,00185 | -0,05938 |
| 4,3 | -10,1007 | 4,357809 | -3,27591E-05 | 1,41334E-05 | -1,66281E-05 | -0,00189 | -0,05884 |
| 4,4 | -10,0077 | 4,317685 | -3,24575E-05 | 1,40033E-05 | -1,69495E-05 | -0,00231 | -0,06996 |
| 4,5 | -9,91473 | 4,277562 | -3,21559E-05 | 1,38732E-05 | -1,72678E-05 | -0,00235 | -0,06931 |
| 4,6 | -9,82173 | 4,237438 | -3,18543E-05 | 1,3743E-05 | -1,75832E-05 | -0,0024 | -0,06866 |
| 4,7 | -9,72873 | 4,197315 | -3,15526E-05 | 1,36129E-05 | -1,78956E-05 | -0,00244 | -0,06801 |
| 4,8 | -9,63573 | 4,157191 | -3,1251E-05 | 1,34828E-05 | -1,8205E-05 | -0,00248 | -0,06736 |
| 4,9 | -9,54273 | 4,117068 | -3,09494E-05 | 1,33527E-05 | -1,85114E-05 | -0,00252 | -0,06671 |
| 5 | -9,44973 | 4,076944 | -3,06478E-05 | 1,32225E-05 | -1,88149E-05 | -0,00256 | -0,06606 |
| 5,1 | -9,35673 | 4,036821 | -3,03462E-05 | 1,30924E-05 | -1,91154E-05 | -0,0026 | -0,06541 |
| 5,2 | -9,26373 | 3,996698 | -3,00445E-05 | 1,29623E-05 | -1,9413E-05 | -0,00264 | -0,06476 |
| 5,3 | -9,17073 | 3,956574 | -2,97429E-05 | 1,28321E-05 | -1,97075E-05 | -0,00268 | -0,06411 |
| 5,4 | -9,07773 | 3,916451 | -2,94413E-05 | 1,2702E-05 | -1,99991E-05 | -0,00272 | -0,06346 |
| 5,5 | -8,98473 | 3,876327 | -2,91397E-05 | 1,25719E-05 | -2,02878E-05 | -0,00276 | -0,06281 |
| 5,6 | -8,89173 | 3,836204 | -2,8838E-05 | 1,24417E-05 | -2,05734E-05 | -0,0028 | -0,06216 |
| 5,7 | -8,79873 | 3,79608 | -2,85364E-05 | 1,23116E-05 | -2,08561E-05 | -0,00284 | -0,06151 |
| 5,8 | -8,70573 | 3,755957 | -2,82348E-05 | 1,21815E-05 | -2,11358E-05 | -0,00288 | -0,06085 |
| 5,9 | -8,61273 | 3,715833 | -2,79332E-05 | 1,20514E-05 | -2,14125E-05 | -0,00292 | -0,0602 |
| 6 | -7,09978 | 3,67571 | -1,59905E-05 | 8,27863E-06 | -1,50599E-05 | -0,00295 | -0,05955 |
| 6,1 | -7,02228 | 3,635586 | -1,58159E-05 | 8,18826E-06 | -1,5248E-05 | -0,00299 | -0,0589 |
| 6,2 | -6,94478 | 3,595463 | -1,56414E-05 | 8,09789E-06 | -1,54339E-05 | -0,00303 | -0,05825 |
| 6,3 | -6,86728 | 3,55534 | -1,54668E-05 | 8,00752E-06 | -1,56179E-05 | -0,00306 | -0,0576 |
| 6,4 | -6,78978 | 3,515216 | -1,52923E-05 | 7,91715E-06 | -1,57997E-05 | -0,0031 | -0,05695 |
| 6,5 | -6,71228 | 3,475093 | -1,51177E-05 | 7,82679E-06 | -1,59795E-05 | -0,00313 | -0,0563 |
| 6,6 | -6,63478 | 3,434969 | -1,49432E-05 | 7,73642E-06 | -1,61572E-05 | -0,00317 | -0,05565 |
| 6,7 | -6,55728 | 3,394846 | -1,47686E-05 | 7,64605E-06 | -1,63329E-05 | -0,0032 | -0,055 |
| 6,8 | -6,47978 | 3,354722 | -1,45941E-05 | 7,55568E-06 | -1,65065E-05 | -0,00324 | -0,05435 |
| 6,9 | -6,40228 | 3,314599 | -1,44195E-05 | 7,46531E-06 | -1,6678E-05 | -0,00327 | -0,0537 |
