Materiał	Siły skupione	Długość przedziałów	Kąty	Obciążenie ciągłe	Moment
N/m­­^2	kN	m	^o	kN/m	kNm
kg	P1	P2	P3	a	b
11,5 ^. 10^7	8	10		0,4	0,3

$$\sum_{}^{}P_{\text{ix}} = 0\overset{\rightarrow}{\ }P_{1x} + R_{\text{Ax}} - P_{2x} = 0\overset{\rightarrow}{\ }P_{1}cos30 + R_{\text{Ax}} - P_{2}\cos 75 = 0$$

P_1x = 6, 9282 kN

P_2x = 2, 5882 kN

R_Ax = −4, 3400 kN

$$\sum_{}^{}P_{\text{iy}} = 0\overset{\rightarrow}{\ }{P_{1y} - Q_{1} + R}_{\text{Ay}} + R_{B} - Q_{2} - P_{2y} = 0\overset{\rightarrow}{\ }$$

$$\overset{\rightarrow}{\ }P_{1}sin30{- qa + R}_{\text{Ay}} + R_{B} - qc - P_{2}\sin 75 = 0$$

P_1y = 4, 0000 kN

Q₁ = 2, 0000 kN

Q₂ = 1, 5000 kN

P_2y = 9, 6593 kN

$$\sum_{}^{}M_{A} = 0\overset{\rightarrow}{\ } - 2M + P_{1y}a - Q_{1}\frac{a}{2} + M - R_{B}b + Q_{2}\left( b + \frac{c}{2} \right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } + P_{2y}\left( b + c \right) = 0$$

R_B = 12, 2352 kN

R_Ay = −3, 0759 kN, $\text{\ \ }R_{A} = \sqrt{{R_{\text{Ax}}}^{2} + {R_{\text{Ay}}}^{2}}$, $\ \ tg\gamma = \frac{R_{\text{Ay}}}{R_{\text{Ax}}}$

R_A = 5, 3195 kN, γ = 3520′

0 ≤ x₁ ≤ a

$$M_{\text{gI}} = - 2M + P_{1y} \bullet x_{1} - q\frac{{x_{1}}^{2}}{2}$$

$$T_{I} = \frac{dM_{\text{gI}}}{\text{dx}} = P_{1y} - q \bullet x_{1}$$

a ≤ x₂ ≤ a + b

$$M_{\text{gII}} = - 2M + P_{1y} \bullet x_{2} - qa\left( x_{2} - \frac{a}{2} \right) + R_{\text{Ay}}\left( x_{2} - a \right)$$

$$T_{\text{II}} = \frac{dM_{\text{gII}}}{\text{dx}} = P_{1y} - qa + R_{\text{Ay}}$$

a + b ≤ x₃ ≤ a + b + c

$$M_{\text{gIII}} = - 2M + P_{1y} \bullet x_{3} - qa\left( x_{3} - \frac{a}{2} \right) + R_{\text{Ay}}\left( x_{3} - a \right) + M\ \ \ \ \ + R_{B}\left( x_{3} - \left( a + b \right) \right)\ - q\frac{\left( x_{3} - \left( a + b \right) \right)^{2}}{2}$$

$$T_{\text{III}} = \frac{dM_{\text{gIII}}}{\text{dx}} = P_{1y} - \ qa + R_{\text{Ay}} + R_{B} - q \bullet \left( x_{3} - \left( a + b \right) \right)$$

$$\left\{ \begin{matrix} \begin{matrix} M_{\text{gI}}\left| \begin{matrix} \ \\ \ _{x_{1} = 0} \\ \end{matrix} = - 8,0000\ kN \bullet m \right.\ \\ M_{\text{gI}}\left| \begin{matrix} \ \\ \ _{x_{1} = a} \\ \end{matrix} = - 6,8000\ kN \bullet m \right.\ \\ \end{matrix} \\ \begin{matrix} T_{I}\left| \begin{matrix} \ \\ \ _{x_{1} = 0} \\ \end{matrix} = 4,0000\text{\ kN} \right.\ \\ T_{I}\left| \begin{matrix} \ \\ \ _{x_{1} = a} \\ \end{matrix} \right.\ = 2,0000\text{\ kN} \\ \end{matrix} \\ \end{matrix} \right.\ $$

$$\left\{ \begin{matrix} M_{\text{gII}}\left| \begin{matrix} \ \\ \ _{x_{2} = a} \\ \end{matrix} = - 6,8000\ kN \bullet m \right.\ \\ M_{\text{gII}}\left| \begin{matrix} \ \\ \ _{x_{2} = a + b} \\ \end{matrix} = - 7,1228\ kN \bullet m \right.\ \\ T_{\text{II}}\left| \begin{matrix} \ \\ \ _{x_{2} = a} \\ \end{matrix} = T_{\text{II}}\left| \begin{matrix} \ \\ \ _{x_{2} = a + b} \\ \end{matrix} = \right.\ - 1,0759\text{\ kN} \right.\ \\ \end{matrix} \right.\ $$

