Metoda MENABREI

$$\sum_{}^{}{M_{A} = 0 = M_{A} + M + P_{1} \bullet cos45 \bullet 2a + P_{1} \bullet sin45 \bullet a - P_{2} \bullet 2a + 3q \bullet (\frac{1}{3} \bullet 3a + 2a)} - H \bullet 5a = M_{A} + 2qa^{2} + \sqrt{2} \bullet qa^{2} + \frac{\sqrt{2}}{2}qa^{2} - 4qa^{2} + 9qa^{2} - H \bullet 5a$$

$$M_{A} = H \bullet 5a - 7qa^{2} - \frac{3\sqrt{2}}{2}qa^{2}$$

$$\sum_{}^{}F_{X} = R_{X} + P_{1} \bullet cos45 = 0$$

$$R_{X} = - \frac{\sqrt{2}}{2}qa$$

$M^{\text{AB}} = M_{A} - R_{X} \bullet x = H \bullet 5a - 7qa^{2} - \frac{3\sqrt{2}}{2}qa^{2} + \frac{\sqrt{2}}{2}q\text{ax}$ $\frac{\partial M^{\text{AB}}}{\partial H} = 5a$

$M^{\text{ED}} = - Hx + 3qa \bullet \frac{1}{3}x = - Hx + \text{qax}$ $\frac{\partial M^{\text{ED}}}{\partial H} = - x$

$$M^{\text{DC}} = - H\left( 3a + xcos45 \right) + 3\text{qa}\left( \frac{1}{3}3a + xcos45 \right) - 2qaxcos45 = \backslash n$$

$\frac{\partial M^{\text{DC}}}{\partial H} = \left( - 3a - \frac{\sqrt{2}}{2}x \right)$

$$M^{\text{CB}} = - H\left( 4a + xcos45 \right) + 3\text{qa}\left( \frac{1}{3}3a + a + xcos45 \right) - 2qa\left( a + xcos45 \right) + \text{qax} =$$

=$- H \bullet 4a - \ H \bullet \frac{\sqrt{2}}{2}x + 6qa^{2} + \frac{3\sqrt{2}}{2}qax - 2qa^{2} - \sqrt{2}qax + qax =$

$= H \bullet \left( - 4a - \frac{\sqrt{2}}{2}x \right) + 4qa^{2} + \frac{\sqrt{2}}{2}\text{qax} + qax$ $\frac{\partial M^{\text{CB}}}{\partial H} = \left( - 4a - \frac{\sqrt{2}}{2}x \right)$

$\frac{\partial V}{\partial H_{i}} = 0 = \frac{1}{\text{EJ}}\sum_{i = 1}^{n}{\int_{0}^{l}{M(x_{i})}}\frac{\partial M_{x_{i}}}{\partial H_{i}}d_{x_{i}}$ - Twierdzenie MENABREI

$$\frac{\partial V}{\partial H_{i}} = 0 = \frac{1}{\text{EJ}}\left\lbrack \left( \int_{0}^{a}{\left( \ H \bullet 5a - 7qa^{2} - \frac{3\sqrt{2}}{2}qa^{2} + \frac{\sqrt{2}}{2}\text{qax} \right) \bullet 5a \bullet \text{dx}} \right) + \left( \int_{0}^{3a}{\left( - Hx + qax\ \right) \bullet \left( - x \right)\text{dx}} \right) + \begin{pmatrix} \int_{0}^{a\sqrt{2}}{\left( - H \bullet 3a - H \bullet \frac{\sqrt{2}}{2}x + 3qa^{2} + \frac{\sqrt{2}}{2}q\text{ax} \right) \bullet \left( - 3a - \frac{\sqrt{2}}{2}x \right)\text{dx}} \\ \\ \end{pmatrix} + \left( \int_{0}^{a\sqrt{2}}\left( - H \bullet 4a - H \bullet \frac{\sqrt{2}}{2}x + 4qa^{2} + \frac{\sqrt{2}}{2}qax + qax \right) \bullet \left( - 4a - \frac{\sqrt{2}}{2}x \right)\text{dx} \right) \right\rbrack$$

