Wytrzymałość materiałów II

Prowadzący:

dr inż. Piotr Paczos

Wykonał:

Jakub Gałka

Przemysław Poszwa

Założenie początkowe:

P_A = P_dod = 0 ∖ nM_E = M_dod = 0

Równania równowagi:

$$\sum_{}^{}{F_{x} = 0\ \overset{\Leftrightarrow}{}\ H_{E} + \ P_{A}}\ \ H_{E} = \ - P_{A}$$

$$\sum_{}^{}{F_{y} = 0\ \overset{\Leftrightarrow}{}\ R_{E} + \ R_{A} - 2aq = 0}\ $$

$$\sum_{}^{}{M_{E} = 0\ \overset{\Leftrightarrow}{}\ M_{E} + \ P_{A}*a - R_{A}*a + 2aq*a = 0}\ $$

Równania momentów:

0  ≤  x₁  ≤ a

$$\mathbf{M}\left( \mathbf{x}_{\mathbf{1}} \right) = \ - R_{A}*\ x_{1} = \ \left( \ - \ \frac{M_{E}}{a} - \ P_{A} - 2aq \right)*\ x_{1}$$

0  ≤  x₂  ≤ a

$$\mathbf{M}\left( \mathbf{x}_{\mathbf{2}} \right) = \ - R_{A}*a\ + \ P_{A}*\ x_{2} = a*\left( \ - \ \frac{M_{E}}{a} - \ P_{A} - 2aq \right) + \ P_{A}*\ x_{2}$$

0  ≤  x₃  ≤ 2a

M(x₃) =  M_E −  H_E * x₃ =  M_E +  P_A * x₃

0  ≤  x₄  ≤ 2a

$$\mathbf{M}\left( \mathbf{x}_{\mathbf{4}} \right) = \ - R_{E}*x_{4} + \ M_{E} - \ H_{E}*2a - \ \frac{q{x_{4}}^{2}}{2} = \ M_{E} + 2a*\ P_{A} + \left( \ \frac{M_{E}}{a} + \ P_{A}\ \right)*x_{4} - \ \frac{q{x_{4}}^{2}}{2}$$

Pochodne momentów gnących względem siły dodanej P_A:

$$\frac{\mathbf{\partial M(}\mathbf{x}_{\mathbf{1}}\mathbf{)}}{\mathbf{P}_{\mathbf{A}}}\mathbf{= \ } - x_{1}$$

$$\frac{\mathbf{\partial M(}\mathbf{x}_{\mathbf{2}}\mathbf{)}}{\mathbf{P}_{\mathbf{A}}}\mathbf{=}\ - a\ + \ x_{2}$$

$$\frac{\mathbf{\partial M(}\mathbf{x}_{\mathbf{3}}\mathbf{)}}{\mathbf{P}_{\mathbf{A}}}\mathbf{=}x_{3}$$

$$\frac{\mathbf{\partial M(}\mathbf{x}_{\mathbf{4}}\mathbf{)}}{\mathbf{P}_{\mathbf{A}}}\mathbf{=}\ 2a\ + \ x_{4}$$

Pochodne momentów gnących względem momentu dodanego M_E:

$$\frac{\mathbf{\partial M}\mathbf{(x}_{\mathbf{1}}\mathbf{)}}{\mathbf{M}_{\mathbf{E}}}\mathbf{=}\ \frac{- x_{1}}{a}$$

$$\frac{\mathbf{\partial M}\mathbf{(x}_{\mathbf{2}}\mathbf{)}}{\mathbf{M}_{\mathbf{E}}}\mathbf{=}\ - 1$$

$$\frac{\mathbf{\partial M}\mathbf{(x}_{\mathbf{3}}\mathbf{)}}{\mathbf{M}_{\mathbf{E}}}\mathbf{=}1$$

$$\frac{\mathbf{\partial M}\mathbf{(x}_{\mathbf{4}}\mathbf{)}}{\mathbf{M}_{\mathbf{E}}}\mathbf{=}\ 1 + \ \frac{x_{4}}{a}$$

Metoda energetyczna:

$\mathbf{u}_{\mathbf{A}}^{\mathbf{\text{poz}}}\mathbf{=}\frac{\mathbf{\partial V}}{\mathbf{\partial P}_{\mathbf{A}}}\mathbf{\ } = \ \frac{1}{\text{EI}}\left( \int_{0}^{a}{M\left( x_{1} \right)*\frac{\partial M\left( x_{1} \right)}{\partial P_{A}}}dx_{1} + \int_{0}^{a}{M\left( x_{2} \right)*\ \frac{\partial M\left( x_{2} \right)}{\partial P_{A}}}dx_{2} + \ + \ \int_{0}^{2a}{M\left( x_{3} \right)*\ \frac{\partial M\left( x_{3} \right)}{\partial P_{A}}}dx_{3} + \ \int_{0}^{2a}{M\left( x_{4} \right)*\ \frac{\partial M\left( x_{4} \right)}{\partial P_{A}}}dx_{4}\ \right)$