| 7 | -6,32478 | 3,274475 | -1,4245E-05 | 7,37494E-06 | -1,68475E-05 | -0,0033 | -0,05305 |
| 7,1 | -6,24728 | 3,234352 | -1,40704E-05 | 7,28458E-06 | -1,70149E-05 | -0,00334 | -0,0524 |
| 7,2 | -6,16978 | 3,194228 | -1,38959E-05 | 7,19421E-06 | -1,71803E-05 | -0,00337 | -0,05175 |
| 7,3 | -6,09228 | 3,154105 | -1,37213E-05 | 7,10384E-06 | -1,73435E-05 | -0,0034 | -0,0511 |
| 7,4 | -6,01478 | 3,113981 | -1,35468E-05 | 7,01347E-06 | -1,75048E-05 | -0,00343 | -0,05045 |
| 7,5 | -5,93728 | 3,073858 | -1,33722E-05 | 6,9231E-06 | -1,76639E-05 | -0,00346 | -0,0498 |
| 7,6 | -5,85978 | 3,033735 | -1,31977E-05 | 6,83274E-06 | -1,7821E-05 | -0,0035 | -0,04915 |
| 7,7 | -5,78228 | 2,993611 | -1,30231E-05 | 6,74237E-06 | -1,7976E-05 | -0,00353 | -0,0485 |
| 7,8 | -5,70478 | 2,953488 | -1,28486E-05 | 6,652E-06 | -1,8129E-05 | -0,00356 | -0,04785 |
| 7,9 | -5,62728 | 2,913364 | -1,2674E-05 | 6,56163E-06 | -1,82799E-05 | -0,00359 | -0,0472 |
| 8 | -5,54978 | 2,873241 | -1,24995E-05 | 6,47126E-06 | -1,84287E-05 | -0,00361 | -0,04655 |
| 8,1 | -5,47228 | 2,833117 | -1,23249E-05 | 6,38089E-06 | -1,85755E-05 | -0,00364 | -0,0459 |
| 8,2 | -5,39478 | 2,792994 | -1,21504E-05 | 6,29053E-06 | -1,87202E-05 | -0,00367 | -0,04525 |
| 8,3 | -5,31728 | 2,75287 | -1,19758E-05 | 6,20016E-06 | -1,88628E-05 | -0,0037 | -0,0446 |
| 8,4 | -5,23978 | 2,712747 | -1,18013E-05 | 6,10979E-06 | -1,90034E-05 | -0,00373 | -0,04395 |
| 8,5 | -5,16228 | 2,672623 | -1,16267E-05 | 6,01942E-06 | -1,91419E-05 | -0,00375 | -0,0433 |
| 8,6 | -5,08478 | 2,6325 | -1,14522E-05 | 5,92905E-06 | -1,92784E-05 | -0,00378 | -0,04265 |
| 8,7 | -5,00728 | 2,592377 | -1,12776E-05 | 5,83869E-06 | -1,94128E-05 | -0,00381 | -0,042 |
| 8,8 | -4,92978 | 2,552253 | -1,11031E-05 | 5,74832E-06 | -1,95451E-05 | -0,00383 | -0,04135 |
| 8,9 | -4,85228 | 2,51213 | -1,09285E-05 | 5,65795E-06 | -1,96753E-05 | -0,00386 | -0,0407 |
| 9 | -4,77478 | 2,472006 | -1,0754E-05 | 5,56758E-06 | -1,98035E-05 | -0,00388 | -0,04005 |
| 9,1 | -4,69728 | 2,431883 | -1,05794E-05 | 5,47721E-06 | -1,99297E-05 | -0,00391 | -0,0394 |
| 9,2 | -4,61978 | 2,391759 | -1,04049E-05 | 5,38685E-06 | -2,00537E-05 | -0,00393 | -0,03875 |
| 9,3 | -4,54228 | 2,351636 | -1,02303E-05 | 5,29648E-06 | -2,01757E-05 | -0,00396 | -0,0381 |
| 9,4 | -4,46478 | 2,311512 | -1,00558E-05 | 5,20611E-06 | -2,02957E-05 | -0,00398 | -0,03745 |
| 9,5 | -4,38728 | 2,271389 | -9,88125E-06 | 5,11574E-06 | -2,04135E-05 | -0,004 | -0,0368 |
| 9,6 | -4,30978 | 2,231265 | -9,7067E-06 | 5,02537E-06 | -2,05293E-05 | -0,00403 | -0,03615 |
| 9,7 | -4,23228 | 2,191142 | -9,53215E-06 | 4,935E-06 | -2,06431E-05 | -0,00405 | -0,0355 |
