$$ \begin{aligned} R_2 &= W, \quad M_x = -Wx, \quad M_{\max} = -W\ell \\[8pt] \delta_x &= \frac{W\ell^3}{3EI} \left(1 - \frac{3x}{2\ell} + \frac{x^3}{2\ell^3}\right) \\[8pt] \delta_{\max} &= \frac{W\ell^3}{3EI} = \frac{2fb\ell^2}{3Eh} \qquad \text{--- 자유단에서} \\[8pt] i_x &= -\frac{W\ell^2}{2EI}\left(1 - \frac{x^2}{\ell^2}\right) \\[8pt] i_{\max} &= -\frac{W\ell^2}{2EI} = -\frac{3}{2\ell}\delta_{\max} \qquad \text{--- 자유단에서} \end{aligned} $$

$$ \begin{aligned} R_2 &= W, \quad M_x = -W(x - \ell_1), \quad M_{\max} = -W\ell_2 \\[8pt] \delta_x &= \frac{W\ell_2^3}{3EI} \left\{ 1 - \frac{3(x-\ell_1)}{2\ell_2} + \frac{(x-\ell_1)^3}{2\ell_2^3} \right\} \\[8pt] \delta_C &= \frac{W\ell_2^3}{3EI} \\[8pt] \delta_{\max} &= \frac{W\ell_2^3}{3EI} \left\{ 1 + \frac{3\ell_1}{2\ell_2} \right\} \qquad \text{--- 자유단에서} \\[8pt] i_x &= -\frac{W\ell_2^2}{2EI} \left\{ 1 - \frac{(x-\ell_1)^2}{\ell_2^2} \right\} \\[8pt] i_{\max} &= -\frac{W\ell_2^2}{2EI} \qquad \text{--- C 점에서 자유단까지 균일} \end{aligned} $$

$$ \begin{aligned} R_2 &= w\ell, \quad M_x = -\frac{wx^2}{2}, \quad M_{\max} = -\frac{w\ell^2}{2} \\[8pt] \delta_x &= \frac{w\ell^4}{8EI} \left(1 - \frac{4x}{3\ell} + \frac{x^4}{3\ell^4}\right) \\[8pt] \delta_{\max} &= \frac{w\ell^4}{8EI} = \frac{fb\ell^2}{2Eh} \qquad \text{--- 자유단에서} \\[8pt] i_x &= -\frac{w\ell^3}{6EI}\left(1 - \frac{x^3}{\ell^3}\right) \\[8pt] i_{\max} &= -\frac{w\ell^3}{6EI} = -\frac{4}{3\ell}\delta_{\max} \qquad \text{--- 자유단에서} \end{aligned} $$

$$ \begin{aligned} R_2 &= \overline{W}, \quad M_x = -\frac{\overline{W} x^3}{3\ell^2}, \quad M_{\max} = -\frac{\overline{W} \ell}{3} \\[8pt] \delta_x &= \frac{\overline{W} \ell^3}{15EI} \left( 1 - \frac{5x}{4\ell} + \frac{x^5}{4\ell^5} \right) \\[8pt] \delta_{\max} &= \frac{\overline{W} \ell^3}{15EI} = \frac{2f\ell^2}{5Eh} \qquad \text{--- 자유단에서} \\[8pt] i_x &= -\frac{\overline{W} \ell^2}{12EI} \left( 1 - \frac{x^4}{\ell^4} \right) \\[8pt] i_{\max} &= -\frac{\overline{W} \ell^2}{12EI} \qquad \text{--- 자유단에서} \end{aligned} $$

$$ \begin{aligned} R_1 &= R_2 = \frac{W}{2}, \quad M_{\max} = \frac{W\ell}{4} \\[8pt] M_x &= \frac{Wx}{2} \quad (0 \le x \le \ell/2), \\ M_x &= \frac{W(\ell - x)}{2} \quad (\ell/2 \le x \le \ell) \\[8pt] \delta_x &= \frac{W\ell^3}{48EI} \left( \frac{3x}{\ell} - \frac{4x^3}{\ell^3} \right) \\ \delta_{\max} &= \frac{W\ell^3}{48EI} = \frac{f\ell^2}{6Eh} \qquad \text{--- 중앙에서} \\[8pt] \end{aligned} $$

