CZĘŚĆ Z PARABOLĄ
OBLICZENIA DLA STANU 2
RYS
$\mathbf{p}_{\mathbf{2}}\left( \mathbf{x} \right)\mathbf{=}\frac{\mathbf{\text{Pα}}}{\mathbf{l}^{\mathbf{2}}}\left( \mathbf{l - x} \right)\mathbf{x}$
Dla x=0,0cm
$p_{2}\left( x \right) = \frac{14\frac{\text{kN}}{cm^{2}}*0,5556}{{(180cm)}^{2}}\left( 180cm - 0cm \right)0cm$
$p_{2}\left( x \right) = \frac{7,7784\frac{\text{kN}}{cm^{2}}}{32400cm^{2}}*180cm*0cm$
$p_{2}\left( x \right) = 0,00024\frac{\text{kN}}{cm^{4}}*0cm^{2}$
$\mathbf{p}_{\mathbf{2}}\left( \mathbf{x} \right)\mathbf{= 0,0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla x=45,0cm
$p_{2}\left( x \right) = \frac{14\frac{\text{kN}}{cm^{2}}*0,5556}{{(180cm)}^{2}}\left( 180cm - 45cm \right)45cm$
$p_{2}\left( x \right) = \frac{7,7784\frac{\text{kN}}{cm^{2}}}{32400cm^{2}}*135cm*45cm$
$p_{2}\left( x \right) = 0,00024\frac{\text{kN}}{cm^{4}}*6075cm^{2}$
$\mathbf{p}_{\mathbf{2}}\left( \mathbf{x} \right)\mathbf{= 1,458}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla x=90cm
$p_{2}\left( x \right) = \frac{14\frac{\text{kN}}{cm^{2}}*0,5556}{{(180cm)}^{2}}\left( 180cm - 90cm \right)90cm$
$p_{2}\left( x \right) = \frac{7,7784\frac{\text{kN}}{cm^{2}}}{32400cm^{2}}*90cm*90cm$
$p_{2}\left( x \right) = 0,00024\frac{\text{kN}}{cm^{4}}*8100cm^{2}$
$\mathbf{p}_{\mathbf{2}}\left( \mathbf{x} \right)\mathbf{= 1,944}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Funkcja Airy’ego:
$$\mathbf{\varphi}^{\left( \mathbf{2} \right)}\left( \mathbf{x,y} \right)\mathbf{= -}\sum_{\mathbf{m = 1}}^{\mathbf{\infty}}{\left( \frac{\mathbf{l}}{\mathbf{m*\pi}} \right)^{\mathbf{2}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}}$$
$$\mathbf{z =}\frac{\mathbf{m*}\pi\mathbf{*y}}{l}$$
Gdzie
ψm(z)=bm*sinh(z)+cm*z * cosh(z)+dm*z * cosh(z)
W tej funkcji, współczynniki bm, cm, dm nie zależą od sposobu obciążenia:
$$\mathbf{b}_{\mathbf{m}}\mathbf{=}\frac{\mathbf{m*\beta*cosh*(m*\beta) + sinh(m*\beta)}}{\mathbf{(sinh\ m*\beta)}^{\mathbf{2}}\mathbf{-}\mathbf{(m*\beta)}^{\mathbf{2}}}$$
$$\mathbf{c}_{\mathbf{m}}\mathbf{= -}\frac{\mathbf{m*\beta*cosh(m*\beta) + sinh*(m*\beta)}}{\mathbf{(sinh\ m*\beta)}^{\mathbf{2}}\mathbf{-}\mathbf{(m*\beta)}^{\mathbf{2}}}$$
$$\mathbf{d}_{\mathbf{m}}\mathbf{=}\frac{\mathbf{m*\beta*s}\mathbf{inh(m*\beta)}}{\mathbf{(sinh\ m*\beta)}^{\mathbf{2}}\mathbf{-}\mathbf{(m*\beta)}^{\mathbf{2}}}$$
Gdzie:
$\mathbf{\beta =}\frac{\mathbf{\pi*h}}{\mathbf{l}}$
$\beta = \frac{\pi*h}{l} = \frac{3,14159*40cm}{180cm} = 0,698132$
Zatem:
$\mathbf{b}_{\mathbf{m}}\mathbf{=}\frac{\mathbf{m*\beta*cosh(m*\beta) + sinh(m*\beta)}}{\mathbf{(sinh\ (m*\beta))}^{\mathbf{2}}\mathbf{-}\mathbf{(m*\beta)}^{\mathbf{2}}}$
Dla m=1
$b_{m} = \frac{1*0,698132*cosh(1*0,698132) + sinh(1*0,698132)}{{(sinh\ (1*0,698132))}^{2} - {(1*0,698132)}^{2}}$
$b_{m} = \frac{0,698132*1,253754 + 0,756240}{0,571899 - 0,487390}$
$b_{m} = \frac{1,631526}{0,084511}$
bm=19, 305565 ok
Dla m=3
$b_{m} = \frac{3*0,698132*cosh(3*0,698132) + sinh(3*0,698132)}{{(sinh\ (3*0,698132))}^{2} - {(3*0,698132)}^{2}}$
$b_{m} = \frac{2,094396*4,121840 + 3,998695}{15,989562 - 4,386495}$
$b_{m} = \frac{12,631460}{11,603067}$
bm=1, 088631
Dla m=5
$b_{m} = \frac{5*0,698132*cosh(5*0,698132) + sinh(5*0,698132)}{{(sinh\ (5*0,698132))}^{2} - {(5*0,698132)}^{2}}$
$b_{m} = \frac{3,490659*16,419013 + 16,388532}{268,583974 - 12,184697}$
$b_{m} = \frac{73,701698}{256,399277}$
bm=0, 287449
$\mathbf{c}_{\mathbf{m}}\mathbf{= -}\frac{\mathbf{m*\beta*cosm*\beta + sinh*(m*\beta)}}{\mathbf{(sinh\ m*\beta)}^{\mathbf{2}}\mathbf{-}\mathbf{(m*\beta)}^{\mathbf{2}}}$
Ponieważ, cm=-bm, to:
Dla m=1
cm = −bm= − 19, 305565
Dla m=3
cm = −bm= − 1, 088631
Dla m=5
cm = −bm= − 0, 287449
$\mathbf{d}_{\mathbf{m}}\mathbf{=}\frac{\mathbf{m*\beta*sinh(m*\beta)}}{\mathbf{(sinh\ m*\beta)}^{\mathbf{2}}\mathbf{-}\mathbf{(m*\beta)}^{\mathbf{2}}}$
Dla m=1
$d_{m} = \frac{1*0,698132*sinh(1*0,698132)}{{(sinh\ (1*0,698132))}^{2} - {(1*0,698132)}^{2}}$
$d_{m} = \frac{0,698132*0,756240}{0,5718990 - 0,487388}$
$d_{m} = \frac{0,527955}{0,084511}$
dm=6, 247205
Dla m=3
$d_{m} = \frac{3*0,698132*sinh(3*0,698132)}{{(sinh\ (3*0,698132))}^{2} - {(3*0,698132)}^{2}}$
$d_{m} = \frac{2,094395*3,998691}{15,989532 - 4,386491}$
$d_{m} = \frac{8,374839}{11,603042}$
dm=0, 721779 ok
Dla m=5
$d_{m} = \frac{5*0,698132*sinh(5*0,698132)}{{(sinh\ (5*0,698132))}^{2} - {(5*0,698132)}^{2}}$
$d_{m} = \frac{3,490659*16,388532}{268,583974 - 12,184697}$
$d_{m} = \frac{57,206768}{256,399277}$
dm=0, 223116 ok
Cm znajdujemy rozwijając funkcję p2(x) w szereg Fourier’a, zatem:
$$\mathbf{C}_{\mathbf{m}}\mathbf{=}\frac{\mathbf{2}}{\mathbf{l}}\mathbf{*}\int_{\mathbf{0}}^{\mathbf{l}}\mathbf{p}_{\mathbf{2(x)}}\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}\mathbf{\text{dx}}$$
Otrzymujemy:
$$\mathbf{C}_{\mathbf{m}}\mathbf{= - 8*P*\alpha}{\mathbf{(}\frac{\mathbf{1}}{\mathbf{m*\pi}}\mathbf{)}}^{\mathbf{3}}$$
Dla
m = 1, 3, 5…
Zatem
Dla m=1
$C_{m} = - 8*14\frac{\text{kN}}{cm^{2}}*0,5556*{(\frac{1}{1*3,141593})}^{3}$
$C_{m} = - 112\frac{\text{kN}}{cm^{2}}*0,5556*{(0,318309)}^{3}$
$C_{m} = - 62,2272\frac{\text{kN}}{cm^{2}}*0,032252$
$\mathbf{C}_{\mathbf{m}}\mathbf{= - 2,006923}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=3
$C_{m} = - 8*14\frac{\text{kN}}{cm^{2}}*0,5556*{(\frac{1}{3*3,141593})}^{3}$
$C_{m} = - 112\frac{\text{kN}}{cm^{2}}*0,5556*{(0,106103)}^{3}$
$C_{m} = - 62,2272\frac{\text{kN}}{cm^{2}}*0,001195$
$\mathbf{C}_{\mathbf{m}}\mathbf{= - 0,074330}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=5
$C_{m} = - 8*14\frac{\text{kN}}{cm^{2}}*0,5556*{(\frac{1}{5*3,141593})}^{3}$
$C_{m} = - 112\frac{\text{kN}}{cm^{2}}*0,5556*{(0,063662)}^{3}$
$C_{m} = - 62,2272\frac{\text{kN}}{cm^{2}}*0,000258$
$\mathbf{C}_{\mathbf{m}}\mathbf{= - 0,016055}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Wzory na naprężenia:
$${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= -}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{''}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$$
Punkt 1 x=0,0cm; y=0,0cm
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*0cm}{180cm}$
z = 0
ψm″(z)=2*dm*cosh(z)+cm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm″(z) = 2 * 6, 247205 * cosh(0) − 19, 305565 * sinh(0) − 19, 305565 * 0 * cosh(0) + 6, 247205 * 0 * sinh(0)
ψm″(z) = 12, 494410 * 1 − 19, 305565 * 0 − 19, 305565 * 0 * 1 + 6, 247205 * 0 * 0
ψm″(z) = 12, 494410 * 1 − 0 − 0 + 0
ψm″(z)=12, 494345 ok
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = - 2,006764\frac{\text{kN}}{cm^{2}}*12,494345*sin\frac{1*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*0cm}{180cm}$
z = 0 ok.
ψm″(z)=2*dm*cosh(z)+cm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm″(z) = 2 * 0, 721779 * cosh(0)−1, 088631 * sinh(0)−1, 088631 * 0 * cosh(0) + 0, 721779 * 0 * sinh(0)
ψm″(z) = 1, 443558 * 1−1, 088631 * 0−1, 088631 * 0 * 1 + 0, 721779 * 0 * 0
ψm″(z) = 1, 443558−0−0 + 0
ψm″(z)=1, 443558 ok
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}*\mathbf{1,443558}*sin\frac{3*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*0cm}{180cm}$
z = 0 ok.