$$\left\{ \begin{matrix} \begin{matrix} M_{\text{gIII}}\left| \begin{matrix} \ \\ \ _{x_{3} = a + b} \\ \end{matrix} = - 3,1228\ kN \bullet m \right.\ \\ M_{\text{gIII}}\left| \begin{matrix} \ \\ \ _{x_{3} = a + b + c} \\ \end{matrix} = 0,0000\ kN \bullet m \right.\ \\ \end{matrix} \\ \begin{matrix} T_{\text{III}}\left| \begin{matrix} \ \\ \ _{x_{3} = a + b} \\ \end{matrix} = 11,1593\ kN \right.\ \\ T_{\text{III}}\left| \begin{matrix} \ \\ \ _{x_{3} = a + b + c} = 9,6593\text{\ kN} \\ \end{matrix} \right.\ \\ \end{matrix} \\ \end{matrix} \right.\ $$

$$\left| M_{g_{\max}} \right| = \left| M_{\text{gI}}\left| \begin{matrix} \ \\ \ _{x_{2} = 0} \\ \end{matrix} \right.\ \right| = 8,0000\ kNm$$

P_ixmax = 6, 9282 kN

S(x_s,y_s), x_s = 0

S_I(x_I,y_I), x_I = 0, y_I = 2a, A_I = 8a²

S_II(x_II,y_II), x_II = 0, $y_{\text{II}} = 4\frac{1}{3}a,$ A_II = a²

S_III(x_III,y_III), x_III = 0, y_III = 2a, A_III = a²

$$y_{s} = \frac{A_{I}y_{I} + A_{\text{II}}y_{\text{II}} - A_{\text{III}}y_{\text{III}}}{A_{I} + A_{\text{II}} - A_{\text{III}}} = 2,2927a$$

I_bxs = I_bxsI + I_bxsII − I_bxsIII

$$I_{bx_{\text{sI}}} = I_{\text{bx}} + A_{I}\left( y_{I} - y_{s} \right)^{2} = \frac{2a\left( 4a \right)^{3}}{12} + 8a^{2}\left( 2a - 2,2927a \right)^{2} = 11,3521a^{4}$$

$$I_{bx_{\text{sII}}} = I_{\text{bx}} + A_{\text{II}}\left( y_{\text{II}} - y_{s} \right)^{2} = \frac{a{\bullet a}^{3}}{18} + a^{2}\left( 4\frac{1}{3}a - 2,2927a \right)^{2} =$$

                     = 4, 2197a⁴

$$I_{bx_{\text{sIII}}} = I_{\text{bx}} + A_{\text{III}}\left( y_{\text{III}} - y_{s} \right)^{2} = \frac{a^{4}}{12} + a^{2}\left( 2a - 2,2927a \right)^{2} =$$

                     = 0, 1690a⁴

I_bxs = 15, 4028a⁴

$$\sigma_{g} = \frac{\left| M_{g_{\max}} \right|}{W_{g}} \leq k_{g}$$

$$W_{g} = \frac{I_{bx_{s}}}{e}$$

e = 5a − 2, 2927a = 2, 7073a

W_g = 5, 6894a³

$$a \geq \sqrt[3]{\frac{\left| M_{g_{\max}} \right|}{5,6894 \bullet k_{g}}} \geq 0,0230\text{\ m}$$

a = 0, 025 m

A = A_I + A_II − A_III = 8a² + a² − a²

     = 5, 0000 • 10⁻³ m²

W_g = 8, 8897 • 10⁻⁵ m³

$$\sigma_{z} = \sigma_{g} \mp \sigma_{\frac{r}{s}} \leq k_{g}$$

$$\sigma_{g} = \frac{\left| M_{g} \right|}{W_{g}}$$

$$\sigma_{\frac{r}{s}} = \frac{P_{\text{ix}}}{A}$$

$$\sigma_{\text{g\ max}}\left| \begin{matrix} \ \\ \ _{x_{1} = 0} \\ \end{matrix} \right.\ = \frac{\left| M_{\text{g\ max}} \right|}{W_{g}} = 8,9992 \bullet 10^{7}\frac{N}{m^{2}} < k_{g}$$

$$\sigma_{\frac{r}{s}\text{\ max}}\left| \begin{matrix} \ \\ \ _{x_{1} = 0} \\ \end{matrix} \right.\ = \frac{6928,2}{A} = 0,1386 \bullet 10^{7}\frac{N}{m^{2}} < k_{g}$$

$$\sigma_{z\text{\ max}}\left| \ _{x_{1} = 0} \right.\ = 9,1378 \bullet 10^{7}\frac{N}{m^{2}} < k_{g}$$