$\frac{1}{\text{EJ}}\left\lbrack \left( \int_{0}^{a}{\left( \ H \bullet 25a^{2} - 35qa^{3} - \frac{15\sqrt{2}}{2}qa^{3} + \frac{5\sqrt{2}}{2}qa^{2}x \right) \bullet \text{dx}} \right) + \left( \int_{0}^{3a}{\left( Hx^{2} - \text{qa}x^{2}\ \right) \bullet \text{dx}} \right) + \left( \int_{0}^{a\sqrt{2}}{\left( 9H \bullet a^{2} + H \bullet \frac{3\sqrt{2}}{2}ax - 9qa^{3} + \frac{3\sqrt{2}}{2}qa^{2}x + H \bullet \frac{3\sqrt{2}}{2}\text{ax} + \frac{1}{2}Hx^{2} - \frac{3\sqrt{2}}{2}qa^{2}x - \frac{1}{2}\text{qa}x^{2} \right) \bullet \text{dx}} \right) + \left( \int_{0}^{a\sqrt{2}}\left( 16Ha^{2} + H \bullet 2\sqrt{2}ax - 16qa^{3} + 2\sqrt{2}qa^{2}x - 4qa^{2}x + H \bullet 2\sqrt{2}\text{ax} + \frac{1}{2}Hx^{2} - 2\sqrt{2}qa^{2}x - \frac{1}{2}\text{qa}x^{2} - \frac{\sqrt{2}}{2}\text{qa}x^{2} \right) \bullet \text{dx} \right) \right\rbrack$=

$= \frac{1}{\text{EJ}}\left\lbrack \left( \left( \ H \bullet 25a^{2}x - 35qa^{3}x - \frac{15\sqrt{2}}{2}qa^{3}x + \frac{5\sqrt{2}}{4}qa^{2}x^{2} \right)_{0}^{a} \right) + \left( \left( H\frac{x^{3}}{3} - \text{qa}\frac{x^{3}}{3} \right)_{0}^{3a} \right) + \left( \left( 9H \bullet a^{2}x + H \bullet \frac{3\sqrt{2}}{4}ax^{2} - 9qa^{3}x + \frac{3\sqrt{2}}{4}qa^{2}x^{2} + H \bullet \frac{3\sqrt{2}}{4}ax^{2} + \frac{1}{6}Hx^{3} - \frac{3\sqrt{2}}{4}qa^{2}x^{2} - \frac{1}{6}\text{qa}x^{3} \right)_{0}^{a\sqrt{2}} \right) + \left( \left( 16Ha^{2}x + H \bullet \sqrt{2}ax^{2} - 16qa^{3}x + \sqrt{2}qa^{2}x^{2} - 2qa^{2}x^{2} + H \bullet \sqrt{2}ax^{2} + \frac{1}{6}Hx^{3} - \sqrt{2}qa^{2}x^{2} - \frac{1}{6}\text{qa}x^{3} - \frac{\sqrt{2}}{6}\text{qa}x^{3} \right)_{0}^{a\sqrt{2}} \right) \right\rbrack$=

$= \frac{1}{\text{EJ}}\left\lbrack \left( \ H \bullet 25a^{3} - 35qa^{4} - \frac{15\sqrt{2}}{2}qa^{4} + \frac{5\sqrt{2}}{4}qa^{4} \right) + \left( 9Ha^{3} - 9qa^{4} \right) + \left( 9H \bullet a^{3}\sqrt{2} + H \bullet \frac{3\sqrt{2}}{2}a^{3} - 9\sqrt{2}qa^{4} + \frac{3\sqrt{2}}{2}qa^{4} + H \bullet \frac{3\sqrt{2}}{2}a^{3} + \frac{\sqrt{2}}{3}Ha^{3} - \frac{3\sqrt{2}}{24}qa^{4} - \frac{\sqrt{2}}{3}qa^{4} \right) + \left( 16\sqrt{2}Ha^{3} + H \bullet 2\sqrt{2}a^{2} - 16\sqrt{2}qa^{4} - 2\sqrt{2}qa^{4} - 4qa^{4} + H \bullet 2\sqrt{2}a^{3} + \frac{\sqrt{2}}{3}Ha^{3} - 2\sqrt{2}qa^{4} - \frac{\sqrt{2}}{3}\text{qa}a^{4} - \frac{2}{3}qa^{4} \right) \right\rbrack$=

=$\frac{1}{\text{EJ}}\left\lbrack \ 25Ha^{3} - 35qa^{4} - \frac{25\sqrt{2}}{4}qa^{4} + 9Ha^{3} - 9qa^{4} + \frac{37}{3}\sqrt{2}Ha^{3} - \frac{37}{3}\sqrt{2}qa^{4} + \frac{61}{3}\sqrt{2}Ha^{3} - \frac{61}{3}\sqrt{2}qa^{4} - \frac{14}{3}qa^{4} \right\rbrack$=

$$34Ha^{3} + \frac{98}{3}\sqrt{2}Ha^{3} - \frac{467}{12}\sqrt{2}qa^{4} - \frac{146}{3}qa^{4} = 0$$

$$Ha^{3}\left( 34 + \frac{98}{3}\sqrt{2} \right) = qa^{4}\left( \frac{467}{12}\sqrt{2} + \frac{146}{3} \right)$$

$$H = \frac{1244}{962}\text{qa} \cong 1,293qa = 1,3qa$$

$$M_{A} = H \bullet 5a - 7qa^{2} - \frac{3\sqrt{2}}{2}qa^{2} = 1,3qa \bullet 5a - 7qa^{2} - \frac{3\sqrt{2}}{2}qa^{2} = - 2,6qa^{2}$$

$$\sum_{}^{}{F_{Y} = R_{Y} - P_{1} \bullet cos45 +}P_{2} - 3qa + H = 0$$

$$R_{Y} = P_{1} \bullet cos45 - P_{2} + 3qa - H = \frac{\sqrt{2}}{2}qa - 2qa + 3qa - 1,3qa = 0,4qa$$