ψm″(z)=2*dm*cosh(z)+cm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm″(z) = 2 * 0, 223116 * cosh(0) − 0, 287449 * sinh(0) − 0, 287449 * 0 * cosh(0) + 0, 223116 * 0 * sinh(0)
ψm″(z) = 0, 446232 * 1 − 0, 287449 * 0 − 0, 287449 * 0 * 1 + 0, 223116 * 0 * 0
ψm″(z) = 0, 446232 − 0 − 0 + 0
ψm″(z)=0, 446232 ok
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*s}\mathbf{\text{in}}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*0,446232*sin\frac{5*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 1 (x=0,0cm; y=0,0cm)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= -}\sum_{\mathbf{m = 1,3,5}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{''}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \ - (0\frac{\text{kN}}{cm^{3}} + \ \ 0\frac{\text{kN}}{cm^{3}} + \ \ \ 0\frac{\text{kN}}{cm^{3}}$)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= \ \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok
Punkt 2 (x=45,0cm; y=0,0cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*0cm}{180cm}$
z = 0
ψm″(z)=jak dla punktu 1 dla m = 1
ψm″(z)=12, 494410 ok
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = - 2,006764\frac{\text{kN}}{cm^{2}}*12,494410*sin\frac{1*3,141593*45cm}{180cm}$
${\sigma_{x}}^{(2)} = - 25,073332\frac{\text{kN}}{cm^{2}}*0,707107$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= - 17,729529}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*0cm}{180cm}$
z = 0 ok.
ψm″(z)=jak dla Punktu 1 dla m = 3
ψm″(z)=1, 443558 ok
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}*\mathbf{1,443558}*sin\frac{3*3,141593*45cm}{180cm}$
${\sigma_{x}}^{(2)} = - 0,107292\frac{\text{kN}}{cm^{2}}*0,707107$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= - 0,075867}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*0cm}{180cm}$
z = 0 ok.
ψm″(z)=jak dla Punktu 1 dla m = 5
ψm″(z)=0, 446232 ok
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*0,446232*sin\frac{5*3,141593*45cm}{180cm}$
${\sigma_{x}}^{(2)} = - 0,0071638808\frac{\text{kN}}{cm^{2}}*( - 0,707107)$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 0,005066}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 2 (x=45,0cm; y=0,0cm)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= -}\sum_{\mathbf{m = 1,3,5}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{''}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \ - (\mathbf{- 17,729529}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,075867}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} + \ \ \ \mathbf{0,005066}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= \ 17,800330}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok
Punktu nr 3 (x=90cm; y=0cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*0cm}{180cm}$
z = 0
ψm″(z)=jak dla Punktu 1 dla m = 1
ψm″(z)=12, 494410 ok
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = - 2,006764\frac{\text{kN}}{cm^{2}}*12,494410*sin\frac{1*3,141593*90cm}{180cm}$
${\sigma_{x}}^{(2)} = - 25,073332\frac{\text{kN}}{cm^{2}}*1$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= -}25,073332\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*0cm}{180cm}$
z = 0 ok.
ψm″(z)=jak dla Punktu 1 dla m = 3
ψm″(z)=1, 443558 ok
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}*\mathbf{1,443558}*sin\frac{3*3,141593*90cm}{180cm}$
${\sigma_{x}}^{(2)} = - 0,107292\frac{\text{kN}}{cm^{2}}*( - 1)$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}0,107292\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*0cm}{180cm}$
z = 0 ok.
ψm″(z)=jak dla Punktu 1 dla m = 5
ψm″(z)=0, 446232 ok
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*0,446232*sin\frac{5*3,141593*90cm}{180cm}$
${\sigma_{x}}^{(2)} = - 0,007164\frac{\text{kN}}{cm^{2}}*1$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}} = - 0,007164\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 3 (x=90,0cm; y=0,0cm)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= -}\sum_{\mathbf{m = 1,3,5}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{''}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \ - (\mathbf{-}25,073332\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+}0,107292\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} - 0,007164\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= \ 24,973204}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Punktu nr 4 (x=0cm; y=20cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,141593*20cm}{180cm}$
z = 0, 349066
ψm″(z)=2*dm*cosh(z)+cm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm″(z) = 2 * 6, 247205 * cosh(0, 349066) − 19, 305565 * sinh(0, 349066) − 19, 305565 * 0, 349066 * cosh(0, 349066) + 6, 247205 * 0, 349066 * sinh(0, 349066)
ψm″(z) = 12, 494410 * 1, 061545 − 19, 305565 * 0, 356198 − 19, 305565 * 0, 349066 * 1, 061545 + 6, 247205 * 0, 349066 * 0, 356198
ψm″(z) = 13, 263378 − 6, 876604 − 7, 153663 + 0, 776756
ψm″(z)=0, 009868 ok
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = - 2,006764\frac{\text{kN}}{cm^{2}}*\mathbf{0,009868}\mathbf{\ }*sin\frac{1*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*20cm}{180cm}$
z = 1, 0471980
ψm″(z)=2*dm*cosh(z)+cm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm″(z) = 2 * 0, 721779 * cosh(1, 0471980 )−1, 088631 * sinh(1, 0471980 )−1, 088631 * 1, 0471980 * cosh(1, 0471980 ) + 0, 721779 * 1, 0471980 * sinh(1, 0471980 )
ψm″(z) = 1, 443558 * 1, 6002870−1, 088631 * 1, 249368−1, 088631 * 1, 0471980 * 1, 6002870 + 0, 721779 * 1, 0471980 * 1, 249368
ψm″(z) = 2, 310107−1, 360101−1, 824347 + 0, 944329
ψm″(z)=0, 069989 ok
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}*\mathbf{0,069989}*sin\frac{3*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*20cm}{180cm}$
z = 1, 745329 ok.
ψm″(z)=2*dm*cosh(z)+cm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm″(z) = 2 * 0, 223116 * cosh(1,745329 ) − 0, 287449 * sinh(1,745329 ) − 0, 287449 * 1, 745329 * cosh(1,745329 ) + 0, 223116 * 1, 745329 * sinh(1,745329 )
ψm″(z) = 0, 446232 * 2, 951187 − 0, 287449 * 2, 776599 − 0, 287449 * 1, 745329 * 2, 951187 + 0, 223116 * 1, 745329 * 2, 776599
ψm″(z) = 1, 316914 − 0, 798130 − 1, 480590 + 1, 081237
ψm″(z)=0, 119431 ok
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*\mathbf{0,119431\ \ }*sin\frac{5*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 4 (x=0,0cm; y=0,0cm)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= -}\sum_{\mathbf{m = 1,3,5}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{''}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \ - (0\frac{\text{kN}}{cm^{3}} + \ \ 0\frac{\text{kN}}{cm^{3}} + \ \ \ 0\frac{\text{kN}}{cm^{3}}$)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= \ \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok.
Punktu nr 5 (x=45cm; y=20cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*20cm}{180cm}$
z = 0, 349066
ψm″(z)=jak dla Punktu 4 dla m = 1
ψm″(z)=0, 009868
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = - 2,006764\frac{\text{kN}}{cm^{2}}*\mathbf{0,009868}*sin\frac{1*3,141593*90cm}{180cm}$
${\sigma_{x}}^{(2)} = - 0,019803\frac{\text{kN}}{cm^{2}}*0,707107$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= - 0,014003}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*20cm}{180cm}$
z = 1, 047198
ψm″(z)=jak dla Punktu 4 dla m = 3
ψm″(z)=0, 069989 ok
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}*\mathbf{0,069989}\mathbf{\ }*sin\frac{3*3,141593*45cm}{180cm}$
${\sigma_{x}}^{(2)} = - 0,005201932\frac{\text{kN}}{cm^{2}}*0,707107$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= - 0,003678}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*20cm}{180cm}$
z = 1, 745329 ok.
ψm″(z)=jak dla Punktu 4 dla m = 5
ψm″(z)=0, 119431 ok
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*\mathbf{0,119431}*sin\frac{5*3,141593*45cm}{180cm}$
${\sigma_{x}}^{(2)} = - 0,01917345808\frac{\text{kN}}{cm^{2}}*( - 0,707107)$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 0,001356}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 5 (x=45,0cm; y=20,0cm)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= -}\sum_{\mathbf{m = 1,3,5}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{''}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \ - (\mathbf{- 0,014003}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{= - 0,003678}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} + \ \ \ \mathbf{0,001356}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= \ 0,01}\mathbf{6325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok
Punktu nr 6 (x=90cm; y=20cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*20cm}{180cm}$
z = 0, 349066
ψm″(z)=jak dla Punktu4 dla m = 1
ψm″(z)=0, 009868 ok
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = - 2,006764\frac{\text{kN}}{cm^{2}}*0,009868*sin\frac{1*3,141593*90cm}{180cm}$
${\sigma_{x}}^{(2)} = - 0,019802747\frac{\text{kN}}{cm^{2}}*1$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=} - 0,019803\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*20cm}{180cm}$
z = 1, 047198
ψm″(z)=jak dla Punktu 4 dla m = 3
ψm″(z)=0, 069989ok
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}*\mathbf{0,069989}*sin\frac{3*3,141593*90cm}{180cm}$
${\sigma_{x}}^{(2)} = - 0,005202\frac{\text{kN}}{cm^{2}}*( - 1)$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}0,005202\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*20cm}{180cm}$
z = 1, 745329 ok.
ψm″(z)=jak dla Punktu 4 dla m = 5
ψm″(z)=0, 119431 ok
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*\mathbf{0,119431}*sin\frac{5*3,141593*90cm}{180cm}$
${\sigma_{x}}^{(2)} = - 0,001917\frac{\text{kN}}{cm^{2}}*1$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}} = - 0,001917\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 6 (x=90,0cm; y=20,0cm)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= -}\sum_{\mathbf{m = 1,3,5}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{''}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \ - ( - 0,019803\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+}0,005202\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} - 0,001917\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= \ 0,016518}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Punktu nr 7 (x=0cm; y=40cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,141593*40cm}{180cm}$
z = 0, 698132
ψm″(z)=2*dm*cosh(z)+cm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm″(z) = 2 * 6, 247205 * cosh(0, 698132) − 19, 305565 * sinh(0, 698132) − 19, 305565 * 0, 698132 * cosh(0, 698132) + 6, 247205 * 0, 698132 * sinh(0, 698132)
ψm″(z) = 12, 494410 * 1, 253754 − 19, 305565 * 0, 756240 − 19, 305565 * 0, 698132 * 1, 253754 + 6, 247205 * 0, 698132 * 0, 756240
ψm″(z) = 15, 664917 − 14, 599640 − 16, 897887 + 3, 298245
ψm″(z)= − 12, 534365 ok
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = - 2,006764\frac{\text{kN}}{cm^{2}}*\mathbf{= - 12,534365}\mathbf{\text{\ \ }}*sin\frac{1*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*40cm}{180cm}$
z = 2, 094395
ψm″(z)=2*dm*cosh(z)+cm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm″(z) = 2 * 0, 721779 * cosh(2, 094395 )−1, 088631 * sinh(2, 094395 )−1, 088631 * 2, 094395 * cosh(2, 094395 ) + 0, 721779 * 2, 094395 * sinh(2, 094395 )
ψm″(z) = 1, 443558 * 4, 121836−1, 088631 * 3, 998691−1, 088631 * 2, 094395 * 4, 121836 + 0, 721779 * 2, 094395 * 3, 998691
ψm″(z) = 5, 950109 − 4, 353010 − 9, 397882 + 6, 044783
ψm″(z)= − 1, 756089 ok
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}*\mathbf{- 1,756089}*sin\frac{3*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*40cm}{180cm}$
z = 3, 490659 ok.
ψm″(z)=2*dm*cosh(z)+cm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm″(z) = 2 * 0, 223116 * cosh(3,490659 ) − 0, 287449 * sinh(3,490659) − 0, 287449 * 3, 490659 * cosh(3,490659) + 0, 223116 * 3, 490659 * sinh(3,490659 )
ψm″(z) = 0, 446232 * 16, 41902 − 0, 287449 * 16, 388540 − 0, 287449 * 3, 490659 * 16, 41902 + 0, 223116 * 3, 490659 * 16, 388540
ψm″(z) = 7, 326692 − 4, 710869 − 16, 474622 + 12, 763753
ψm″(z)= − 1, 095046 ok
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*\mathbf{- 1,095046}*sin\frac{5*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 7 (x=0,0cm; y=40,0cm)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= -}\sum_{\mathbf{m = 1,3,5}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{''}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \ - (0\frac{\text{kN}}{cm^{3}} + \ \ 0\frac{\text{kN}}{cm^{3}} + \ \ \ 0\frac{\text{kN}}{cm^{3}}$)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= \ \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok.