$$E = 2,1 \bullet 10^{11}\frac{N}{m^{2}}$$

I_bxs = 6, 0167 • 10⁻⁶ m⁴

y dla x = 0

$$EI_{\text{bx}}y" = - Mg$$

$$- 2M + P_{1y} \bullet x_{3} - qa\left( x_{3} - \frac{a}{2} \right) + R_{\text{Ay}}\left( x_{3} - a \right) + M\ \ \ \ \ + R_{B}\left( x_{3} - \left( a + b \right) \right)\ - q\frac{\left( x_{3} - \left( a + b \right) \right)^{2}}{2}$$

$$EI_{\text{bx}}y^{''} = - \left\lbrack - 2M + P_{1y} \bullet x_{1} - q\frac{{x_{1}}^{2}}{2}\left| \begin{matrix} \ \\ \ _{I} \\ \end{matrix} \right.\ + R_{\text{Ay}}\left( x_{2} - a \right) + q\frac{\left( x_{2} - a \right)^{2}}{2}\left| \begin{matrix} \ \\ \ _{\text{II}} \\ \end{matrix} \right.\ + M + R_{B}\left( x_{3} - \left( a + b \right) \right)\ - q\frac{\left( x_{3} - \left( a + b \right) \right)^{2}}{2}\left| \begin{matrix} \ \\ \ _{\text{III}} \\ \end{matrix} \right.\ \right\rbrack$$

$$EI_{\text{bx}}y' = - \left\lbrack C - 2Mx_{1} + P_{1y}\frac{{x_{1}}^{2}}{2} - q\frac{{x_{1}}^{3}}{6}\left| \begin{matrix} \ \\ \ _{I} \\ \end{matrix} \right.\ + R_{\text{Ay}}\frac{\left( x_{2} - a \right)^{2}}{2} + q\frac{\left( x_{2} - a \right)^{3}}{6}\left| \begin{matrix} \ \\ \ _{\text{II}} \\ \end{matrix} \right.\ + M\left( x_{3} - \left( a + b \right) \right) + R_{B}\frac{\left( x_{3} - \left( a + b \right) \right)^{2}}{2}\ - q\frac{\left( x_{3} - \left( a + b \right) \right)^{3}}{6}\left| \begin{matrix} \ \\ \ _{\text{III}} \\ \end{matrix} \right.\ \right\rbrack$$

$$EI_{\text{bx}}y = - \left\lbrack D + Cx - 2M\frac{{x_{1}}^{2}}{2} + P_{1y}\frac{{x_{1}}^{3}}{6} - q\frac{{x_{1}}^{4}}{24}\left| \begin{matrix} \ \\ \ _{I} \\ \end{matrix} \right.\ + R_{\text{Ay}}\frac{\left( x_{2} - a \right)^{3}}{6} + q\frac{\left( x_{2} - a \right)^{4}}{24}\left| \begin{matrix} \ \\ \ _{\text{II}} \\ \end{matrix} \right.\ + M\frac{\left( x_{3} - \left( a + b \right) \right)^{2}}{2} + R_{B}\frac{\left( x_{3} - \left( a + b \right) \right)^{3}}{6}\ - q\frac{\left( x_{3} - \left( a + b \right) \right)^{4}}{24}\left| \begin{matrix} \ \\ \ _{\text{III}} \\ \end{matrix} \right.\ \right\rbrack$$

dla x = a,  y = 0 

dla x = a + b,  y = 0 

C = 3, 9695kNm²

D = −0, 9851 kNm³

$$y\left| \begin{matrix} \ \\ \ _{x = 0} \\ \end{matrix} \right.\ = \frac{- 1}{EI_{\text{bx}}}\left\lbrack 1,4655 - 2,1571x - 2M\frac{{x_{1}}^{2}}{2} + P_{1y}\frac{{x_{1}}^{3}}{6} - q\frac{{x_{1}}^{4}}{24}\left| \begin{matrix} \ \\ \ _{I} \\ \end{matrix} \right.\ + R_{\text{Ay}}\frac{\left( x_{2} - a \right)^{3}}{6} + q\frac{\left( x_{2} - a \right)^{4}}{24}\left| \begin{matrix} \ \\ \ _{\text{II}} \\ \end{matrix} \right.\ + M\frac{\left( x_{3} - \left( a + b \right) \right)^{2}}{2} + R_{B}\frac{\left( x_{3} - \left( a + b \right) \right)^{3}}{6}\ - q\frac{\left( x_{3} - \left( a + b \right) \right)^{4}}{24}\left| \begin{matrix} \ \\ \ _{\text{III}} \\ \end{matrix} \right.\ \right\rbrack$$

y = 0, 8 • 10⁻³ m