| 9,8 | -4,15478 | 2,151019 | -9,3576E-06 | 4,84464E-06 | -2,07548E-05 | -0,00407 | -0,03485 |
| 9,9 | -4,07728 | 2,110895 | -9,18305E-06 | 4,75427E-06 | -2,08644E-05 | -0,00409 | -0,0342 |
| 10 | -3,99978 | 2,070772 | -9,0085E-06 | 4,6639E-06 | -2,09719E-05 | -0,00411 | -0,03355 |
| 10,1 | -3,92228 | 2,030648 | -8,83395E-06 | 4,57353E-06 | -2,10774E-05 | -0,00413 | -0,0329 |
| 10,2 | -3,84478 | 1,990525 | -8,6594E-06 | 4,48316E-06 | -2,11809E-05 | -0,00415 | -0,03225 |
| 10,3 | -3,76728 | 1,950401 | -8,48485E-06 | 4,3928E-06 | -2,12822E-05 | -0,00417 | -0,0316 |
| 10,4 | -3,68978 | 1,910278 | -8,3103E-06 | 4,30243E-06 | -2,13815E-05 | -0,00419 | -0,03095 |
| 10,5 | -3,61228 | 1,870154 | -8,13575E-06 | 4,21206E-06 | -2,14787E-05 | -0,00421 | -0,0303 |
| 10,6 | -3,53478 | 1,830031 | -7,9612E-06 | 4,12169E-06 | -2,15739E-05 | -0,00423 | -0,02965 |
| 10,7 | -3,45728 | 1,789907 | -7,78666E-06 | 4,03132E-06 | -2,1667E-05 | -0,00425 | -0,029 |
| 10,8 | -3,37978 | 1,749784 | -7,61211E-06 | 3,94095E-06 | -2,17581E-05 | -0,00427 | -0,02835 |
| 10,9 | -3,30228 | 1,70966 | -7,43756E-06 | 3,85059E-06 | -2,1847E-05 | -0,00535 | -0,03462 |
| 11 | -3,22478 | 1,669537 | -7,26301E-06 | 3,76022E-06 | -2,19339E-05 | -0,00537 | -0,03381 |
| 11,1 | -3,14728 | 1,629414 | -7,08846E-06 | 3,66985E-06 | -2,20188E-05 | -0,00539 | -0,033 |
| 11,2 | -3,06978 | 1,58929 | -6,91391E-06 | 3,57948E-06 | -2,21016E-05 | -0,00541 | -0,03219 |
| 11,3 | -2,99228 | 1,549167 | -6,73936E-06 | 3,48911E-06 | -2,21823E-05 | -0,00543 | -0,03137 |
| 11,4 | -2,91478 | 1,509043 | -6,56481E-06 | 3,39875E-06 | -2,2261E-05 | -0,00545 | -0,03056 |
| 11,5 | -2,83728 | 1,46892 | -6,39026E-06 | 3,30838E-06 | -2,23376E-05 | -0,00547 | -0,02975 |
| 11,6 | -2,75978 | 1,428796 | -6,21571E-06 | 3,21801E-06 | -2,24121E-05 | -0,00548 | -0,02894 |
| 11,7 | -2,68228 | 1,388673 | -6,04116E-06 | 3,12764E-06 | -2,24846E-05 | -0,0055 | -0,02812 |
| 11,8 | -2,60478 | 1,348549 | -5,86661E-06 | 3,03727E-06 | -2,2555E-05 | -0,00552 | -0,02731 |
| 11,9 | -2,52728 | 1,308426 | -5,69206E-06 | 2,94691E-06 | -2,26233E-05 | -0,00554 | -0,0265 |
| 12 | -2,44978 | 1,268302 | -5,51751E-06 | 2,85654E-06 | -2,26896E-05 | -0,00555 | -0,02568 |
| 12,1 | -2,37228 | 1,228179 | -5,34296E-06 | 2,76617E-06 | -2,27538E-05 | -0,00557 | -0,02487 |
| 12,2 | -2,29478 | 1,188056 | -5,16841E-06 | 2,6758E-06 | -2,28159E-05 | -0,00558 | -0,02406 |
| 12,3 | -2,21728 | 1,147932 | -4,99386E-06 | 2,58543E-06 | -2,2876E-05 | -0,0056 | -0,02325 |
| 12,4 | -2,13978 | 1,107809 | -4,81931E-06 | 2,49506E-06 | -2,2934E-05 | -0,00561 | -0,02243 |