Punktu nr 8 (x=45cm; y=40cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,141593*40cm}{180cm}$
z = 0, 698132
ψm″(z)=jak dla Punktu 7 dla m = 1
ψm″(z)= − 12, 534365 ok
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = - 2,006764\frac{\text{kN}}{cm^{2}}*\mathbf{- 12,534365}*sin\frac{1*3,141593*45cm}{180cm}$
${\sigma_{x}}^{(2)} = 25,153512\frac{\text{kN}}{cm^{2}}*0,707107$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 17,786225}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*40cm}{180cm}$
z = 2, 094395
ψm″(z)=jak dla Punktu 7 dla m = 3
ψm″(z)= − 1, 756089 ok
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}*\mathbf{- 1,756089}\mathbf{\text{\ \ \ }}*sin\frac{3*3,141593*45cm}{180cm}$
${\sigma_{x}}^{(2)} = 1,756089\frac{\text{kN}}{cm^{2}}*0,707107$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= 0,092293}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*40cm}{180cm}$
z = 3, 490659 ok.
ψm″(z)=jak dla Punktu 4 dla m = 5
ψm″(z)= − 1, 095046 ok
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*\mathbf{- 1,095046}*sin\frac{5*3,141593*45cm}{180cm}$
${\sigma_{x}}^{(2)} = 0,017580\frac{\text{kN}}{cm^{2}}*( - 0,707107)$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= - 0,12431}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 8 (x=45,0cm; y=40,0cm)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= -}\sum_{\mathbf{m = 1,3,5}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{''}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \ - (\mathbf{17,786225}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,092293}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,12431}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= \ - 17,866087}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok
Punktu nr 9 (x=90cm; y=40cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,141593*40cm}{180cm}$
z = 0, 698132
ψm″(z)=jak dla Punktu 7 dla m = 1
ψm″(z)= − 12, 534365 ok
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = - 2,006764\frac{\text{kN}}{cm^{2}}*\mathbf{- 12,534365}*sin\frac{1*3,141593*90cm}{180cm}$
${\sigma_{x}}^{(2)} = 25,153512\frac{\text{kN}}{cm^{2}}*1$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}25,153512\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*40cm}{180cm}$
z = 2, 094395
ψm″(z)=jak dla Punktu 7 dla m = 3
ψm″(z)= − 1, 756089 ok
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}*\mathbf{( - 1,756089}\mathbf{\ \ )}*sin\frac{3*3,141593*90cm}{180cm}$
${\sigma_{x}}^{(2)} = 0,130521\frac{\text{kN}}{cm^{2}}*( - 1)$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=} - 0,130521\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*40cm}{180cm}$
z = 3, 490659 ok.
ψm″(z)=jak dla Punktu 4 dla m = 5
ψm″(z)= − 1, 095046 ok
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{"}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*\mathbf{- 1,095046\ \ }*sin\frac{5*3,141593*90cm}{180cm}$
${\sigma_{x}}^{(2)} = 0,017580\frac{\text{kN}}{cm^{2}}*1$
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}} = 0,017580\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 9 (x=90,0cm; y=40,0cm)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= -}\sum_{\mathbf{m = 1,3,5}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{''}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\sigma_{x}}^{(2)} = \ - (25,153512\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} - 0,130521\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} + 0,017580\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$)
${\mathbf{\sigma}_{\mathbf{x}}}^{\mathbf{(2)}}\mathbf{= - 25,040571}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
$${\mathbf{\sigma}_{\mathbf{\text{yy}}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$$
Punktu nr 1 (x=0cm; y=0cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*0cm}{180cm}$
z = 0
ψm(z)=bm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm(z)=19, 305565 * sinh(0)−19, 305565 * 0 * cosh(0)+6, 247205 * 0 * sin(0)
ψm(z)=19, 305565 * 0 − 19, 305565 * 0 * 1 + 6, 247205 * 0 * 0
ψm(z)=0
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = - 2,006764\frac{\text{kN}}{cm^{2}}*0*sin\frac{1*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*0cm}{180cm}$
z = 0 ok.
ψm(z)=bm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm(z)=0, 088631 * sinh(0)−0, 088631 * 0 * cosh(0)+0, 721779 * 0 * sinh(0)
ψm(z)=0, 088631 * 0 − 0, 088631 * 0 * 1 + 0, 721779 * 0 * 0
ψm(z)=0
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}*\mathbf{0}*sin\frac{3*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*0cm}{180cm}$
z = 0 ok.
ψm(z)=bm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm(z)=0, 287449 * sinh(0)−0, 287449 * 0 * cosh(0)+0, 223116 * 0 * sinh(0)
ψm(z)=0, 287449 * 0 − 0, 287449 * 0 * 1 + 0, 223116 * 0 * 0
ψm(z)=0
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*0*sin\frac{5*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 1 (x=0,0cm; y=0,0cm)
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \ 0\frac{\text{kN}}{cm^{3}} + \ \ 0\frac{\text{kN}}{cm^{3}} + \ \ \ 0\frac{\text{kN}}{cm^{3}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= \ \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok
Punktu nr 2 (x=45cm; y=0cm)
Punkt 2 (x=45,0cm; y=0,0cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*0cm}{180cm}$
z = 0
ψm(z)=jak dla punktu 1 dla m = 1
ψm(z)=0
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = - 2,006764\frac{\text{kN}}{cm^{2}}*0*sin\frac{1*3,141593*45cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*0cm}{180cm}$
z = 0 ok.
ψm(z)=jak dla punktu 1 dla m = 3
ψm(z)=0
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,074325}\frac{\text{kN}}{cm^{2}}*0*sin\frac{3*3,141593*45cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*0cm}{180cm}$
z = 0 ok.
ψm(z)=jak dla punktu 1 dla m = 5
ψm(z)=0
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*0*sin\frac{5*3,141593*45cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 2 (x=45,0cm; y=0,0cm)
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \ 0\frac{\text{kN}}{cm^{3}} + \ \ 0\frac{\text{kN}}{cm^{3}} + \ \ \ 0\frac{\text{kN}}{cm^{3}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= \ \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok.
Punktu nr 3 (x=90cm; y=0cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*0cm}{180cm}$
z = 0
ψm(z)=jak dla punktu 1 dla m = 1
ψm(z)=0
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{-}2,006764\frac{\text{kN}}{cm^{2}}*0*sin\frac{1*3,141593*90cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*0cm}{180cm}$
z = 0 ok.
ψm(z)=jak dla punktu 1 dla m = 3
ψm(z)=0
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,074325}\frac{\text{kN}}{cm^{2}}*0*sin\frac{3*3,141593*90cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*0cm}{180cm}$
z = 0 ok.
ψm(z)=jak dla punktu 1 dla m = 1
ψm(z)=0
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*0*sin\frac{5*3,141593*90cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 3 (x=90,0cm; y=0,0cm)
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \ 0\frac{\text{kN}}{cm^{3}} + \ \ 0\frac{\text{kN}}{cm^{3}} + \ \ \ 0\frac{\text{kN}}{cm^{3}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= \ \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok.
Punktu nr 4 (x=0cm; y=20cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,141593*20cm}{180cm}$
z = 0, 349066
ψm(z)=bm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm(z)=19, 305565 * sinh(0, 349066)−19, 305565 * 0, 349066 * cosh(0, 349066)+6, 247205 * 0, 349066 * sinh(0, 349066)
ψm(z)=19, 305565 * 0, 356198 − 19, 305565 * 0, 349066 * 1, 061545 + 6, 247205 * 0, 349066 * 0, 356198
ψm(z)=6, 876604 − 7, 153663 + 0, 776756
ψm(z)=0, 499697
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*si}\mathbf{n}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 2,006764}\frac{\text{kN}}{cm^{2}}*\mathbf{0,499697}\mathbf{\ }*sin\frac{1*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*20cm}{180cm}$
z = 1, 0471980
ψm(z)=bm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm(z)=1, 088631 * sinh(1, 0471980)−1, 088631 * 1, 0471980 * cosh(1, 0471980)+0, 721779 * 1, 0471980 * sinh(1, 0471980)
ψm(z)=1, 088631 * 1, 249368 − 1, 088631 * 1, 0471980 * 1, 600287 + 0, 721779 * 1, 0471980 * 1, 249368
ψm(z)=1, 360101 − 1, 824347 + 0, 944329
ψm(z)=0, 480083
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,074325}\frac{\text{kN}}{cm^{2}}*\mathbf{0,499697}\mathbf{\ }*sin\frac{3*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*20cm}{180cm}$
z = 1, 745329 ok.
ψm(z)=bm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm(z)=0, 287449 * sinh(1,745329)−0, 287449*1, 745329*cosh(1,745329)+0, 223116*1, 745329*sinh(1, 745329)
ψm(z)=0, 287449 * 2, 776599 − 0, 287449*1, 745329*2, 951187 + 0, 223116*1, 745329*2, 776599
ψm(z)=0, 798131 − 1, 480590 + 1, 081238
ψm(z)=0, 398778
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*\mathbf{0,398778}\mathbf{\ }*sin\frac{5*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 4 (x=0,0cm; y=20,0cm)
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \ 0\frac{\text{kN}}{cm^{3}} + \ \ 0\frac{\text{kN}}{cm^{3}} + \ \ \ 0\frac{\text{kN}}{cm^{3}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= \ \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok.
Punktu nr 5 (x=45cm; y=20cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*20cm}{180cm}$
z = 0, 349066
ψm(z)=jak dla Punktu 4 dla m = 1
ψm(z)=0, 499697
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 2,006764}\frac{\text{kN}}{cm^{2}}*\mathbf{0,499697}\mathbf{\ }*sin\frac{1*3,141593*45cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = - 1,002774\frac{\text{kN}}{cm^{2}}*\mathbf{0,707107}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 0,709068}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*20cm}{180cm}$
z = 1, 047198
ψm(z)=jak dla Punktu 4 dla m = 3
ψm(z)=0, 480083
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,074325}\frac{\text{kN}}{cm^{2}}*\mathbf{0,480083}\mathbf{\text{\ \ }}*sin\frac{3*3,141593*45cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = - 0,035682\frac{\text{kN}}{cm^{2}}*\mathbf{0,707107}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 0,025231}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*20cm}{180cm}$
z = 1, 745329 ok.
ψm(z)=jak dla Punktu 4 dla m = 3
ψm(z)=0, 398778
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*\mathbf{0,398778}\mathbf{\text{\ \ \ }}*sin\frac{5*3,141593*45cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = - 0,006402\frac{\text{kN}}{cm^{2}}*( - \mathbf{0,707107)}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0,004527}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 5 (x=45,0cm; y=20,0cm)
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \ \mathbf{- 0,709068}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,02523}\mathbf{1}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} + \ \ \mathbf{0,004527}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= \ - 0,729772}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok.