Metoda MAXWELLA – MOHRA

Stan “0”

Stan “1”

$$\sum_{}^{}{M_{A} = 0 = M_{A} + M + P_{1} \bullet cos45 \bullet 2a + P_{1} \bullet sin45 \bullet a - P_{2} \bullet 2a + 3q \bullet (\frac{1}{3} \bullet 3a + 2a)}$$

$$M_{A} + 2qa^{2} + \sqrt{2} \bullet qa^{2} + \frac{\sqrt{2}}{2}qa^{2} - 4qa^{2} + 9qa^{2} = 0$$

$$M_{A} = - 7qa^{2} - \frac{3\sqrt{2}}{2}qa^{2}$$

$$\sum_{}^{}{M_{A} = 0 = M_{A}} - H \bullet 5a$$

M_A = H • 5a

$$\sum_{}^{}F_{X} = R_{\text{AX}} + P_{1} \bullet cos45 = 0$$

$$R_{\text{AX}} = - \frac{\sqrt{2}}{2} \bullet \text{qa}$$

$M^{\text{AB}} = M_{A} - R_{X} \bullet x = - 7qa^{2} - \frac{3\sqrt{2}}{2}qa^{2} + \frac{\sqrt{2}}{2}\text{qax}$

$$M^{\text{ED}} = 3qa \bullet \frac{1}{3}x = qax$$

$$M^{\text{DC}} = 3\text{qa}\left( \frac{1}{3}3a + \text{xcos}45 \right) - 2\text{qaxcos}45 = \backslash n$$

$$M^{\text{CB}} = 3qa\left( \frac{1}{3}3a + a + xcos45 \right) - 2qa\left( a + xcos45 \right) + qax =$$

=$6qa^{2} + \frac{3\sqrt{2}}{2}qax - 2qa^{2} - \sqrt{2}qax + qax =$

$= 4qa^{2} + \frac{\sqrt{2}}{2}\text{qax} + qax$

$$\sum_{}^{}F_{X} = R_{\text{AX}} = 0$$

R_AX = 0

M^AB = M_A = H • 5a

M^ED = −Hx

$M^{\text{DC}} = - H\left( 3a + \text{xcos}45 \right) = - H \bullet 3a - H \bullet \frac{\sqrt{2}}{2}x$

$$M^{\text{CB}} = - H\left( 4a + xcos45 \right) = - H \bullet 4a - \ H \bullet \frac{\sqrt{2}}{2}x$$

Równanie Maxwella – Mohra

α₁₀ + α₁₁H = 0

$$\alpha_{10} = \frac{1}{\text{EJ}}\left\lbrack \left( \int_{0}^{a}{\left( 5a \right) \bullet \left( - 7qa^{2} - \frac{3\sqrt{2}}{2}qa^{2} + \frac{\sqrt{2}}{2}\text{qax} \right)\text{dx}} \right) + \left( \int_{0}^{3a}{\left( - x \right) \bullet \left( \text{qa}x \right)\text{dx}} \right) + \left( \int_{0}^{a\sqrt{2}}{\left( - 3a - \frac{\sqrt{2}}{2}x \right) \bullet \left( 3qa^{2} + \frac{\sqrt{2}}{2}\text{qax} \right)\text{dx}} \right) + \left( \int_{0}^{a\sqrt{2}}{\left( - 4a - \ \frac{\sqrt{2}}{2}x \right) \bullet \left( 4qa^{2} + \frac{\sqrt{2}}{2}\text{qax} + qax \right)\text{dx}} \right) \right\rbrack = \frac{1}{\text{EJ}}\left\lbrack \left( \int_{0}^{a}{\left( - 35qa^{3} - \frac{15\sqrt{2}}{2}qa^{3} + \frac{5\sqrt{2}}{2}qa^{2}x \right)\text{dx}} \right) + \left( \int_{0}^{3a}{\text{qa}x^{2}\text{dx}} \right) + \left( \int_{0}^{a\sqrt{2}}{\left( - 9qa^{3} - \frac{3\sqrt{2}}{2}qa^{2}x - \frac{3\sqrt{2}}{2}qa^{2}x - \frac{1}{2}\text{qa}x^{2} \right)\text{dx}} \right) + \left( \int_{0}^{a\sqrt{2}}{\left( 16qa^{3} - 2\sqrt{2}qa^{2}x - 4qa^{2}x - 2\sqrt{2}qa^{2}x - \frac{1}{2}\text{qa}x^{2} - \frac{\sqrt{2}}{2}\text{qa}x^{2} \right)\text{dx}} \right) \right\rbrack =$$

