Appendix B: Slope and Deflection Tables

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Page ID: 145531

James K. Lord, Amy Richardson, Sneha Davison, and David A. Dillard
Virginial Tech

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B.1 Simply Supported Beams

Slope	Max. Deflection	Elastic Curve
$θ_{\max} = - \frac{{PL}^{2}}{16 EI}$	$y_{max} = - \frac{{PL}^{3}}{48 EI}$	$y = - \frac{Px}{48 EI} ({3 L}^{2} - {4 x}^{2})$ Valid for x ≤ L/2
$\begin{array}{l} θ_{A} & = & - Pab (L + b) \\ 6 EIL \\ θ_{B} \\ = & \frac{Pab (L + a)}{6 EIL} \end{array}$	$\begin{matrix} y_{max} = - {Pb(L}^{2} - b^{2})^{3/} \\ \frac{2}{9 \sqrt{3} EIL} \\ At x \\ = \sqrt{\frac{L^{2} -}{b^{2}}} \end{matrix}$	$\begin{array}{l} y = - \frac{Pbx}{6 EIL} {(L}^{2} - b^{2} - x^{2}) \\ Valid for x < a \\ At x = a, y = - \frac{{Pa}^{2} b^{2}}{3 EIL} \end{array}$
$θ_{\max} = - \frac{{wL}^{3}}{24 EL}$	$y_{max} = - \frac{{5 wL}^{4}}{384 EI}$	$y = - \frac{wx}{24 EI} (x^{3} - {2 Lx}^{2} + L^{3})$
$\begin{array}{l} θ_{A} & = & - \frac{3 {wL}^{3}}{128 EI} \\ θ_{B} & = & \frac{7 {wL}^{3}}{384 EI} \end{array}$	$\begin{matrix} y_{max} = - \frac{{wL}^{4}}{152.37EI} \\ At x = 0.4598 L \end{matrix}$	$\begin{array}{l} y = - \frac{wx}{384 EI} {(16x}^{3} - 24 {Lx}^{2} + 9 L^{3}) \\ Valid for 0 \leq x \leq \frac{L}{2} \\ At x = \frac{L}{2}, y = - \frac{5 {wL}^{4}}{768 EI} \end{array}$
$\begin{array}{l} θ_{A} & = & - \frac{7 {wL}^{3}}{360 EI} \\ θ_{B} & = & \frac{{wL}^{3}}{45 EI} \end{array}$	$\begin{matrix} y_{max} = - \frac{w_{0} L^{4}}{153.37EI} \\ At x = 0.5193 L \end{matrix}$	$y = - \frac{wx}{360 EIL} ({3 x}^{4} - {10 L}^{2} x^{2} + {7 L}^{4})$
$\begin{array}{l} θ_{A} & = & - \\ \frac{ML}{6 EI} \\ θ_{B} \\ = & \frac{ML}{3 EI} \end{array}$	$\begin{matrix} y_{max} = - \frac{{ML}^{2}}{9 \sqrt{3} EI} \\ At x = \frac{L}{\sqrt{3}} \end{matrix}$	$y = - \frac{Mx}{6 EIL} {(L}^{2} - x^{2})$

B.2 Cantilever Beams

Slope	Max. Deflection	Elastic Curve
$θ_{max} = - \frac{{PL}^{2}}{2 EI}$	$y_{max} = - \frac{{PL}^{3}}{3 EI}$	$y = - \frac{{Px}^{2}}{6 EI} (3L - x)$
$θ_{max} = - \frac{{PL}^{2}}{8 EI}$	$y_{max} = - \frac{{5 PL}^{3}}{48 EI}$	$\begin{array}{l} y = - \frac{{Px}^{2}}{12 EI} (3L - 2x) \\ Valid for 0 \leq x \leq \\ \frac{L}{2} \\ y \\ = - \frac{{PL}^{2}}{48 EI} (6x - L) \\ Valid for \frac{L}{2} \leq x \leq L \end{array}$
$θ_{max} = - \frac{{wL}^{3}}{6 EI}$	$y_{max} = - \frac{{wL}^{4}}{8 EI}$	$y = - \frac{{wx}^{2}}{24 EI} (x^{2} - 4 Lx + {6 L}^{2})$
$θ_{max} = - \frac{{wL}^{3}}{48 EI}$	$y_{max} = - \frac{{7 wL}^{4}}{384 EI}$	$\begin{array}{l} y = - \frac{w x^{2}}{24 E I} (x^{2} - 2 L x + 1.5 L^{2}) \\ Valid for 0 \leq x \leq \frac{L}{2} \\ 0 \leq x \leq \frac{L}{2} \\ Valid for \frac{L}{2} \leq x \leq L \end{array}$
$θ_{max} = - \frac{{wL}^{3}}{24 EI}$	$y_{max} = - \frac{w_{0} L^{4}}{30 EI}$	$y = - \frac{{wx}^{2}}{120 EIL} (10 L^{3} - {10 L}^{2} x + 5 {Lx}^{2} - x^{3})$
$θ_{max} = \frac{ML}{EI}$	$y_{max} = \frac{{ML}^{2}}{2 EI}$	$y = \frac{{Mx}^{2}}{2 EI}$

References

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All figures in this appendix were created by Kindred Grey in 2025 and released under a CC BY license.

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