Punktu nr 6 (x=90cm; y=20cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*20cm}{180cm}$
z = 0, 349066
ψm(z)=jak dla Punktu 4 dla m = 1
ψm(z)=0, 499697
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 2,006764}\frac{\text{kN}}{cm^{2}}*\mathbf{0,499697}\mathbf{\ }*sin\frac{1*3,141593*90cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = - 1,002774\frac{\text{kN}}{cm^{2}}*\mathbf{1}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= -}1,002774\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*20cm}{180cm}$
z = 1, 047198
ψm(z)=jak dla Punktu 4 dla m = 3
ψm(z)=0, 480083
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,074325}\frac{\text{kN}}{cm^{2}}*\mathbf{0,480083}\mathbf{\text{\ \ }}*sin\frac{3*3,141593*90cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = - 0,035682\frac{\text{kN}}{cm^{2}}*\mathbf{( - 1)}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}0,035682\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*20cm}{180cm}$
z = 1, 745329 ok.
ψm(z)=jak dla Punktu 4 dla m = 3
ψm(z)=0, 398778
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*\mathbf{0,398778}\mathbf{\text{\ \ \ }}*sin\frac{5*3,141593*90cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = - 0,006402\frac{\text{kN}}{cm^{2}}*1$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= -}0,006402\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 6 (x=90,0cm; y=20,0cm)
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \ \mathbf{-}1,002774\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+}0,035682\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\text{\ \ }\mathbf{-}0,006402\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 0,973494\ }\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok.
Punktu nr 7 (x=0cm; y=40cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,141593*40cm}{180cm}$
z = 0, 698132
ψm(z)=bm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm(z)=19, 305565 * sinh(0, 698132 )−19, 305565 * 0, 698132 *cosh(0, 698132 )+6, 247205 * 0, 698132 *sinh(0, 698132 )
ψm(z)=19, 305565 * 0, 756240 − 19, 305565 * 0, 698132 *1, 253754 + 6, 247205 * 0, 698132 *0, 756240
ψm(z)=14, 599640 − 16, 897887 + 3, 298245
ψm(z)=0, 999999
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 2,006764}\frac{\text{kN}}{cm^{2}}*\mathbf{0,999999}\mathbf{\ }*sin\frac{1*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*40cm}{180cm}$
z = 2, 094395
ψm(z)=bm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm(z)=1, 088631 * sinh(2, 094395 )−1, 0886315 * 2, 094395 *cosh(2, 094395 )+0, 721779 * 2, 094395 *sinh(2, 094395 )
ψm(z)=1, 088631 * 3, 998691 − 1, 0886315 * 2, 094395 *4, 121836 + 0, 721779 * 2, 094395 *3, 998691
ψm(z)=4, 353099 − 9, 397887 + 6, 044783
ψm(z)=0, 999999
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,074325764}\frac{\text{kN}}{cm^{2}}*\mathbf{0,999999}\mathbf{\ }*sin\frac{3*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*40cm}{180cm}$
z = 3, 490659 ok.
ψm(z)=bm*sinh(z)+cm*z * cosh(z)+dm*z * sinh(z)
ψm(z)=0, 287449 * sinh(3,490659)−0, 287449*3, 490659*cosh(3,490659)+0, 223116*3, 490659*sinh(3, 490659)
ψm(z)=0, 287449 * 16, 388540 − 0, 287449*3, 490659*16, 419021 + 0, 223116*3, 490659*16, 388540
ψm(z)=4, 710869 − 16, 474623 + 12, 763753
ψm(z)=0, 999999
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*\mathbf{0,999999}\mathbf{\ }*sin\frac{5*3,141593*0cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Zatem dla Punktu 7 (x=0,0cm; y=40,0cm)
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \ \mathbf{0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+}0\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\ \mathbf{+}0\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0\ }\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok.
Punktu nr 8 (x=45cm; y=40cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,141593*40cm}{180cm}$
z = 0, 698132
ψm(z)=jak dla Punktu 7 dla m = 1
ψm(z)=0, 999999
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 2,006764}\frac{\text{kN}}{cm^{2}}*\mathbf{0,999999}\mathbf{\ }*sin\frac{1*3,141593*45cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 2,006762}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,707107}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 1,418995}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*40cm}{180cm}$
z = 2, 094395
ψm(z)=jak dla Punktu 7 dla m = 1
ψm(z)=0, 999999
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,074325764}\frac{\text{kN}}{cm^{2}}*\mathbf{0,999999}\mathbf{\ }*sin\frac{3*3,141593*45cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 0,074326}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,707107}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 0,052556}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*40cm}{180cm}$
z = 3, 490659 ok.
ψm(z)=jak dla Punktu 7 dla m = 1
ψm(z)=0, 999999
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*\mathbf{0,99999}\mathbf{9}\mathbf{\ }*sin\frac{5*3,141593*45cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*( - 0,707107)}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0,011352}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 8 (x=45,0cm; y=40,0cm)
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \ \mathbf{- 1,418995}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,052556}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\ \mathbf{+ 0,011352}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 1,460199\ }\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok.
Punktu nr 9 (x=90cm; y=40cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,141593*40cm}{180cm}$
z = 0, 698132
ψm(z)=jak dla Punktu 7 dla m = 1
ψm(z)=0, 999999
Zatem dla m=1
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 2,006764}\frac{\text{kN}}{cm^{2}}*\mathbf{0,999999}\mathbf{\ }*sin\frac{1*3,141593*90cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 2,006762}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*1}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 2,006762}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*40cm}{180cm}$
z = 2, 094395
ψm(z)=jak dla Punktu 7 dla m = 1
ψm(z)=0, 999999
Zatem dla m=3
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,074325764}\frac{\text{kN}}{cm^{2}}*\mathbf{0,999999}\mathbf{\ }*sin\frac{3*3,141593*90cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*( - 1)}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*40cm}{180cm}$
z = 3, 490659 ok.
ψm(z)=jak dla Punktu 7 dla m = 1
ψm(z)=0, 999999
Zatem dla m=5
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \mathbf{- 0,016054}\frac{\text{kN}}{cm^{2}}*\mathbf{0,999999}\mathbf{\ }*sin\frac{5*3,141593*90cm}{180cm}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*1}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 9 (x=90,0cm; y=40,0cm)
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}}\left( \mathbf{z} \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}} = \ \mathbf{- 2,006762}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\ \mathbf{+ - 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
${\mathbf{\sigma}_{\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 1,946491\ }\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
$${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi}_{\mathbf{m}}\mathbf{'}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$$
Punktu nr 1 (x=0cm; y=0cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*0cm}{180cm}$
z = 0
ψm′(z)=dm*sinh(z)+dm*z * cosh(z)+cm*z * sinh(z)
ψm′(z)=6, 247205 * sinh(0)+6, 247205 * 0 * cosh(0)−19, 305565 * 0 * sinh(0)
ψm′(z)=6, 247205 * 0 + 6, 247205 * 0 * 1 − 19, 305565 * 0 * 0
ψm′(z)=0
Zatem dla m=1
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*0cm}{180cm}$
z = 0 ok.
ψm′(z)=dm*sinh(z)+dm*z * cosh(z)+cm*z * sinh(z)
ψm′(z)=0, 721779 * sinh(0)+0, 721779 * 0 * cosh(0)−1, 088631 * 0 * sinh(0)
ψm′(z)=0, 721779 * 0 + 0, 721779 * 0 * 1 − 1, 088631 * 0 * 0
ψm′(z)=0
Zatem dla m=3
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{3*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*0cm}{180cm}$
z = 0 ok.
ψm′(z)=dm*sinh(z)+dm*z * cosh(z)+cm*z * sinh(z)
ψm′(z)=0, 223116 * sinh(0)+0, 223116 * 0 * cosh(0)−0, 287449 * 0 * sinh(0)
ψm′(z)=0, 223116 * 0 + 0, 223116 * 0 * 1 − 0, 287449 * 0 * 0
ψm′(z)=0
Zatem dla m=5
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,0}\mathbf{16054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{5*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 1 (x=0,0cm; y=0,0cm)
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}} = \ 0\frac{\text{kN}}{cm^{3}} + \ \ 0\frac{\text{kN}}{cm^{3}} + \ \ \ 0\frac{\text{kN}}{cm^{3}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= \ \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok
Punktu nr 2 (x=45cm; y=0cm)
Punkt 2 (x=45,0cm; y=0,0cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*0cm}{180cm}$
z = 0
ψm′(z)=jak dla punktu 1 dla m = 1
ψm′(z)=0
Zatem dla m=1
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{1*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*0cm}{180cm}$
z = 0 ok.
ψm(z)=jak dla punktu 1 dla m = 3
ψm(z)=0
Zatem dla m=3
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{3*3,14159}\mathbf{3*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*0cm}{180cm}$
z = 0 ok.
ψm(z)=jak dla punktu 1 dla m = 5
ψm(z)=0
Zatem dla m=5
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{5*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 2 (x=45,0cm; y=0,0cm)
${\mathbf{\tau}_{\mathbf{x}\mathbf{y}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}} = \ 0\frac{\text{kN}}{cm^{3}} + \ \ 0\frac{\text{kN}}{cm^{3}} + \ \ \ 0\frac{\text{kN}}{cm^{3}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= \ \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok
Punktu nr 3 (x=90cm; y=0cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*0cm}{180cm}$
z = 0
ψm′(z)=jak dla punktu 1 dla m = 1
ψm′(z)=0
Zatem dla m=1
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{1*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*0cm}{180cm}$
z = 0 ok.
ψm′(z)=jak dla punktu 1 dla m = 3
ψm′(z)=0
Zatem dla m=3
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{x}\mathbf{y}}}^{\mathbf{(2)}}\mathbf{= - 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{3*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*0cm}{180cm}$
z = 0 ok.
ψm(z)=jak dla punktu 1 dla m = 1
ψm(z)=0
Zatem dla m=5
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{5*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 3 (x=90,0cm; y=0,0cm)
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}} = \ 0\frac{\text{kN}}{cm^{3}} + \ \ 0\frac{\text{kN}}{cm^{3}} + \ \ \ 0\frac{\text{kN}}{cm^{3}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= \ \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok
Punktu nr 4 (x=0cm; y=20cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,141593*20cm}{180cm}$
z = 0, 349066
ψm′(z)=dm*sinh(z)+dm*z * cosh(z)+cm*z * sinh(z)
ψm′(z)=6, 247205 * sinh(0, 349066 )+6, 247205 * 0, 349066 * cosh(0, 349066)−19, 305565 * 0, 349066 * sinh(0, 349066)
ψm′(z)=6, 247205 * 0, 356198 + 6, 247205 * 0, 349066 * 1, 061545 − 19, 305565 * 0, 349066 * 0, 356198
ψm′(z)=2, 225341 + 2, 314897 − 2, 400359
ψm′(z)=2, 139751
Zatem dla m=1
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*2,139751*cos}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 4,293975}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*1}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 4,293975}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*20cm}{180cm}$
z = 1, 0471980
ψm′(z)=dm*sinh(z)+dm*z * cosh(z)+cm*z * sinh(z)
ψm′(z)=0, 721779 * sinh(1, 0471980)+0, 721779 * 1, 0471980 * cosh(1, 0471980)−1, 088631 * 1, 0471980 * sinh(1, 0471980)
ψm′(z)=0, 721779 * 1, 249368 + 0, 721779 * 1, 0471980 * 1, 600287 − 1, 088631 * 1, 0471980 * 1, 249368
ψm′(z)=0, 901768 + 1, 209569 − 1, 424296
ψm′(z)=0, 687043
Zatem dla m=3
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,687043}\mathbf{\ }\mathbf{*cos}\frac{\mathbf{3*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\left( \mathbf{2} \right)}\mathbf{= - 0,051064}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*1}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,051064}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*20cm}{180cm}$
z = 1, 745329 ok.