$$ \begin{aligned} i_x &= -\frac{W\ell^2}{16EI} \left( 1 - \frac{4x^2}{\ell^2} \right) \\[8pt] i_{\max} &= \pm \frac{W\ell^2}{16EI} = \pm \frac{3}{\ell} \delta_{\max} \qquad \text{--- 양단에서} \end{aligned} $$

$$ \begin{aligned} R_1 &= \frac{W\ell_2}{\ell}, \quad R_2 = \frac{W\ell_1}{\ell}, \quad M_{\max} = \frac{W\ell_1\ell_2}{\ell} \\[8pt] M_x &= \frac{W\ell_2 x}{\ell} \quad (0 \le x \le \ell_1) \\[8pt] M_x &= \frac{W\ell_1(\ell - x)}{\ell} \quad (\ell_1 \le x \le \ell) \\[8pt] \end{aligned}

$$ \begin{aligned} \delta_{x_1} &= \frac{W\ell_1^2\ell_2^2}{6EI\ell} \left( \frac{2x}{\ell_1} + \frac{x}{\ell_2} - \frac{x^3}{\ell_1^2\ell_2} \right) \qquad (0 \le x \le \ell_1) \\[8pt] \delta_{x_2} &= \frac{W\ell_1^2\ell_2^2}{6EI\ell} \left( \frac{2(\ell-x)}{\ell_2} + \frac{(\ell-x)}{\ell_1} - \frac{(\ell-x)^3}{\ell_1\ell_2^2} \right) \qquad (\ell_1 \le x \le \ell) \\ \delta_c &= \frac{W\ell_1^2\ell_2^2}{3EI\ell} \\ \delta_{\max} &= \frac{W\ell_2}{3EI\ell} \left\{ \frac{\ell_1(\ell+\ell_2)}{3} \right\}^{\frac{3}{2}} \qquad \left( x_1 = \ell_1 \sqrt{\frac{1}{3} + \frac{2\ell_2}{3\ell_1}} \right) \\[8pt] \delta_{\max} &= \frac{W\ell_1}{3EI\ell} \left\{ \frac{\ell_2(\ell+\ell_1)}{3} \right\}^{\frac{3}{2}} \qquad \left( x_2 = \ell_2 \sqrt{\frac{1}{3} + \frac{2\ell_1}{3\ell_2}} \right) \end{aligned} $$

$$ \begin{aligned} R_1 &= R_2 = W \\[8pt] M_x &= -W\ell_1 \qquad \text{--- } \ell_2 \text{ 구간일정} \\[8pt] \end{aligned} $$

$$ \begin{aligned} 0 \le x \le \ell_1 \text{ 에서} \quad \delta &= \frac{W\ell_1^3}{6EI} \left\{ \frac{x^3}{\ell_1^3} - \frac{3(\ell_1+\ell_2)}{\ell_1^2}x + \frac{3\ell_2}{\ell_1} + 2 \right\} \\[8pt] \ell_1 \le x \le (\ell_1 + \ell_2) \text{ 에서} \quad \delta &= \frac{W\ell_1^3}{6EI} \left\{ \frac{x^3}{\ell_1^3} - \frac{3(\ell_1+\ell_2)}{\ell_1^2}x + \frac{3\ell_2}{\ell_1} + 2 \right\} - \frac{W(x-\ell_1)^3}{6EI} \\[8pt] \delta_1 &= \frac{W\ell_1^3}{6EI} \left( \frac{3\ell_2}{\ell_1} + 2 \right), \quad \delta_2 = -\frac{W\ell_1^3}{8EI} \end{aligned} $$

$$ \begin{aligned} R_1 &= R_2 = \ell w / 2 \\[8pt] M_x &= (wx/2) \cdot (\ell - x), \quad M_{\max} = w\ell^2 / 8 \\[8pt] \delta_x &= \frac{w\ell^4}{24EI} \left( \frac{x}{\ell} - \frac{2x^3}{\ell^3} + \frac{x^4}{\ell^4} \right) \\[8pt] \delta_{\max} &= \frac{5w\ell^4}{384EI} = \frac{5fb\ell^2}{24Eh} \\[8pt] i_x &= -\frac{w\ell^3}{24EI} \left( 1 - \frac{6x^2}{\ell^2} + \frac{4x^3}{\ell^3} \right) \\[8pt] i_{\max} &= -\frac{\pm W\ell^2}{24EI} = \frac{\pm 3.2}{\ell} \delta_{\max} \qquad \text{--- 양단 에서} \end{aligned} $$