$\mathbf{\theta}_{\mathbf{E}}\mathbf{= \ }\frac{\mathbf{\partial V}}{\mathbf{\partial M}_{\mathbf{E}}}\ = \frac{1}{\text{EI}}\left( \int_{0}^{a}{M\left( x_{1} \right)*\frac{\partial M\left( x_{1} \right)}{\partial M_{E}}}dx_{1} + \int_{0}^{a}{M\left( x_{2} \right)*\ \frac{\partial M\left( x_{2} \right)}{\partial M_{E}}}dx_{2} + \ + \ \int_{0}^{2a}{M\left( x_{3} \right)*\ \frac{\partial M\left( x_{3} \right)}{\partial M_{E}}}dx_{3} + \ \int_{0}^{2a}{M\left( x_{4} \right)*\ \frac{\partial M\left( x_{4} \right)}{\partial M_{E}}}dx_{4}\ \right)\ $

Obliczenie przemieszczenia punktu A z wykorzystaniem metody energetycznej:

$\mathbf{u}_{\mathbf{A}}^{\mathbf{\text{poz}}} = \frac{\partial V}{{\partial P}_{A}}\ = \ \frac{1}{\text{EI}}\ \left\lbrack \ \left( \int_{0}^{a}{\left\lbrack \left( \ - \ \frac{M_{E}}{a} - \ P_{A} - 2aq \right)*\ x_{1} \right\rbrack*\left( - x_{1} \right)}dx_{1} + \ \int_{0}^{a}{\left\lbrack a*\left( \ - \ \frac{M_{E}}{a} - \ P_{A} - 2aq \right) + \ P_{A}*\ x_{2}\ \right\rbrack*\left( - a\ + \ x_{2} \right)}dx_{2}\ + \ \int_{0}^{2a}{\left( M_{E} + \ P_{A} \right)*\ x_{3}}\text{\ d}x_{3} + \ \int_{0}^{2a}{\left\lbrack \ M_{E} + 2a*\ P_{A} + \left( \ \frac{M_{E}}{a} + \ P_{A}\ \right)*x_{4} - \ \frac{q{x_{4}}^{2}}{2}\ \right\rbrack*\left( 2a\ + \ x_{4} \right)}dx_{4}\ \right)\ \right\rbrack = \frac{27a^{2}*M_{E}}{2EI} + \ \frac{58a^{3}*P_{A}}{3EI} + \frac{2a*\left( M_{E} + P_{A} \right)}{\text{EI}} - \ \frac{3a^{4}q}{\text{EI}}$

z założenia:

$\left. \ \begin{matrix} P_{A} = P_{\text{dod}} = 0 \\ M_{E} = \ M_{\text{dod}} = 0 \\ \end{matrix} \right\}\ \rightarrow \ u_{A}^{\text{poz}} = \frac{\partial V}{{\partial P}_{A}} = \ - \ \frac{3a^{4}q}{\text{EI}}$ }zał. ycznej:omentu dodanego

Obliczenie kątu obrotu w punkcie E z wykorzystaniem metody energetycznej:

$\mathbf{\theta}_{\mathbf{E}} = \frac{\partial V}{{\partial M}_{E}}\ = \ \frac{1}{\text{EI}}\ \left\lbrack \ \left( \int_{0}^{a}{\left\lbrack \left( \ - \ \frac{M_{E}}{a} - \ P_{A} - 2aq \right)*\ x_{1} \right\rbrack*\left( \frac{- x_{1}}{a} \right)}dx_{1} + \ \int_{0}^{a}{\left\lbrack a*\left( \ - \ \frac{M_{E}}{a} - \ P_{A} - 2aq \right) + \ P_{A}*\ x_{2}\ \right\rbrack*\left( - 1 \right)}dx_{2} + \ + \ \int_{0}^{2a}{\left( M_{E} + \ P_{A} \right)*\ 1}\text{\ d}x_{3} + \ \int_{0}^{2a}{\left\lbrack \ M_{E} + 2a*\ P_{A} + \left( \ \frac{M_{E}}{a} + \ P_{A}\ \right)*x_{4} - \ \frac{q{x_{4}}^{2}}{2}\ \right\rbrack*\left( 1 + \ \frac{x_{4}}{a} \right)}dx_{4}\ \right)\ \right\rbrack = \frac{10a*M_{E}}{\text{EI}} + \frac{27a^{2}*P_{A}}{2EI} + \ \frac{2a*\left( M_{E} + P_{A} \right)}{\text{EI}} - \ \frac{2a^{3}q}{3EI}$

z założenia:

$\left. \ \begin{matrix} P_{A} = P_{\text{dod}} = 0 \\ M_{E} = \ M_{\text{dod}} = 0 \\ \end{matrix} \right\}\ \rightarrow \ \theta_{E} = \frac{\partial V}{{\partial M}_{E}} = \ - \ \frac{2a^{3}q}{3EI}$ }zał. ycznej:omentu dodanego

Odpowiedź:

$$\mathbf{u}_{\mathbf{A}}^{\mathbf{\text{poz}}} = \ - \ \frac{3a^{4}q}{\text{EI}}$$

$$\mathbf{\theta}_{\mathbf{E}}\mathbf{=}\ - \ \frac{2a^{3}q}{3EI}$$

Uwaga:

Wartości ujemne ugięcia oraz kąta ugięcia wynikają z założenia ugięcia belki - odwrotnego do rzeczywistego.