ψm′(z)=dm*sinh(z)+dm*z * cosh(z)+cm*z * sinh(z)
ψm′(z)=0, 223116 * sinh(1,745329 )+0, 223116*1, 745329 *cosh(1,745329 )−0, 287449*1, 745329 *sinh(1, 745329 )
ψm′(z)=0, 223116 * 2, 776599 + 0, 223116*1, 745329 *2, 951187 − 0, 287449*1, 745329 *2, 776599
ψm′(z)=0, 619503 + 1, 149224 − 1, 393000
ψm′(z)=0, 375727
Zatem dla m=5
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,375727*cos}\frac{\mathbf{5*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,006032}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{**1}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,00603}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 4 (x=0,0cm; y=20,0cm)
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}} = \mathbf{- 4,293975}\frac{\text{kN}}{cm^{3}}\mathbf{- 0,051064}\frac{\text{kN}}{cm^{3}}\mathbf{- 0,00603}\frac{\text{kN}}{cm^{3}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= \ - 4,351071}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{3}}}$ ok
Punktu nr 5 (x=45cm; y=20cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*20cm}{180cm}$
z = 0, 349066
ψm(z)=jak dla Punktu 4 dla m = 1
ψm(z)=0, 499697
Zatem dla m=1
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*2,139751*cos}\frac{\mathbf{1*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 4,293975}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,707107}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 3,036300}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*20cm}{180cm}$
z = 1, 047198
ψm(z)=jak dla Punktu 4 dla m = 3
ψm(z)=0, 687043
Zatem dla m=3
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,687043}\mathbf{\ }\mathbf{*cos}\frac{\mathbf{3*3,141}\mathbf{593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\left( \mathbf{2} \right)}\mathbf{= - 0,051064}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*( - 0,707107)}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0,035108}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*20cm}{180cm}$
z = 1, 745329 ok.
ψm(z)=jak dla Punktu 4 dla m = 3
ψm(z)=0, 375727
Zatem dla m=5
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,375727*cos}\frac{\mathbf{5*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,006032}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*( - 0,707107)}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0,004265}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 5 (x=45,0cm; y=20,0cm)
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}} = \mathbf{- 3,036300}\frac{\text{kN}}{cm^{2}}\mathbf{+ 0,036108}\frac{\text{kN}}{cm^{2}}\mathbf{+ 0,004265}\frac{\text{kN}}{cm^{2}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= \ - 2,995927}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Punktu nr 6 (x=90cm; y=20cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,14159265359*20cm}{180cm}$
z = 0, 349066
ψm(z)=jak dla Punktu 4 dla m = 1
ψm(z)=2, 139751
Zatem dla m=1
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*2,139751*cos}\frac{\mathbf{1*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 4,293975}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*20cm}{180cm}$
z = 1, 047198
ψm′(z)=jak dla Punktu 4 dla m = 3
ψm′(z)=0, 687043
Zatem dla m=3
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,687043}\mathbf{\ }\mathbf{*cos}\frac{\mathbf{3*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\left( \mathbf{2} \right)}\mathbf{= - 0,051064}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*20cm}{180cm}$
z = 1, 745329 ok.
ψm(z)=jak dla Punktu 4 dla m = 3
ψm(z)=0, 398778
Zatem dla m=5
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,375727*cos}\frac{\mathbf{5*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,006032}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 6 (x=90,0cm; y=20,0cm)
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}} = \mathbf{0}\frac{\text{kN}}{cm^{2}}\mathbf{+ 0}\frac{\text{kN}}{cm^{2}}\mathbf{+ 0}\frac{\text{kN}}{cm^{2}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Punktu nr 7 (x=0cm; y=40cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,141593*40cm}{180cm}$
z = 0, 698132
ψm′(z)=dm*sinh(z)+dm*z * cosh(z)+cm*z * sinh(z)
ψm′(z)=6, 247205 * sinh(0, 698132 )+6, 247205 * 0, 698132 *cosh(0, 698132 )−19, 305565 * 0, 698132 *sinh(0, 698132 )
ψm′(z)=6, 247205 * 0, 756240 + 6, 247205 * 0, 698132 *1, 253754 − 19, 305565 * 0, 698132 *0, 756240
ψm′(z)=4, 724386 + 5, 468089 − 10, 192476
ψm′(z)=0
Zatem dla m=1
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*40cm}{180cm}$
z = 2, 094395
ψm′(z)=dm*sinh(z)+dm*z * cosh(z)+cm*z * sinh(z)
ψm′(z)=0, 721779 * sinh(2, 094395 )+0, 721779 * 2, 094395 *cosh(2, 094395 )−1, 088631 * 2, 094395 *sinh(2, 094395 )
ψm′(z)=0, 721779 * 3, 998691 + 0, 721779 * 2, 094395 *4, 121836 − 1, 088631 * 2, 094395 *3, 998691
ψm′(z)=2, 886215 + 6, 230939 − 9, 117109
ψm′(z)=0
Zatem dla m=3
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0}\mathbf{\ }\mathbf{*cos}\frac{\mathbf{3*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*40cm}{180cm}$
z = 3, 490659 ok.
ψm′(z)=dm*sinh(z)+dm*z * cosh(z)+cm*z * sinh(z)
ψm′(z)=0, 223116 * sinh(3,490659 )+0, 223116*3, 490659 *cosh(3,490659)−0, 287449*3, 490659 *sinh(3, 490659)
ψm′(z)=0, 223116 * 16, 388540 + 0, 223116*3, 490659 *16, 419021 − 0, 287449*3, 490659*16, 388540
ψm′(z)=3, 656545 + 12, 787494 − 16, 444039
ψm′(z)=0
Zatem dla m=5
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{5*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 7 (x=0,0cm; y=40,0cm)
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}} = \mathbf{0}\frac{\text{kN}}{cm^{2}}\mathbf{+ 0}\frac{\text{kN}}{cm^{2}}\mathbf{+ 0}\frac{\text{kN}}{cm^{2}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Punktu nr 8 (x=45cm; y=40cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,141593*40cm}{180cm}$
z = 0, 698132
ψm(z)=jak dla Punktu 7 dla m = 1
ψm′(z)=0
Zatem dla m=1
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{1*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*40cm}{180cm}$
z = 2, 094395
ψm(z)=jak dla Punktu 7 dla m = 1
ψm′(z)=0
Zatem dla m=3
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0}\mathbf{\ }\mathbf{*cos}\frac{\mathbf{3*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*40cm}{180cm}$
z = 3, 490659 ok.
ψm(z)=jak dla Punktu 7 dla m = 1
ψm′(z)=0
Zatem dla m=5
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{5*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 8 (x=45,0cm; y=40,0cm)
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}} = \mathbf{0}\frac{\text{kN}}{cm^{2}}\mathbf{+ 0}\frac{\text{kN}}{cm^{2}}\mathbf{+ 0}\frac{\text{kN}}{cm^{2}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Punktu nr 9 (x=90cm; y=40cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{1*3,141593*40cm}{180cm}$
z = 0, 698132
ψm(z)=jak dla Punktu 7 dla m = 1
ψm′(z)=0
Zatem dla m=1
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{1*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{3*3,141593*40cm}{180cm}$
z = 2, 094395
ψm(z)=jak dla Punktu 7 dla m = 1
ψm′(z)=0
Zatem dla m=3
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0}\mathbf{\ }\mathbf{*cos}\frac{\mathbf{3*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$z = \frac{5*3,141593*40cm}{180cm}$
z = 3, 490659 ok.
ψm(z)=jak dla Punktu 7 dla m = 1
ψm′(z)=0
Zatem dla m=5
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= - 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0*cos}\frac{\mathbf{5*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 9 (x=90,0cm; y=40,0cm)
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{=}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\mathbf{C}_{\mathbf{m}}\mathbf{*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}} = \mathbf{0}\frac{\text{kN}}{cm^{2}}\mathbf{+ 0}\frac{\text{kN}}{cm^{2}}\mathbf{+ 0}\frac{\text{kN}}{cm^{2}}$
${\mathbf{\tau}_{\mathbf{\text{xy}}}}^{\mathbf{(2)}}\mathbf{= \ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$ ok.
Przemieszczenia
$$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$$
Punktu nr 1 (x=0cm; y=0cm)
Dla m=1
ψ″m(z)=jak dla sigma x dla Punktu 1 dla m = 1
ψ″m(z)=12, 49441
ψm(z)=jak dla sigma y dla Punktu 1 dla m = 1
ψm(z)=0
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{12,49441 + 0,34*0} \right)\mathbf{*cos}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*14,49441*1}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 25,073332}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$$
ψ″m(z)=jak dla sigma x dla Punktu 1 dla m = 3
ψ″m(z)=1, 443558
ψm(z)=jak dla sigma y dla Punktu 1 dla m = 3
ψm(z)=0
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{1,443558 + 0,34*0} \right)\mathbf{*cos}\frac{\mathbf{3*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*1,443558*1}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,035764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
ψ″m(z)=jak dla sigma x dla Punktu 1 dla m = 5
ψ″m(z)=0, 446252
ψm(z)=jak dla sigma y dla Punktu 1 dla m = 5
ψm(z)=0
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*} \right)\mathbf{*}\left( \mathbf{0,446252 + 0,34*0} \right)\mathbf{*cos}\frac{\mathbf{5*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,0032108}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,446252*1}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,001433}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 1 (x=0,0 cm; y=0,0 cm)
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*( - 25,073332}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,035764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,001433}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*( - 25,110529}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
u(2)=0, 191819cm
Punktu nr 2 (x=45cm; y=0cm)
Dla m=1
ψ″m(z)=jak dla sigma x dla Punktu 1 dla m = 1
ψ″m(z)=12, 49441
ψm(z)=jak dla sigma y dla Punktu 1 dla m = 1
ψm(z)=0
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{12,49441 + 0,34*0} \right)\mathbf{*cos}\frac{\mathbf{1*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*12,49441*0,707107}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 17,729529}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
ψ″m(z)=jak dla sigma x dla Punktu 1 dla m = 3
ψ″m(z)=1, 443558
ψm(z)=jak dla sigma y dla Punktu 1 dla m = 3
ψm(z)=0
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{1,443558 + 0,34*0} \right)\mathbf{*cos}\frac{\mathbf{3*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*1,443558*( - 0,707107)}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,025289}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
ψ″m(z)=jak dla sigma x dla Punktu 1 dla m = 5
ψ″m(z)=0, 446252
ψm(z)=jak dla sigma y dla Punktu 1 dla m = 5
ψm(z)=0
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*} \right)\mathbf{*}\left( \mathbf{0,446252 + 0,34*0} \right)\mathbf{*cos}\frac{\mathbf{5*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,0032108}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,446252*( - 0,707107}$ )
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,001013}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 2 (x=45,0 cm; y=0,0 cm)
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*( - 17,729529}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,025289}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,001013}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*( - 17,703227}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
u(2)= − 0, 135235cm
Punktu nr 3 (x=90cm; y=0cm)
Dla m=1
ψ″m(z)=jak dla sigma x dla Punktu 1 dla m = 1
ψ″m(z)=12, 49441
ψm(z)=jak dla sigma y dla Punktu 1 dla m = 1
ψm(z)=0
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{12,49441 + 0,34*0} \right)\mathbf{*cos}\frac{\mathbf{1*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*12,49441*0}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