$$ \begin{aligned} R_1 &= R_2 = \ell w / 2 \\[8pt] M_x &= -\frac{w}{2} \cdot \{ x \cdot (x - \ell) + \ell \cdot \ell_1 \} \quad (\ell_1 \le x \le \ell_1 + \ell_2 \text{ 에서})\\[8pt] M_A &= M_B = -\frac{w\ell_1^2}{2}, \quad M_C = -\frac{w\ell^2}{2} \left( \frac{1}{4} - \frac{\ell_1}{\ell} \right), \quad x = \frac{\ell}{2} \\[8pt] \ell_1 &\lt \left( \sqrt{\frac{1}{2}} - \frac{1}{2} \right) \ell \text{ 일때 } M_C \text{ 최대} \\[8pt] \ell_1 &> \left( \sqrt{\frac{1}{2}} - \frac{1}{2} \right) \ell \text{ 일때 } M_A, M_B \text{ 최대} \\[8pt] \end{aligned} $$

$$ \begin{aligned} & 0 \le x \le \ell_1 \text{ 에서} \\ & \delta = \frac{w\ell^4}{24EI} \left\{ \left( \frac{6\ell_1^3}{\ell^3} - \frac{6\ell_1}{\ell^2}x + \frac{1}{\ell} \right)x + \frac{x^4}{\ell^4} - \frac{\ell_1^4}{\ell^4} - \frac{6\ell_1^3}{\ell^3} + \frac{6\ell_1^2}{\ell^2} - \frac{\ell_1}{\ell} \right\} \\[8pt] & \ell_1 \le x \le \ell_1 + \ell_2 \text{ 에서} \\ & \delta = \frac{w\ell^4}{24EI} \left\{ \left( 1 - \frac{6\ell_1}{\ell} \right)\frac{x}{\ell} + \frac{6\ell_1 x^2}{\ell^3} - \frac{2x^3}{\ell^3} + \frac{x^4}{\ell^4} - \frac{\ell_1^4}{\ell^4} + \frac{4\ell_1^3}{\ell^3} + \frac{6\ell_1^2}{\ell^2} - \frac{\ell_1}{\ell} \right\} \\[8pt] & \delta_0 = \frac{w\ell_1}{24EI} (3\ell_1^3 + 6\ell_1^2\ell^2 - \ell_2^3), \quad \delta_{\ell/2} = \frac{w\ell_2^2}{384EI} (5\ell_2^2 - 24\ell_1^2) \end{aligned} $$

$$ \begin{aligned} R_1 &= \frac{w\ell_1}{\ell} \left( \ell_3 + \frac{\ell_1}{2} \right), \quad R_2 = \frac{w\ell_1}{\ell} \left( \ell_2 + \frac{\ell_1}{2} \right) \\[8pt] \ell_2 \le x \le (\ell_2 &+ \ell_1) \text{ 에서} \\ M_x &= w \left\{ \frac{\ell_1}{\ell} \left( \ell_3 + \frac{\ell_1}{2} \right)x - \frac{(x - \ell_2)^2}{2} \right\} \\[8pt] \end{aligned} $$

$$ \begin{aligned} x &= \ell_2 + \frac{\ell_1}{\ell} \left( \ell_3 + \frac{\ell_1}{2} \right) \text{ 에서} \\ M_{\max} &= \frac{w\ell_1}{\ell} \left( \ell_3 + \frac{\ell_1}{2} \right) \left( \ell_2 + \frac{2\ell_1\ell_3 + \ell_1^2}{4\ell} \right) \\[8pt] \delta_x &= \frac{w}{6EI} \left[ \frac{\ell_1 x}{\ell} \left( \ell_3 + \frac{\ell_1}{2} \right) \left\{ \left( \ell_2 + \frac{\ell_1}{2} \right) \times \left( \ell + \ell_3 + \frac{\ell_1}{2} \right) - \frac{\ell_1^2}{4} - x^2 \right\} + \frac{(x - \ell_2)^4}{4} \right] \end{aligned} $$

