The following moment, rotation and deflection diagrams are commonly encountered in real life design situations and are worth memorizing to have at your finger tips.

Remember the basical integral relationship between loading, shear, moment, curvature, rotation and deflection. The following diagrams form a rich palette of graphical analysis tools that can be combined using the principal of superposition to idealize more complex cases.

Deflection & Rotation Diagrams

Problems that test knowledge of rotation are fairly rare, however, deflection problems are some of the highest frequency problems of all. Moreover, deflection diagrams are also important for solving a number of problems.

Cantilevered Beams

Case	Deflection (\(\delta_{max}\))	Rotation (\(\theta_{max}\))
	$$\frac{M\ell^2}{2EI}$$	$$\frac{M\ell}{EI}$$
	$$\frac{P\ell^3}{3EI}$$	$$\frac{P\ell^2}{2EI}$$
	$$\frac{w\ell^4}{8EI}$$	$$\frac{w\ell^3}{6EI}$$

Simple Beams

Case	Deflection (\(\delta_{max}\))	Rotation (\(\theta_{max}\))
	$$\frac{M\ell^2}{9\sqrt{3}EI}$$	$$\theta_{near}=\frac{M\ell}{3EI}$$	$$\theta_{far}=\frac{M\ell}{6EI}$$
	$$\frac{M\ell^2}{8EI}$$	$$\frac{M\ell}{2EI}$$
	$$\frac{P\ell^3}{48EI}$$	$$\frac{P\ell^2}{16EI}$$
	$$\frac{5w\ell^4}{384EI}$$	$$\frac{w\ell^3}{24EI}$$

Fixed Beams

Case	Deflection (\(\delta_{max}\))
	$$\frac{P\ell^3}{192EI}$$
	$$\frac{w\ell^4}{384EI}$$

Moment Diagrams

The following cases are ones that every structural engineer and first-class architect should know by heart. The moment diagrams below are shown on the tension face of the member. The dashed lines indicate the deflected shape.

Cantilevered Beams

Case	Moment (\(M_{max}\))
	$$M$$
	$$P\ell$$
	$$\frac{w\ell^2}{2}$$

Propped Cantilever Beams

Case	Moment (\(M_{max}\))
	$$M_{wall}=\frac{M}{2}$$	$$M_{max}=M$$
	$$M_{wall}=\frac{3P\ell}{16}$$ $$R_{wall}=\frac{11P}{16}$$	$$M_{max}=\frac{5P\ell}{32}$$ $$R_{prop}=\frac{5P}{16}$$
	$$M_{wall}=\frac{w\ell^2}{8}$$ $$R_{wall}=\frac{5w\ell}{8}$$	$$M_{max}=\frac{9w\ell^2}{128}$$ $$R_{prop}=\frac{3w\ell}{8}$$

Simple Beams

Case	Moment (\(M_{max}\))
	$$M$$
	$$\frac{P\ell}{4}$$
	$$\frac{w\ell^2}{8}$$

Additional Simple Beam Moment Cases

Case	Moment (\(M_{max}\))
	$$M$$
	$$M$$
	$$\frac{M}{2}$$

Fixed Beams

Case	Moment (\(M_{max}\))
	$$M_{wall}=\frac{M}{4}$$	$$M_{mid}=\frac{M}{2}$$
	$$M_{wall}=\frac{p\ell}{8}$$	$$M_{mid}=\frac{p\ell}{8}$$
	$$M_{wall}=\frac{w\ell^2}{12}$$	$$M_{mid}=\frac{w\ell^2}{24}$$