ψ″m(z)=jak dla sigma x dla Punktu 1 dla m = 3
ψ″m(z)=1, 443558
ψm(z)=jak dla sigma y dla Punktu 1 dla m = 3
ψm(z)=0
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{1,443558 + 0,34*0} \right)\mathbf{*cos}\frac{\mathbf{3*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*1,443558*0}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
ψ″m(z)=jak dla sigma x dla Punktu 1 dla m = 5
ψ″m(z)=0, 446252
ψm(z)=jak dla sigma y dla Punktu 1 dla m = 5
ψm(z)=0
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*} \right)\mathbf{*}\left( \mathbf{0,446252 + 0,34*0} \right)\mathbf{*cos}\frac{\mathbf{5*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,0032108}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,446252*0}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 3 (x=90,0 cm; y=0,0 cm)
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,1415}\mathbf{93}}\mathbf{*(0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
u(2)=0cm
Punktu nr 4 (x=0cm; y=20cm)
Dla m=1
ψ″m(z)=jak dla sigma x dla Punktu 4 dla m = 1
ψ″m(z)=0, 009868
ψm(z)=jak dla sigma y dla Punktu 4 dla m = 1
ψm(z)=0, 499697
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{0,009868 + 0,34*0,499697} \right)\mathbf{*cos}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,179765*1}$
$\mathbf{u}^{\left( \mathbf{2} \right)}\mathbf{= - 0,360746}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
ψ″m(z)=jak dla sigma x dla Punktu 4 dla m = 3
ψ″m(z)=0, 069989
ψm(z)=jak dla sigma y dla Punktu 4 dla m = 3
ψm(z)=0, 480083
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{0,069989 + 0,34*0,480083}\mathbf{\ } \right)\mathbf{*cos}\frac{\mathbf{3*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,233217*1}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,005778}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
ψ″m(z)=jak dla sigma x dla Punktu 4 dla m = 5
ψ″m(z)=0, 119431
ψm(z)=jak dla sigma y dla Punktu 4 dla m = 5
ψm(z)=0, 398778
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*} \right)\mathbf{*}\left( \mathbf{0,119431}\mathbf{\ }\mathbf{+ 0,34*0,398778} \right)\mathbf{*cos}\frac{\mathbf{5*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,003211}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,255016*1}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,000819}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 4 (x=0,0 cm; y=20,0 cm)
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*( - 0,360746}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,005778}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,000819}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*( - 0,367343}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
u(2)= − 0, 002806cm
Punktu nr 5 (x=45cm; y=20cm)
Dla m=1
ψ″m(z)=jak dla sigma x dla Punktu 4 dla m = 1
ψ″m(z)=0, 009868
ψm(z)=jak dla sigma y dla Punktu 4 dla m = 1
ψm(z)=0, 499697
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{0,009868 + 0,34*0,499697} \right)\mathbf{*cos}\frac{\mathbf{1*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,179765*0,707107}$
$\mathbf{u}^{\left( \mathbf{2} \right)}\mathbf{= - 0,255086}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
ψ″m(z)=jak dla sigma x dla Punktu 4 dla m = 3
ψ″m(z)=0, 069989
ψm(z)=jak dla sigma y dla Punktu 4 dla m = 3
ψm(z)=0, 480083
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{0,069989 + 0,34*0,480083}\mathbf{\ } \right)\mathbf{*cos}\frac{\mathbf{3*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,233217*( - 0,707107)}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,004086}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
ψ″m(z)=jak dla sigma x dla Punktu 4 dla m = 5
ψ″m(z)=0, 119431
ψm(z)=jak dla sigma y dla Punktu 4 dla m = 5
ψm(z)=0, 398778
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*} \right)\mathbf{*}\left( \mathbf{0,119431}\mathbf{\ }\mathbf{+ 0,34*0,398778} \right)\mathbf{*cos}\frac{\mathbf{5*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,003211}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,255016*( - 0,707107)}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,000579}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 5 (x=45,0 cm; y=20,0 cm)
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*( - 0,255086}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,004086}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,00}\mathbf{0579}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*( - 0,250421}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
u(2)= − 0, 001913cm
Punktu nr 6 (x=90cm; y=20cm)
Dla m=1
ψ″m(z)=jak dla sigma x dla Punktu 4 dla m = 1
ψ″m(z)=0, 009868
ψm(z)=jak dla sigma y dla Punktu 4 dla m = 1
ψm(z)=0, 499697
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,0067}\mathbf{64}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{0,009868 + 0,34*0,499697} \right)\mathbf{*cos}\frac{\mathbf{1*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,179765*0}$
$\mathbf{u}^{\left( \mathbf{2} \right)}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
ψ″m(z)=jak dla sigma x dla Punktu 4 dla m = 3
ψ″m(z)=0, 069989
ψm(z)=jak dla sigma y dla Punktu 4 dla m = 3
ψm(z)=0, 480083
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{0,069989 + 0,34*0,480083}\mathbf{\ } \right)\mathbf{*cos}\frac{\mathbf{3*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,233217*0}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
ψ″m(z)=jak dla sigma x dla Punktu 4 dla m = 5
ψ″m(z)=0, 119431
ψm(z)=jak dla sigma y dla Punktu 4 dla m = 5
ψm(z)=0, 398778
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*} \right)\mathbf{*}\left( \mathbf{0,119431}\mathbf{\ }\mathbf{+ 0,34*0,398778} \right)\mathbf{*cos}\frac{\mathbf{5*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,003211}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,255016*0}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 6 (x=90,0 cm; y=20,0 cm)
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*(0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*(0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
u(2)=0cm
Punktu nr 7 (x=0cm; y=40cm)
Dla m=1
ψ″m(z)=jak dla sigma x dla Punktu 7 dla m = 1
ψ″m(z)= − 12, 5343658
ψm(z)=jak dla sigma y dla Punktu 7 dla m = 1
ψm(z)=0, 999999
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006}\mathbf{764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{- 12,5343658 + 0,34*0,999999} \right)\mathbf{*cos}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*( - 12,194366)*1}$
$\mathbf{u}^{\left( \mathbf{2} \right)}\mathbf{= 24,471213}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
ψ″m(z)=jak dla sigma x dla Punktu 7 dla m = 3
ψ″m(z)= − 1, 756089
ψm(z)=jak dla sigma y dla Punktu 7 dla m = 3
ψm(z)=0, 999999
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{- 1,756089 + 0,34*0,999999}\mathbf{\text{\ \ }} \right)\mathbf{*cos}\frac{\mathbf{3*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*( - 1,416089)*1}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,035084}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
ψ″m(z)=jak dla sigma x dla Punktu 7 dla m = 5
ψ″m(z)= − 1, 095046
ψm(z)=jak dla sigma y dla Punktu 7 dla m = 5
ψm(z)=0, 999999
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*} \right)\mathbf{*}\left( \mathbf{- 1,095046}\mathbf{\text{\ \ }}\mathbf{+ 0,34*0,999999}\mathbf{\ } \right)\mathbf{*cos}\frac{\mathbf{5*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,00}\mathbf{3211}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*( - 0,755046)*1}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,002424}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 7 (x=0,0 cm; y=40,0 cm)
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*(24,471213}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,035084}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,002424}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*24,508721}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
u(2)=0, 187222cm
Punktu nr 8 (x=45cm; y=40cm)
Dla m=1
ψ″m(z)=jak dla sigma x dla Punktu 7 dla m = 1
ψ″m(z)= − 12, 534366
ψm(z)=jak dla sigma y dla Punktu 7 dla m = 1
ψm(z)=0, 999999
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{- 12,5343658 + 0,34*0,999999} \right)\mathbf{*cos}\frac{\mathbf{1*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*}\left( \mathbf{- 12,194366} \right)\mathbf{*0,707107}$
$\mathbf{u}^{\left( \mathbf{2} \right)}\mathbf{= 17,303766}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
ψ″m(z)=jak dla sigma x dla Punktu 7 dla m = 3
ψ″m(z)= − 1, 756089
ψm(z)=jak dla sigma y dla Punktu 7 dla m = 3
ψm(z)=0, 999999
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{- 1,756089 + 0,34*0,9}\mathbf{99999}\mathbf{\text{\ \ }} \right)\mathbf{*cos}\frac{\mathbf{3*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*}\left( \mathbf{- 1,416089} \right)\mathbf{*( - 0,707107)}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,024508}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
ψ″m(z)=jak dla sigma x dla Punktu 7 dla m = 5
ψ″m(z)= − 1, 095046
ψm(z)=jak dla sigma y dla Punktu 7 dla m = 5
ψm(z)=0, 999999
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*} \right)\mathbf{*}\left( \mathbf{- 1,095046}\mathbf{\text{\ \ }}\mathbf{+ 0,34*0,999999}\mathbf{\ } \right)\mathbf{*cos}\frac{\mathbf{5*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,003211}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*}\left( \mathbf{- 0,755046} \right)\mathbf{*( - 0,707107)}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0,000735}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 8 (x=45,0 cm; y=40,0 cm)
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*(17,303766}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,024508}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,000735}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{u}^{\left( \mathbf{2} \right)}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*17,279693}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
u(2)=0, 1319999cm
Punktu nr 9 (x=90cm; y=40cm)
Dla m=1
ψ″m(z)=jak dla sigma x dla Punktu 7 dla m = 1
ψ″m(z)= − 12, 534366
ψm(z)=jak dla sigma y dla Punktu 7 dla m = 1
ψm(z)=0, 999999
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{- 12,5343658 + 0,34*0,999999} \right)\mathbf{*cos}\frac{\mathbf{1*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*}\left( \mathbf{- 12,194366} \right)\mathbf{*0}$
$\mathbf{u}^{\left( \mathbf{2} \right)}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
ψ″m(z)=jak dla sigma x dla Punktu 7 dla m = 3
ψ″m(z)= − 1, 756089
ψm(z)=jak dla sigma y dla Punktu 7 dla m = 3
ψm(z)=0, 999999
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{- 1,756089 + 0,34*0,999999}\mathbf{\text{\ \ }} \right)\mathbf{*cos}\frac{\mathbf{3*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*}\left( \mathbf{- 1,416089} \right)\mathbf{*0}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
ψ″m(z)=jak dla sigma x dla Punktu 7 dla m = 5
ψ″m(z)= − 1, 095046
ψm(z)=jak dla sigma y dla Punktu 7 dla m = 5
ψm(z)=0, 999999
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*} \right)\mathbf{*}\left( \mathbf{- 1,095046}\mathbf{\text{\ \ }}\mathbf{+ 0,34*0,999999}\mathbf{\ } \right)\mathbf{*cos}\frac{\mathbf{5*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{c}\mathbf{m}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= - 0,003211}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*}\left( \mathbf{- 0,755046} \right)\mathbf{*0}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 8 (x=45,0 cm; y=40,0 cm)