$$ \begin{aligned} R_1 &= \frac{w}{2\ell} (2\ell\ell_2 + \ell_3^2 - \ell_2^2) \\[8pt] R_2 &= \frac{w}{2\ell} (2\ell\ell_3 + \ell_2^2 - \ell_3^2) \end{aligned} $$

$$ \begin{aligned} &(\ell_2 + \ell_1) \le x_3 \le \ell \text{ 에서} \\ & M_{x_3} = \frac{w}{2\ell} \left[ -\ell\ell_1(\ell_1 + 2\ell_2) + \{ 2\ell(\ell_1 + \ell_2) + (\ell_3^2 - \ell_2^2) \} x_3 - \ell x_3^2 \right] \\[8pt] & \ell_2 \lt \ell_3 \text{ 일때, } \quad x_3 = (\ell_1 + \ell_2) + \frac{\ell_3^2 - \ell_2^2}{2\ell} \text{ 에서} \\ & M_{\max} = \frac{w}{8\ell^2} \left[ -4\ell\ell_3 \{ \ell_3(\ell_1 + \ell_2) + \ell_2^2 \} + (\ell_3^2 - \ell_2^2)^2 \right] \\[8pt] & \delta_{x_3} = \frac{w}{24EI\ell} \left[ \ell \{ (\ell_1 + \ell_2)^4 - \ell_2^4 \} + \{ 2\ell^2(\ell_3^2 + 2\ell_2^2) \right. \\ & \qquad \left. - 4\ell(\ell_1 + \ell_2)^3 - (\ell_3^4 - \ell_2^4) \} x_3 + 6\ell \{ (\ell_1 + \ell_2)^2 - \ell_2^2 \} x_3^2 \right. \\ & \qquad \left. - 2 \{ 2\ell(\ell_1 + \ell_2) + (\ell_3^2 - \ell_2^2) \} x_3^3 + \ell x_3^4 \right] \end{aligned} $$

$$ \begin{aligned} R_1 &= \frac{\overline{W}}{3}, \quad R_2 = \frac{2\overline{W}}{3} \\[8pt] M_x &= \frac{\overline{W} x}{3} \left( 1 - \frac{x^2}{\ell^2} \right) \\[8pt] M_{\max} &= \frac{2}{9\sqrt{3}} \overline{W} \ell = 0.128 \overline{W} \ell, \quad x = 0.5774\ell \text{ 에서} \\[8pt] \delta_x &= \frac{\overline{W} \ell^3}{180EI} \left( \frac{7x}{\ell} - \frac{10x^3}{\ell^3} + \frac{3x^5}{\ell^5} \right) \\[8pt] x &= \ell \sqrt{1 - \sqrt{8/15}} = 0.5193\ell \text{ 에서} \\ \delta_{\max} &= 0.01304 \frac{\overline{W} \ell^3}{EI} \end{aligned} $$

$$ \begin{aligned} R_1 &= R_2 = \frac{\overline{W}}{2} \\[8pt] M_x &= \overline{W}x \left( \frac{1}{2} - \frac{x}{\ell} + \frac{2x^2}{3\ell^2} \right) \\[8pt] M_{\max} &= \frac{\overline{W}\ell}{12} \\[8pt] \delta_x &= \frac{\overline{W}\ell^3}{12EI} \left( \frac{3x}{8\ell} - \frac{x^3}{\ell^3} + \frac{x^4}{\ell^4} - \frac{2x^5}{5\ell^5} \right) \\[8pt] \delta_{\max} &= \frac{3\overline{W}\ell^3}{320EI} = \frac{9fb\ell^2}{40Eh} \end{aligned} $$

$$ \begin{aligned} R_1 &= R_2 = \frac{\overline{W}}{2} \\[8pt] M_x &= \overline{W}x \left( \frac{1}{2} - \frac{2x^2}{3\ell^2} \right) \\[8pt] M_{\max} &= \frac{\overline{W}\ell}{6} \\[8pt] \delta_x &= \frac{\overline{W}\ell^3}{12EI} \left( \frac{3x}{8\ell} - \frac{x^3}{\ell^3} + \frac{2x^5}{5\ell^5} \right) \\[8pt] \delta_{\max} &= \frac{\overline{W}\ell^3}{60EI} = \frac{fb\ell^2}{5Eh} \end{aligned} $$