$$\alpha_{10} = \frac{1}{\text{EJ}}\left\lbrack - \frac{467}{12}\sqrt{2}qa^{4} - \frac{146}{3}qa^{4} \right\rbrack$$

$$\alpha_{11} = \frac{1}{\text{EJ}}\left\lbrack \left( \int_{0}^{a}{\left( 5a \right) \bullet \left( 5a \right)\text{dx}} \right) + \left( \int_{0}^{3a}{\left( - x \right) \bullet \left( - x \right)\text{dx}} \right) + \left( \int_{0}^{a\sqrt{2}}{\left( - 3a - \frac{\sqrt{2}}{2}x \right) \bullet \left( - 3a - \frac{\sqrt{2}}{2}x \right)\text{dx}} \right) + \left( \int_{0}^{a\sqrt{2}}{\left( - 4a - \ \frac{\sqrt{2}}{2}x \right) \bullet \left( - 4a - \ \frac{\sqrt{2}}{2}x \right)\text{dx}} \right) \right\rbrack = \frac{1}{\text{EJ}}\left\lbrack \left( \int_{0}^{a}{\left( 25a^{2} \right)\text{dx}} \right) + \left( \int_{0}^{3a}{x^{2}\text{dx}} \right) + \left( \int_{0}^{a\sqrt{2}}{\left( 9a^{2} + \frac{3\sqrt{2}}{2}ax + \frac{3\sqrt{2}}{2}ax + \frac{1}{2}x^{2} \right)\text{dx}} \right) + \left( \int_{0}^{a\sqrt{2}}{\left( 16a^{2} + \ 2\sqrt{2}x^{2} + 2\sqrt{2}x^{2} + \frac{1}{2}x^{2} \right)\text{dx}} \right) \right\rbrack = \frac{1}{\text{EJ}}\left\lbrack \left( \left( 25a^{2}x \right)_{0}^{a} \right) + \left( \left( \frac{x^{3}}{3} \right)_{0}^{3a} \right) + \left( \left( 9a^{2}x + \frac{3\sqrt{2}}{4}ax^{2} + \frac{3\sqrt{2}}{4}ax^{2} + \frac{1}{6}x^{3} \right)_{0}^{a\sqrt{2}} \right) + \left( \left( 16a^{2}x + \ \frac{2\sqrt{2}}{3}x^{3} + \frac{2\sqrt{2}}{3}x^{3} + \frac{1}{6}x^{3} \right)_{0}^{a\sqrt{2}} \right) \right\rbrack = \frac{1}{\text{EJ}}\left\lbrack 25a^{3} + 9a^{3} + 9{\sqrt{2}a}^{3} + \frac{3\sqrt{2}}{2}a^{3} + \frac{3\sqrt{2}}{2}a^{3} + \frac{\sqrt{2}}{3}a^{3} + 16\sqrt{2}a^{3} + \ 2\sqrt{2}a^{3} + 2\sqrt{2}a^{3} + \frac{\sqrt{2}}{3}a^{3} \right\rbrack = \frac{1}{\text{EJ}}\left\lbrack 34a^{3} + \frac{98\sqrt{2}}{3}a^{3} \right\rbrack$$

$$\alpha_{11} = \frac{1}{\text{EJ}}\left\lbrack 34a^{3} + \frac{98\sqrt{2}}{3}a^{3} \right\rbrack$$

$$\frac{1}{\text{EJ}}\left\lbrack - \frac{467}{12}\sqrt{2}qa^{4} - \frac{146}{3}qa^{4} \right\rbrack + \frac{1}{\text{EJ}}\left\lbrack 34a^{3} + \frac{98\sqrt{2}}{3}a^{3} \right\rbrack \bullet H = 0$$

$$34Ha^{3} + \frac{98}{3}\sqrt{2}Ha^{3} - \frac{467}{12}\sqrt{2}qa^{4} - \frac{146}{3}qa^{4} = 0$$

$$Ha^{3}\left( 34 + \frac{98}{3}\sqrt{2} \right) = qa^{4}\left( \frac{467}{12}\sqrt{2} + \frac{146}{3} \right)$$

$$H = \frac{1244}{962}qa \cong 1,293qa = 1,3qa$$

Dwie metody dają identyczne rozwiązanie.