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( {\mathbf{\psi}^{\mathbf{''}}}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*cos}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{u}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*(0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{u}^{\left( \mathbf{2} \right)}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
u(2)=0cm
$$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$$
Punkt 1
X=0,0cm
Y=0,0cm
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0
Ψm(z)=(bm−cm)*cosh z − dmsinh z+dmz cosh z+cm z sinh z
Ψm(z)=(19, 305565 + 19, 305565)*cosh (0)− 6, 247205 * sinh (0)+6, 247205 * 0 * cosh(0)−19, 305565 * 0 * sinh (0)
Ψm(z)=38, 61113 * 1 − 6, 247205 * 0 + 6, 247205 * 0 * 1 − 19, 305565 * 0 * 0
Ψm(z)=38, 61113
ψ′m(z)= jak dla tau xy (2)… dla Punktu 1 dla m=1
ψ′m(z)=0
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764} \right)\mathbf{*}\left( \mathbf{38,61113 + 0,34*0} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
v(2)= − 2, 006764 * 38, 61113 * 0
v(2)=0
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{3*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0
Ψm(z)=(bm−cm)*cosh z − dmsinh z+dmz cosh z+cm z sinh z
Ψm(z)=(1, 0886310 + 1, 0886310)*cosh (0)− 0, 721779 * sinh (0)+0, 721779 * 0 * cosh(0)−1, 0886310 * 0 * sinh (0)
Ψm(z)=2, 177262 * 1 − 0, 721779 * 0 + 0, 721779 * 0 * 1 − 1, 0886310 * 0 * 0
Ψm(z)=2, 177262
ψ′m(z)= jak dla tau xy (2)… dla Punktu 1 dla m=3
ψ′m(z)=0
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,07432}\mathbf{5} \right)\mathbf{*}\left( \mathbf{2,177262 + 0,34*0} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
v(2)= − 0, 024775 * 2, 177262 * 0
v(2)=0
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{5*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0
Ψm(z)=(bm−cm)*cosh z − dmsinh z+dmz cosh z+cm z sinh z
Ψm(z)=(0, 287449 + 0, 287449)*cosh (0)− 0, 223116 * sinh (0)+0, 223116 * 0 * cosh(0)−0, 287449 * 0 * sinh (0)
Ψm(z)=0, 574898 * 1 − 0, 223116 * 0 + 0, 223116 * 0 * 1 − 0, 287449 * 0 * 0
Ψm(z)=0, 574898
ψ′m(z)= jak dla tau xy (2)… dla Punktu 1 dla m=5
ψ′m(z)=0
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054} \right)\mathbf{*}\left( \mathbf{0,574898}\mathbf{\ }\mathbf{+ 0,34*0} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
v(2)= − 0, 003211 * 0, 574898 * 0
v(2)=0
Zatem dla Punktu 1 (x=0,0cm; y=0,0 cm)
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*(0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{v}^{\left( \mathbf{2} \right)}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
v(2)=0cm
Punktu nr 2 (x=45cm; y=0cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0
Ψm(z)=jak dla Punktu 1 dla m = 1
Ψm(z)=38, 61113
ψ′m(z)= jak dla tau xy (2)… dla Punktu 1 dla m=1
ψ′m(z)=0
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764} \right)\mathbf{*}\left( \mathbf{38,61113 + 0,34*0} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
v(2)= − 2, 006764 * 38, 611130*0, 707107
v(2)= − 54, 789073
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{3*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0
Ψm(z)=jak dla Punktu 1 dla m = 3
Ψm(z)=2, 177262
ψ′m(z)= jak dla tau xy (2)… dla Punktu 1 dla m=3
ψ′m(z)=0
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325} \right)\mathbf{*}\left( \mathbf{2,177262 + 0,34*0} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593}\mathbf{*}\mathbf{45cm}}{\mathbf{180}\mathbf{\text{cm}}}$
v(2)= − 0, 024775 * 2, 177262 * 0, 707107
v(2)= − 0, 038143
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{5*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0
Ψm(z)=jak dla Punktu 1 dla m = 5
Ψm(z)=0, 574898
ψ′m(z)= jak dla tau xy (2)… dla Punktu 1 dla m=5
ψ′m(z)=0
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054} \right)\mathbf{*}\left( \mathbf{0,574898}\mathbf{\ }\mathbf{+ 0,34*0} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
v(2)= − 0, 003211 * 0, 574898 * ( − 0, 707107)
v(2)=0, 001305
Zatem dla Punktu 2 (x=45,0cm; y=0,0 cm)
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*}\mathbf{( - 54,789073}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,038143}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,001305}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{v}^{\left( \mathbf{2} \right)}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*( - 54,}\mathbf{825911}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
v(2)= − 0, 418815cm
Punktu nr 3 (x=90cm; y=0cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0
Ψm(z)=jak dla Punktu 1 dla m = 1
Ψm(z)=38, 61113
ψ′m(z)= jak dla tau xy (2)… dla Punktu 1 dla m=1
ψ′m(z)=0
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764} \right)\mathbf{*}\left( \mathbf{38,61113 + 0,34*0} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
v(2)= − 2, 006764 * 38, 61113 * 1
v(2)= − 77, 483426
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{3*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0
Ψm(z)=jak dla Punktu 1 dla m = 3
Ψm(z)=2, 177262
ψ′m(z)= jak dla tau xy (2)… dla Punktu 1 dla m=3
ψ′m(z)=0
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325} \right)\mathbf{*}\left( \mathbf{2,177262 + 0,34*0} \right)\mathbf{*sin}\frac{\mathbf{1*3,1415934*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
v(2)= − 0, 024775 * 2, 177262 * ( − 1)
v(2)=0, 053942
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{5*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0
Ψm(z)=jak dla Punktu 1 dla m = 5
Ψm(z)=0, 574898
ψ′m(z)= jak dla tau xy (2)… dla Punktu 1 dla m=5
ψ′m(z)=0
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054} \right)\mathbf{*}\left( \mathbf{0,574898}\mathbf{\ }\mathbf{+ 0,34*0} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
v(2)= − 0, 003211 * 0, 574898 * 1
v(2)= − 0, 001846
Zatem dla Punktu 3 (x=90,0cm; y=0,0 cm)
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*( - 77,483426}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,053942}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,001846}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{v}^{\left( \mathbf{2} \right)}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*( - 77,431330}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
v(2)= − 0, 591497cm
Punktu nr 4 (x=0cm; y=20cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{1*3,1415}\mathbf{93*20}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0, 349066
Ψm(z)=(bm−cm)*cosh z − dmsinh z+dmz cosh z+cm z sinh z
Ψm(z)=(19, 305565 + 19, 305565)*cosh (0, 349066 )− 6, 247205 * sinh (0, 349066 )+6, 247205 * 0, 349066 * cosh(0, 349066 )−19, 305565 * 0, 349066 * sinh (0, 349066 )
Ψm(z)=38, 611130*1, 061545 − 6, 247205 * 0, 356198 + 6, 247205 * 0, 349066 * 1, 061545 − 19, 305565 * 0, 349066 *0, 356198
Ψm(z)=40, 987452 − 2, 225242 + 2, 314897 − 2, 400389
Ψm(z)=38, 676704
ψ′m(z)= jak dla tau xy (2)… dla Punktu 4 dla m=1
ψ′m(z)=2, 139751
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764} \right)\mathbf{*}\left( \mathbf{38,676704 + 0,34*2,139751} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
v(2)= − 2, 006764 * 39, 404219 * 0
v(2)=0
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{3*3,141593*20}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 1, 047198
Ψm(z)=(bm−cm)*cosh z − dmsinh z+dmz cosh z+cm z sinh z
Ψm(z)=(1, 0886310 + 1, 0886310)*cosh (1, 047198 )− 0, 721779 * sinh (1, 047198 )+0, 721779 * 1, 047198 * cosh(1, 047198 )−1, 0886310 * 1, 047198 * sinh (1, 047198 )
Ψm(z)=(1, 0886310 + 1, 0886310)*1, 6002874 − 0, 721779 * 1, 2493678 + 0, 721779 * 1, 047198 * 1, 6002874 − 1, 0886310 * 1, 047198 * 1, 2493678
Ψm(z)=3, 484245 − 0, 901767 + 1, 209570 − 1, 424295
Ψm(z)=2, 367751
ψ′m(z)= jak dla tau xy (2)… dla Punktu 4 dla m=3
ψ′m(z)=0, 687043
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{2,367751 + 0,34*}\mathbf{0,687043}\mathbf{\ } \right)\mathbf{*sin}\frac{\mathbf{3*3,1415934*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*}\mathbf{2,601346}\mathbf{*0}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{5*3,141593*20}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 1, 745329
Ψm(z)=(bm−cm)*cosh z − dmsinh z+dmz cosh z+cm z sinh z
Ψm(z)=(0, 287449 + 0, 287449)*cosh (1, 745329 )− 0, 223116 * sinh (1, 745329 )+0, 223116 * 1, 745329 * cosh(1, 745329 )−0, 287449 * 1, 745329 * sinh (1, 745329 )
Ψm(z)=0, 574898 * 2, 951187 − 0, 223116 * 2, 776599 + 0, 223116 * 1, 745329 * 2, 951187 − 0, 287449 * 1, 745329 * 2, 776599
Ψm(z)=1, 696632 − 0, 619504 + 1, 149224 − 1, 393000
Ψm(z)=0, 833352
ψ′m(z)= jak dla tau xy (2)… dla Punktu 4 dla m=5
ψ′m(z)=0, 375727
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{0,833352}\mathbf{\text{\ \ }}\mathbf{+ 0,34*0,375727}\mathbf{\ } \right)\mathbf{*sin}\frac{\mathbf{5*3,141593*0}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 0,003211}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*}\mathbf{0,961099}\mathbf{*}\mathbf{0}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 4 (x=0,0cm; y=20,0 cm)
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*(}\mathbf{0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+}\mathbf{0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{v}^{\left( \mathbf{2} \right)}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*}\mathbf{0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
v(2)=0cm
Punktu nr 5 (x=45cm; y=20cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{1*3,1415}\mathbf{93*20}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0, 349066
Ψm(z)=jak dla Punktu 4 dla m = 1
Ψm(z)=38, 676719
ψ′m(z)= jak dla tau xy (2)… dla Punktu 4 dla m=1
ψ′m(z)=2, 139751
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{38,676719 + 0,34*2,139751} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*45cm}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*39,404234*0,707107}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 55,914485}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{3*3,141593*20}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 1, 047198
Ψm(z)=jak dla Punktu 4 dla m=3
Ψm(z)=2, 367751
ψ′m(z)= jak dla tau xy (2)… dla Punktu 4 dla m=3
ψ′m(z)=0, 687043
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{2,367751 + 0,34*0,687043}\mathbf{\ } \right)\mathbf{*sin}\frac{\mathbf{3*3,1415934*45cm}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*2,601346*0,707107}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\mathbf{- 0,045572}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{5*3,141593*20}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 1, 745329
Ψm(z)=jak dla Punktu 4 dla m = 5
Ψm(z)=0, 833352
ψ′m(z)= jak dla tau xy (2)… dla Punktu 4 dla m=5
ψ′m(z)=0, 375727
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{0,833352}\mathbf{\text{\ \ }}\mathbf{+ 0,34*0,375727}\mathbf{\ } \right)\mathbf{*sin}\frac{\mathbf{5*3,141593*}\mathbf{45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 0,003211}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,961099*}\mathbf{( - 0,707107)}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\mathbf{0,002182}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 5 (x=45,0cm; y=20,0 cm)
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*(}\mathbf{- 55,914485}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,045572}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+}\mathbf{0,002182}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{v}^{\left( \mathbf{2} \right)}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*}\mathbf{( - 55,957875}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
v(2)=−0, 427462cm
Punktu nr 6 (x=90cm; y=20cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{1*3,1415}\mathbf{93*20}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0, 349066
Ψm(z)=jak dla Punktu 4 dla m = 1
Ψm(z)=38, 676719
ψ′m(z)= jak dla tau xy (2)… dla Punktu 4 dla m=1
