$\mathbf{\text{wsp}}\mathbf{ol}\mathbf{\text{czynnik}}\mathbf{\ }\mathbf{\text{gi}}\mathbf{e}\mathbf{\text{tno}}\mathbf{-}\mathbf{\text{skr}}\mathbf{e}\mathbf{\text{tny}}\ \alpha = \sqrt{\frac{G*K_{s}}{E*I_{\omega}}}$, $\mathbf{\text{Napr}}\mathbf{ez}\mathbf{\text{enia}}\mathbf{\ }\mathbf{\text{normalne}}\mathbf{\ }\mathbf{\sigma}_{\mathbf{x}} = \frac{M_{y}}{I_{y}}*z - \frac{M_{z}}{I_{z}}*y + \frac{B}{I_{\omega}}*\omega,$ ${\mathbf{\text{napr}}\mathbf{ez}\mathbf{\text{enia}}\mathbf{\ }\mathbf{\text{styczne}}\mathbf{\ }\mathbf{\tau}}_{\mathbf{s}} = \frac{M_{s}}{K_{s}}*\delta$

$$\mathbf{\text{Wydluzenie}}\mathbf{\ }\mathbf{\text{preta}}\mathbf{\ }\mathbf{\text{od}}\mathbf{\ }\mathbf{\text{ciezaru}}\mathbf{\ }\mathbf{\text{wlasnego}}\ \int_{0}^{l}\frac{q*\left( l - x \right)}{E*A}\text{dx}\ $$

$\mathbf{\text{Sily}}\mathbf{\ }\mathbf{w}\mathbf{\ }\mathbf{\text{slupie}}\mathbf{\ }\mathbf{\text{sciskanym}}\mathbf{\ }\Sigma Y = 0;\ \Delta l = \frac{R*l}{E*A};\ \Delta l_{R1} = \Delta l_{R2}\mathbf{\ }\mathbf{\text{Wyznaczyc}}\mathbf{\ }\mathbf{\text{kat}}\mathbf{\ }\mathbf{\text{skrecenia}}\mathbf{\ }\Sigma M_{x} = 0;\ \varphi = \frac{M*L}{G*I}\ \mathbf{\text{Polaczenie}}\mathbf{\ }\mathbf{\text{nitowane}}\mathbf{\ }A = xA_{1};\ \tau = \frac{P}{A} = < k_{t};\ \frac{P}{A} = < d\ \mathbf{\text{Polaczenie}}\mathbf{\ }\mathbf{\text{spawane}}\ \tau = \frac{P}{A} = < k_{t};\frac{P}{A} = < l\ \mathbf{\text{Dopuszczalny}}\mathbf{\ }\mathbf{\text{moment}}\mathbf{\ }\mathbf{\text{zginajacy}}\ W = I*\frac{2}{d};\ \sigma = \frac{( - )M}{W} = < k_{c};\ \sigma = \frac{( + )M}{W} = < k_{r}$

$\mathbf{\text{Skr}}\mathbf{o}\mathbf{\text{cenie}}\mathbf{\ }\mathbf{\text{wymiaru}}\mathbf{\ }\mathbf{i}\mathbf{\ }\mathbf{\text{wydlozenie}}\mathbf{:\ }\varepsilon = \frac{\Delta l}{b},\ \varepsilon_{p} = - \nu\varepsilon,\ \ \nu = - \frac{\varepsilon_{p}}{\varepsilon},\ \sigma = E\varepsilon;\ \varepsilon^{'} = \frac{1}{E}\left( \sigma - \nu\sigma^{'} \right),\ \Delta l^{'} = \varepsilon^{'}*b,\ \varepsilon^{'} = \frac{1}{E}\left( \sigma^{'} - \text{νσ} \right),\ \Delta l^{'} = \varepsilon_{1}^{'}*\text{a\ }\mathbf{\text{Naprezenia}}\mathbf{:\ }\sigma = \frac{N}{A},\ \Delta l_{i} = \frac{N_{i}l_{i}}{E_{i}A_{i}}\mathbf{\ }\mathbf{\text{Pret}}\mathbf{\ }\mathbf{\text{sto}}\mathbf{z}\mathbf{\text{kowy}}\mathbf{:}N\left( x \right) = \frac{1}{3}A\left( x \right)x\gamma,\ \sigma = \frac{N(x)}{A(x)} = \frac{1}{3}x\gamma,\ \varepsilon = \frac{\sigma}{E} = \frac{\gamma}{3E}x,\ \Delta l = \int_{0}^{a}{\varepsilon\left( x \right)dx = \frac{\gamma}{3E}\int_{0}^{a}{xdx =}}\frac{\gamma l^{2}}{\sigma E},\ a = l = > \Delta l = \frac{\gamma l^{2}}{\sigma E}\mathbf{,\ \ Wydluzenie\ preta:}\ \Delta l = \int_{l}^{}{\varepsilon_{x}dx + \alpha l\Delta T + \delta,\ \ \ Ra - Rb = 0,\ \Delta l = 0,\ \Delta l = \frac{\text{Rb}*\text{lo}}{\text{EA}}} + \delta,\ Rb = - \frac{\delta\text{EA}}{\text{Lo}}\ \mathbf{\text{momenty}}\mathbf{\ }\mathbf{\text{bezwladnosci}}\mathbf{:}\ yo = \sum_{i}^{}{yoi,\ \ \ yoi = \text{Ιη}i + A\eta^{2}i}\ $