$$ \begin{aligned} R_1 &= R_2 = \frac{W}{2} \\[8pt] M_x &= \frac{W\ell}{2} \left( \frac{x}{\ell} - \frac{1}{4} \right), \quad M_{x_1} = -\frac{W\ell}{2} \left( \frac{x_1}{\ell} - \frac{3}{4} \right)\\ M_{\max} &= \pm \frac{W\ell}{8} \\[8pt] \delta_x &= \frac{W\ell^3}{16EI} \left( \frac{x^2}{\ell^2} - \frac{4x^3}{3\ell^3} \right) \\[8pt] \delta_{\max} &= \frac{W\ell^3}{192EI} = \frac{fb\ell^2}{12Eh} \qquad \text{--- 중앙에서} \\[8pt] i_x &= \frac{W\ell^2}{8EI} \left( \frac{x}{\ell} - \frac{2x^2}{\ell^2} \right) \end{aligned} $$

$$ \begin{aligned} R_1 &= \frac{W\ell_2^2(3\ell_1 + \ell_2)}{\ell^3}, \quad R_2 = \frac{W\ell_1^2(\ell_1 + 3\ell_2)}{\ell^3} \\[8pt] M_A &= \frac{W\ell_1\ell_2^2}{\ell^2}, \quad M_B = \frac{W\ell_1^2\ell_2}{\ell^2} \\[8pt] M_x &= -\frac{W\ell_1\ell_2^2}{\ell^2} + \frac{W\ell_2^2 x(3\ell_1 + \ell_2)}{\ell^3} \\[8pt] M_{x_1} &= \frac{W\ell_1^2 x(\ell_1 + 2\ell_2)}{\ell^2} - \frac{W\ell_1^2 x_1(\ell_1 + 3\ell_2)}{\ell^3} \\[8pt] M_C &= \frac{W\ell_1^2\ell_2^2}{\ell^3} \\[8pt] \end{aligned} $$

$$ \begin{aligned} \delta_x &= \frac{W\ell_2^2 x^2}{6EI\ell} \left\{ \frac{3\ell_1}{\ell} - \frac{(3\ell_1 + \ell_2)x}{\ell^2} \right\} \\[8pt] \delta_{x_1} &= \frac{W\ell_2^2 x_1^2}{6EI\ell} \left\{ \frac{3\ell_1}{\ell} - \frac{(3\ell_1 + \ell_2)x_1}{\ell^2} \right\} + \frac{W(x_1 - \ell_1)^3}{6EI} \\[8pt] \delta_C &= \frac{W\ell_1^3\ell_2^3}{3EI\ell^3} \\[8pt] \ell_1 &> \ell_2 \text{ 에서} \quad \delta_{\max} = \frac{2W\ell_1^3\ell_2^2}{3EI(3\ell_1 + \ell_2)^2}, \quad x = \frac{\ell^2}{\ell_1 + 3\ell_2} \\[8pt] \ell_1 &\lt \ell_2 \text{ 에서} \quad \delta_{\max} = \frac{2W\ell_1^2\ell_2^3}{3EI(\ell_1 + 3\ell_2)^2}, \quad x = \frac{2\ell_1\ell}{3\ell_1 + \ell_2} \end{aligned} $$

$$ \begin{aligned} R_1 &= R_2 = \frac{w\ell}{2} \\[8pt] M_x &= \frac{w\ell^2}{2} \left( -\frac{1}{6} + \frac{x}{\ell} - \frac{x^2}{\ell^2} \right) \\[8pt] M_A &= M_B = -\frac{w\ell^2}{12} \\[8pt] M_C &= \frac{w\ell^2}{24} \\[8pt] \delta_x &= \frac{w\ell^4}{24EI} \left( \frac{x^2}{\ell^2} - \frac{2x^3}{\ell^3} + \frac{x^4}{\ell^4} \right) \\[8pt] \delta_{\max} &= \frac{w\ell^4}{384EI} \end{aligned} $$