ψ′m(z)=2, 139751
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{38,676719 + 0,34*2,139751} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*}\mathbf{90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*39,404234*}\mathbf{1}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= -}\mathbf{79,074999}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{3*3,141593*20}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 1, 047198
Ψm(z)=jak dla Punktu 4 dla m=3
Ψm(z)=2, 367751
ψ′m(z)= jak dla tau xy (2)… dla Punktu 4 dla m=3
ψ′m(z)=0, 687043
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{2,367751 + 0,34*0,687043}\mathbf{\ } \right)\mathbf{*sin}\frac{\mathbf{3*3,1415934*}\mathbf{90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*2,601346*}\mathbf{( - 1)}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\mathbf{0,064448}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{5*3,141593*20}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 1, 745329
Ψm(z)=jak dla Punktu 4 dla m = 5
Ψm(z)=0, 833352
ψ′m(z)= jak dla tau xy (2)… dla Punktu 4 dla m=5
ψ′m(z)=0, 375727
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{0,833352}\mathbf{\text{\ \ }}\mathbf{+ 0,34*0,375727}\mathbf{\ } \right)\mathbf{*sin}\frac{\mathbf{5*3,141593*}\mathbf{90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 0,003211}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*0,961099*}\mathbf{1}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\mathbf{-}\mathbf{0,00}\mathbf{3086}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 6 (x=90,0cm; y=20,0 cm)
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*( -}\mathbf{79,074999}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,064448}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{-}\mathbf{0,00}\mathbf{3086}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{v}^{\left( \mathbf{2} \right)}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*( -}\mathbf{79,013528}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
v(2)=−0, 603585cm
Punktu nr 7 (x=0cm; y=40cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{1*3,141593*40}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0, 698132
Ψm(z)=(bm−cm)*cosh z − dmsinh z+dmz cosh z+cm z sinh z
Ψm(z)=(19, 305565 + 19, 305565)*cosh (0, 698132)− 6, 247205 * sinh (0, 698132)+6, 247205 * 0, 698132 * cosh(0, 698132)−19, 305565 * 0, 698132 * sinh (0, 698132)
Ψm(z)=38, 611130 * 1, 253754 − 6, 247205 * 0, 756240 + 6, 247205 * 0, 698132 * 1, 253754 − 19, 305565 * 0, 698132 * 0, 756240
Ψm(z)=48, 408859 − 4, 724386 + 5, 468089 − 10, 192476
Ψm(z)=38, 960086
ψ′m(z)= jak dla tau xy (2)… dla Punktu 7 dla m=1
ψ′m(z)=0
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{38,960086\ + 0,34*0} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*0cm}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*38,960086*0}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\mathbf{0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{3*3,141593*40}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 2, 094395
Ψm(z)=(bm−cm)*cosh z − dmsinh z+dmz cosh z+cm z sinh z
Ψm(z)=(1, 0886310 + 1, 0886310)*cosh (2, 094395 )− 0, 721779 * sinh (2, 094395 )+0, 721779 * 2, 094395 * cosh(2, 094395 )−1, 0886310 * 2, 094395 * sinh (2, 094395 )
Ψm(z)=2, 177262 * 4, 121836 − 0, 721779 * 3, 998691 + 0, 721779 * 2, 094395 * 4, 121836 − 1, 0886310 * 2, 094395 *3, 998691
Ψm(z)=8, 974318 − 2, 886171 + 6, 230939 − 9, 117109
Ψm(z)=3, 201977
ψ′m(z)= jak dla tau xy (2)… dla Punktu 7 dla m=3
ψ′m(z)=0
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{3,201977\ }\mathbf{\ }\mathbf{+ 0,34*0}\mathbf{\ } \right)\mathbf{*sin}\frac{\mathbf{3*3,1415934*0cm}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,201977*0}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{5*3,141593*40}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 3, 490659
Ψm(z)=(bm−cm)*cosh z − dmsinh z+dmz cosh z+cm z sinh z
Ψm(z)=(0, 287449 + 0, 287449)*cosh (3, 490659 )− 0, 223116 * sinh (3, 490659 )+0, 223116 * 3, 490659 * cosh(3, 490659 )−0, 287449 * 3, 490659 * sinh (3, 490659 )
Ψm(z)=0, 574898 * 16, 419021 − 0, 223116 * 16, 388540 + 0, 223116 * 3, 490659 *16, 419021 − 0, 287449 * 3, 490659 * 16, 388540
Ψm(z)=9, 439262 − 3, 656545 + 12, 787493 − 16, 444039
Ψm(z)=2, 126171
ψ′m(z)= jak dla tau xy (2)… dla Punktu 7 dla m=5
ψ′m(z)=0
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{2,126171}\mathbf{\text{\ \ \ }}\mathbf{+ 0,34*0}\mathbf{\ } \right)\mathbf{*sin}\frac{\mathbf{5*3,141593*0cm}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 0,003211}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*2,126171}\mathbf{\text{\ \ \ }}\mathbf{*0}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 7 (x=0,0cm; y=40,0 cm)
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*(0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+}\mathbf{0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{v}^{\left( \mathbf{2} \right)}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*0}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
v(2)=0cm
Punktu nr 8 (x=45cm; y=40cm)
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{1*3,141593*40}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0, 698132
Ψm(z)=jak dla Punktu 7 dla m = 1
Ψm(z)=38, 960086
ψ′m(z)= jak dla tau xy (2)… dla Punktu 7 dla m=1
ψ′m(z)=0
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{38,960086\ + 0,34*0} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*}\mathbf{45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*38,960086*0}\mathbf{,707107}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\mathbf{- 55,284240}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{3*3,141593*40}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 2, 094395
Ψm(z)=jak dla Punktu 7 dla m = 3
Ψm(z)=3, 201977
ψ′m(z)= jak dla tau xy (2)… dla Punktu 7 dla m=3
ψ′m(z)=0
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{3,201977\ }\mathbf{\ }\mathbf{+ 0,34*0}\mathbf{\ } \right)\mathbf{*sin}\frac{\mathbf{3*3,1415934*}\mathbf{45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,201977*0}\mathbf{,707107}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\mathbf{- 0,056074}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{5*3,141593*40}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 3, 490659
Ψm(z)=jak dla Punktu 7 dla m = 3
Ψm(z)=2, 126171
ψ′m(z)= jak dla tau xy (2)… dla Punktu 7 dla m=5
ψ′m(z)=0
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{2,126171}\mathbf{\text{\ \ \ }}\mathbf{+ 0,34*0}\mathbf{\ } \right)\mathbf{*sin}\frac{\mathbf{5*3,141593*}\mathbf{45}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 0,003211}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*2,126171}\mathbf{\text{\ \ \ }}\mathbf{*}\mathbf{( - 0,707107)}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\mathbf{0,004827}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 8 (x=45,0cm; y=40,0 cm)
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*(}\mathbf{- 55,28424}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,056094}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0}\mathbf{,004827}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{v}^{\left( \mathbf{2} \right)}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*}\mathbf{( - 55,335507}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
v(2)=−0, 4227079cm
Punktu nr 9 (x=90cm; y=40cm)
Dla m=1
Dla m=1
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{1*3,141593*40}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 0, 698132
Ψm(z)=jak dla Punktu 7 dla m = 1
Ψm(z)=38, 960086
ψ′m(z)= jak dla tau xy (2)… dla Punktu 7 dla m=1
ψ′m(z)=0
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}}\mathbf{*}\left( \mathbf{- 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{38,960086\ + 0,34*0} \right)\mathbf{*sin}\frac{\mathbf{1*3,141593*}\mathbf{90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 2,006764}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*38,960086*}\mathbf{1}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\mathbf{- 78,183698}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=3
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{3*3,141593*40}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 2, 094395
Ψm(z)=jak dla Punktu 7 dla m = 3
Ψm(z)=3, 201977
ψ′m(z)= jak dla tau xy (2)… dla Punktu 7 dla m=3
ψ′m(z)=0
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{*}\left( \mathbf{- 0,074325}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{3,201977\ }\mathbf{\ }\mathbf{+ 0,34*0}\mathbf{\ } \right)\mathbf{*sin}\frac{\mathbf{3*3,1415934*}\mathbf{90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 0,024775}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,201977*}\mathbf{( - 1)}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\mathbf{0,079329}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Dla m=5
$\mathbf{z =}\frac{\mathbf{m*\pi*y}}{\mathbf{l}}$
$\mathbf{z =}\frac{\mathbf{5*3,141593*40}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
z = 3, 490659
Ψm(z)=jak dla Punktu 7 dla m = 3
Ψm(z)=2, 126171
ψ′m(z)= jak dla tau xy (2)… dla Punktu 7 dla m=5
ψ′m(z)=0
Zatem
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\mathbf{*}\left( \mathbf{- 0,016054}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}} \right)\mathbf{*}\left( \mathbf{2,126171}\mathbf{\text{\ \ \ }}\mathbf{+ 0,34*0}\mathbf{\ } \right)\mathbf{*sin}\frac{\mathbf{5*3,141593*}\mathbf{90}\mathbf{\text{cm}}}{\mathbf{180}\mathbf{\text{cm}}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{= - 0,003211}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*2,126171}\mathbf{\text{\ \ \ }}\mathbf{*}\mathbf{1}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\mathbf{- 0,006827}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}$
Zatem dla Punktu 9 (x=90,0cm; y=40,0 cm)
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{E*\pi}}\mathbf{*}\sum_{\mathbf{m = 1,3,5\ldots}}^{\mathbf{\infty}}{\frac{\mathbf{1}}{\mathbf{m}}\mathbf{*}\mathbf{C}_{\mathbf{m}}}\mathbf{*}\left( \mathbf{\Psi}_{\mathbf{m}}\left( \mathbf{z} \right)\mathbf{+ \nu*}\mathbf{\psi'}_{\mathbf{m}}\left( \mathbf{z} \right) \right)\mathbf{*sin}\frac{\mathbf{m*\pi*x}}{\mathbf{l}}$
$\mathbf{v}^{\mathbf{(2)}}\mathbf{=}\frac{\mathbf{180}\mathbf{\text{cm}}}{\mathbf{7500}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{*3,141593}}\mathbf{*(}\mathbf{- 78,183669}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{+ 0,079329}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{- 0,006827}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
$\mathbf{v}^{\left( \mathbf{2} \right)}\mathbf{= 0,007639}\frac{\mathbf{c}\mathbf{m}^{\mathbf{3}}}{\mathbf{\text{kN}}}\mathbf{*}\mathbf{( - 78,111167}\frac{\mathbf{\text{kN}}}{\mathbf{c}\mathbf{m}^{\mathbf{2}}}\mathbf{)}$
v(2)=−0, 596694m
Na podstawie zasady superpozycji, rozwiązania naszego zadania jest sumą poszczególnych rozwiązań:
Punkt 1
σxx=σxx(1)+σxx(2)
σyy=σyy(1)+σyy(2)
σxy=σxy(1)+σxy(2)
u=u(1)+u(2)
v=v(1)+v(2)
Punkt 2
σxx=σxx(1)+σxx(2)
σyy=σyy(1)+σyy(2)
σxy=σxy(1)+σxy(2)
u=u(1)+u(2)
v=v(1)+v(2)
Punkt 3
σxx=σxx(1)+σxx(2)
σyy=σyy(1)+σyy(2)
σxy=σxy(1)+σxy(2)
u=u(1)+u(2)
v=v(1)+v(2)
Punkt 4
σxx=σxx(1)+σxx(2)
σyy=σyy(1)+σyy(2)
σxy=σxy(1)+σxy(2)
u=u(1)+u(2)
v=v(1)+v(2)
Punkt 5
σxx=σxx(1)+σxx(2)
σyy=σyy(1)+σyy(2)
σxy=σxy(1)+σxy(2)
u=u(1)+u(2)
v=v(1)+v(2)
Punkt 6
σxx=σxx(1)+σxx(2)
σyy=σyy(1)+σyy(2)
σxy=σxy(1)+σxy(2)
u=u(1)+u(2)
v=v(1)+v(2)
Punkt 7
σxx=σxx(1)+σxx(2)
σyy=σyy(1)+σyy(2)
σxy=σxy(1)+σxy(2)
u=u(1)+u(2)
v=v(1)+v(2)
Punkt 8
σxx=σxx(1)+σxx(2)
σyy=σyy(1)+σyy(2)
σxy=σxy(1)+σxy(2)
u=u(1)+u(2)
v=v(1)+v(2)
Punkt 9
σxx=σxx(1)+σxx(2)
σyy=σyy(1)+σyy(2)
σxy=σxy(1)+σxy(2)
u=u(1)+u(2)
v=v(1)+v(2)