# Overhanging Beam - Uniformly Distributed Load

on . Posted in Structural Engineering

### diagram Symbols

• Bending moment diagram (BMD)  -  Used to determine the bending moment at a given point of a structural element.  The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
• Free body diagram (FBD)  -  Used to visualize the applied forces, moments, and resulting reactions on a structure in a given condition.
• Shear force diagram (SFD)  -  Used to determine the shear force at a given point of a structural element.  The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
• Uniformly distributed load (UDL)  -  A load that is distributed evenly across the entire length of the support area.

### Overhanging Beam - Uniformly Distributed Load formulas

$$R_1 \;=\; V_1 \;=\; (w\;/\;2\;L) \; ( L^2 - a^2 )$$

$$R_2 \;=\; V_2 + V_3 \;=\; ( w \;/\;2\;L) \; ( L + a )^2$$

$$V_2 \;=\; w\;a$$

$$V_3 \;=\; ( w \;/\;2\;L) \; ( L^2 + a^2 )$$

$$V_x \; (between\; supports ) \;=\; R_1 - w\;x$$

$$V_{x_1} \; (for \;overhang ) \;=\; w \; ( a - x_1 )$$

$$M_x \; (between\; supports ) \;=\; (w\;x \;/\;2\;L) \; ( L^2 - a^2 - x\;L )$$

$$M_{x_1} \; (overhang ) \;=\; ( w \;/\;2) \; ( a - x_1 )^2$$

$$M_1 \; [\;at\; x = \frac{L}{2} (1 - \frac{a^2}{L^2} )\;] \;=\; (w \;/\;8\; L^2) \; (L + a)^2 \; (L - a)^2$$

$$M_2 \; (at\; R_2 ) \;=\; w\;a^2 \;/\;2$$

$$\Delta_x \; (between \;supports ) \;=\; \frac { w \;x} { 24 \;\lambda \;I \;L} \; ( L^4 - 2\;L^2\;x^2 + L\;x^3 - 2\;a^2\;L^2 + 2\;a^2\;x^2 )$$

$$\Delta_{x_1} \; (for\; overhang ) \;=\; \frac { w\; x_1} { 24 \;\lambda\; I } \; ( 4\;a^2\;L - L^3 + 6\;a^2\;x_1 - 4\;a\;x_{1}{^2} + x_{1}{^3} )$$

Symbol English Metric
$$\Delta$$ = deflection or deformation $$in$$ $$mm$$
$$x$$ = horizontal distance from reaction to point on beam $$in$$ $$mm$$
$$w$$ = load per unit length $$lbf\;/\;in$$ $$N\;/\;m$$
$$M$$ = maximum bending moment $$lbf-in$$ $$N-mm$$
$$V$$ = maximum shear force $$lbf$$ $$N$$
$$\lambda$$   (Greek symbol lambda) = modulus of elasticity $$lbf\;/\;in^2$$ $$Pa$$
$$R$$ = reaction load at bearing point $$lbf$$ $$N$$
$$I$$ = second moment of area (moment of inertia) $$in^4$$ $$mm^4$$
$$L$$ = span length of the bending member $$in$$ $$mm$$

Tags: Beam Support