| 12,5 | -2,06228 | 1,067685 | -4,64476E-06 | 2,4047E-06 | -2,299E-05 | -0,00563 | -0,02162 |
| 12,6 | -1,98478 | 1,027562 | -4,47021E-06 | 2,31433E-06 | -2,30439E-05 | -0,00564 | -0,02081 |
| 12,7 | -1,90728 | 0,987438 | -4,29566E-06 | 2,22396E-06 | -2,30957E-05 | -0,00565 | -0,02 |
| 12,8 | -1,82978 | 0,947315 | -4,12111E-06 | 2,13359E-06 | -2,31455E-05 | -0,00566 | -0,01918 |
| 12,9 | -1,75228 | 0,907191 | -3,94657E-06 | 2,04322E-06 | -2,31932E-05 | -0,00568 | -0,01837 |
| 13 | -1,67478 | 0,867068 | -3,77202E-06 | 1,95286E-06 | -2,32388E-05 | -0,00569 | -0,01756 |
| 13,1 | -1,59728 | 0,826944 | -3,59747E-06 | 1,86249E-06 | -2,32824E-05 | -0,0057 | -0,01675 |
| 13,2 | -1,51978 | 0,786821 | -3,42292E-06 | 1,77212E-06 | -2,33239E-05 | -0,00571 | -0,01593 |
| 13,3 | -1,44228 | 0,746698 | -3,24837E-06 | 1,68175E-06 | -2,33633E-05 | -0,00572 | -0,01512 |
| 13,4 | -1,36478 | 0,706574 | -3,07382E-06 | 1,59138E-06 | -2,34007E-05 | -0,00573 | -0,01431 |
| 13,5 | -1,28728 | 0,666451 | -2,89927E-06 | 1,50101E-06 | -2,3436E-05 | -0,00574 | -0,0135 |
| 13,6 | -1,20978 | 0,626327 | -2,72472E-06 | 1,41065E-06 | -2,34693E-05 | -0,00574 | -0,01268 |
| 13,7 | -1,13228 | 0,586204 | -2,55017E-06 | 1,32028E-06 | -2,35004E-05 | -0,00575 | -0,01187 |
| 13,8 | -1,05478 | 0,54608 | -2,37562E-06 | 1,22991E-06 | -2,35296E-05 | -0,00576 | -0,01106 |
| 13,9 | -0,97727 | 0,505957 | -2,20107E-06 | 1,13954E-06 | -2,35566E-05 | -0,00577 | -0,01025 |
| 14 | -0,89978 | 0,465833 | -2,02652E-06 | 1,04917E-06 | -2,35816E-05 | -0,00577 | -0,00943 |
| 14,1 | -0,82228 | 0,42571 | -1,85197E-06 | 9,58806E-07 | -2,36045E-05 | -0,00578 | -0,00862 |
| 14,2 | -0,74478 | 0,385586 | -1,67742E-06 | 8,68438E-07 | -2,36254E-05 | -0,00578 | -0,00781 |
| 14,3 | -0,66727 | 0,345463 | -1,50287E-06 | 7,7807E-07 | -2,36442E-05 | -0,00579 | -0,007 |
| 14,4 | -0,58977 | 0,30534 | -1,32832E-06 | 6,87702E-07 | -2,3661E-05 | -0,00579 | -0,00618 |
| 14,5 | -0,51228 | 0,265216 | -1,15377E-06 | 5,97333E-07 | -2,36756E-05 | -0,00579 | -0,00537 |
| 14,6 | -0,43478 | 0,225093 | -9,79223E-07 | 5,06965E-07 | -2,36882E-05 | -0,0058 | -0,00456 |
| 14,7 | -0,35728 | 0,184969 | -8,04673E-07 | 4,16597E-07 | -2,36988E-05 | -0,0058 | -0,00375 |
| 14,8 | -0,27977 | 0,144846 | -6,30124E-07 | 3,26229E-07 | -2,37073E-05 | -0,0058 | -0,00293 |
| 14,9 | -0,20227 | 0,104722 | -4,55574E-07 | 2,35861E-07 | -2,37137E-05 | -0,0058 | -0,00212 |
| 15 | -0,12478 | 0,064599 | -2,81025E-07 | 1,45493E-07 | -2,3718E-05 | -0,0058 | -0,00131 |
| 15,1 | -0,04728 | 0,024475 | -1,06475E-07 | 5,51246E-08 | -2,37203E-05 | -0,00581 | -0,0005 |
| 15,161 | 0 | 0 | 0 | 0 | -2,37207E-05 | -0,00581 | 0 |