$$ \begin{aligned} R_1 &= \frac{5W}{16}, \quad R_2 = \frac{11W}{16} \\[8pt] M_x &= \frac{5Wx}{16}, \quad M_C = \frac{5W\ell}{32} \\[8pt] M_{x_1} &= W\ell \left( \frac{1}{2} - \frac{11x_1}{16\ell} \right), \quad M_{\max} = -\frac{3W\ell}{16} \\[8pt] \delta_x &= \frac{W\ell^3}{32EI} \left( \frac{x}{\ell} - \frac{5x^3}{3\ell^3} \right) \\[8pt] \delta_{x_1} &= \frac{W\ell^3}{32EI} \left( -\frac{2}{3} + \frac{5x_1}{\ell} - \frac{8x_1^2}{\ell^2} + \frac{11x_1^3}{3\ell^3} \right) \\[8pt] \delta_C &= \frac{7W\ell^3}{768EI}, \quad \delta_{\max} = \sqrt{\frac{1}{5}} \cdot \frac{W\ell^3}{48EI}, \quad x = \ell \sqrt{\frac{1}{5}} \text{ 에서} \end{aligned} $$

$$ \begin{aligned} R_1 &= \frac{W\ell_2^2}{2\ell^3}(3\ell_1 + 2\ell_2), \quad R_2 = \frac{W\ell_1}{2\ell^3}(2\ell_1^2 + 6\ell_1\ell_2 + 3\ell_2^2) \\[8pt] M_x &= \frac{W\ell_2^2(3\ell_1 + 2\ell_2)x}{2\ell^3} \\[8pt] M_{x_1} &= \frac{W\ell_2^2}{2\ell^3}(3\ell_1 + 2\ell_2)x_1 - W(x_1 - \ell_1) \\[8pt] M_B &= -\frac{W\ell_1\ell_2}{2\ell^2}(2\ell_1 + \ell_2), \quad M_C = \frac{W\ell_1\ell_2^2}{2\ell^3}(3\ell_1 + 2\ell_2) \\[8pt] \end{aligned} $$

$$ \begin{aligned} \delta_x &= \frac{W\ell_2^2}{12EI} \left\{ \frac{3\ell_1 x}{\ell} - \frac{(3\ell_1 + 2\ell_2)x^3}{\ell^3} \right\} \\ \delta_{x_1} &= \frac{W\ell_2^2}{12EI} \left\{ \frac{3\ell_1 x_1}{\ell} - \frac{(3\ell_1 + 2\ell_2)x_1^3}{\ell^3} \right\} + \frac{W(x_1 - \ell_1)^3}{12EI} \\ \delta_C &= \frac{W\ell_1^2\ell_2^3(4\ell_1 + 3\ell_2)}{12EI\ell^3} \\ \delta_{\max} &\text{ 의 위치는 } \ell_2 = \sqrt{2}\ell_1 \text{ 일 때 } x = \ell_1 \end{aligned} $$

$$ \begin{aligned} R_1 &= \frac{3w\ell}{8}, \quad R_2 = \frac{5w\ell}{8} \\ M_x &= \frac{w\ell x}{2} \left( \frac{3}{4} - \frac{x}{\ell} \right) \\ M_{\max} &= -\frac{w\ell^2}{8} \dots \text{고정단에서} \\[8pt] \delta_x &= \frac{w\ell^4}{48EI} \left( \frac{x}{\ell} - \frac{3x^3}{\ell^3} + \frac{2x^4}{\ell^4} \right) \\[8pt] \delta_{\max} &= \frac{w\ell^4}{185EI}, \quad x = 0.4215\ell \text{ 에서} \end{aligned} $$

$$ \begin{aligned} R_1 &= \frac{\overline{W}}{5}, \quad R_2 = \frac{4\overline{W}}{5} \\ M_x &= \overline{W}x \left( \frac{1}{5} - \frac{x^2}{3\ell^2} \right) \\ M_{\max} &= -\frac{\overline{W}\ell}{15} \dots \text{고정단에서} \\ \delta_x &= \frac{\overline{W}\ell^3}{60EI} \left( \frac{x}{\ell} - \frac{2x^3}{\ell^3} + \frac{x^5}{\ell^5} \right) \\ \delta_{\max} &= \frac{4\overline{W}\ell^3}{375\sqrt{5}EI} \fallingdotseq 0.00477\frac{\overline{W}\ell^3}{EI}, \quad x = \ell / \sqrt{5} \text{ 에서} \end{aligned} $$