TENSOR ODKSZTAŁCENIA COUCHEGO: $E_{\text{xx}} = \frac{\partial u_{i}}{\partial x}$; $E_{\text{yy}} = \frac{\partial v}{\partial y}$ ; $E_{\text{zz}} = \frac{\partial w}{\partial z}$ ; $E_{\text{xy}} = \frac{1}{2}\left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right);$ $E_{\text{xz}} = \frac{1}{2}\left( \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \right);\ E_{\text{yz}} = \frac{1}{2}\left( \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} \right);\mathbf{\ }\mathbf{\text{Sztywny}}\mathbf{\ }\mathbf{\text{blat}}\ :\ N_{1} = \frac{P}{4}\left( 1 + \frac{- 2s - r}{a} \right),\ N_{2} = \frac{P}{4}\left( 1 + \frac{2s - r}{a} \right),\ N_{3} = \frac{P}{4}\left( 1 + \frac{2s + r}{a} \right),\ N_{4} = \frac{P}{4}\left( 1 + \frac{- 2s + r}{a} \right)\text{\ Pr}e\text{ty\ podg}zane:\ R_{2} = - \propto t*EA,\ \sigma = - t*E,\ Naprez\text{enie\ w\ s}lupie\ AB:R_{1} + R_{2} - P = 0,\ l = 0,\ \left( \frac{R_{2}\frac{1}{3}l}{\text{EA}} + \frac{\left( R_{2} - P \right)\frac{2}{3}l}{\text{EA}} \right) + R\alpha = 0,\mathbf{\ }\mathbf{\text{po}}\mathbf{la}\mathbf{\text{czenie}}\mathbf{\ }\mathbf{\text{nitowane}}\mathbf{:}\ \tau = \frac{P}{\text{Amn}} \leq k_{t}^{n},\ \sigma_{d} = \frac{P}{A_{d}n} \leq k_{d}^{b},\ g_{\min}\left\{ 2g_{2},\ g_{1} \right\},\ \sigma_{d},\ A_{d} = d_{n}*g_{\min},\ \sigma = \frac{P}{\text{Anetto}},\ Anetto = l*g_{a},\ \ \mathbf{\text{Pod}}\mathbf{\ }\mathbf{\text{jakim}}\mathbf{\ }\mathbf{\text{katem}}\mathbf{\ }\mathbf{\text{nalezy}}\mathbf{\ }\mathbf{\text{wykonac}}\mathbf{\ }\mathbf{\text{spoine}}\mathbf{\ }\mathbf{\sigma}_{\mathbf{0}} = \frac{P}{b*g},\ \sigma = \sigma_{0}*\sin\varphi,\ \sigma_{y} = \sigma\sin\varphi = \frac{P\sin^{2}\varphi}{\text{bg}} \leq k_{r}^{s},\ \tau_{\varphi} = \sigma\cos\varphi = \frac{\text{Psin}\varphi\cos\varphi}{\text{bg}}\ \leq k_{r}^{s},\ \mathbf{\text{Polaczenie}}\mathbf{\ }\mathbf{\text{drewniane}}\mathbf{:}\ \tau_{1 - 1} = \frac{P}{l_{2}b} \leq k_{t}^{s} \rightarrow l_{2,},\ \tau_{2 - 2} = \frac{P}{l_{1}b} \leq k_{t}^{d},\ docisk:\ \sigma_{d} = \frac{P}{A_{d}} = \frac{P}{\text{tb}} \leq k_{t}^{s} \rightarrow t..,\ rozciaganie:\ \sigma = \frac{P}{bh_{2}} \leq k_{r}^{s} \rightarrow h,\ \sigma = \frac{P}{bh_{1}} \leq k_{r}^{d} \rightarrow h,\ \mathbf{\text{wyznacz}}\mathbf{\ }\mathbf{\text{srednice}}\mathbf{\ }\mathbf{\text{belki}}\mathbf{\ }\mathbf{\text{kolowej}}\mathbf{:}\ I_{y} = \frac{\pi d^{4}}{64},\ \sigma_{x} = \frac{\text{Mmax}}{\text{Wy}},\ II\ czesc:\ A_{o} = \frac{\pi d^{2}}{4},\ \frac{A_{1}}{A_{0}}$ extr. napr. stycz. i kąt skr. przek. I₀=$\frac{\pi d^{4}}{32}$; W₀=$\frac{2}{d}$*I₀; τ₁=$\frac{Mx_{1}}{W_{1}}$; ₀=$\int_{0}^{l}\frac{\text{Mx}\left( x \right)\text{dx}}{GI_{0}(x)}$

porów. noś. i kąt skr. $\frac{d_{2w}}{d_{2z}}$=…=>d_2w = …; $\frac{\pi{d_{1}}^{2}}{4}$=$\frac{\pi}{4}$(d_2z² − d_2w²); $\frac{M_{2}}{M_{1}}$=$\frac{W_{2}*k_{t}}{W_{1}*k_{t}}$; W₁=$\frac{\pi{d_{1}}^{3}}{16}$; W₂=$\frac{I_{2}*2}{d_{2z}}$=$\frac{\pi*2}{32*d_{2z}}$(d_2z⁴ − d_2w⁴); I₂=I_z-I_w; $\frac{_{2}}{_{1}}$=$\frac{M*l}{G*I_{2}}$*$\frac{G{*I}_{1}}{M*l}$=$\frac{I_{1}}{I_{2}}$

wyzn. M dop. M=m*a; I₀=$\frac{\pi*d^{4}}{32}$

napr. stycz. max i kąt skrec. M_s(x)=m_x; M_smax=m*l; τ=$\frac{M_{s}}{W_{s}}$; W_s=$\frac{k_{s}}{\sigma}$; k_s=$\frac{1}{3}$n*Σσ_i³*h_i; τ_max=$\frac{M_{s}\max*\sigma}{k_{s}}$