# Overhanging Beam - Uniformly Distributed Load on Overhang

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 on Overhang formulas

$$R_1 \;=\; V_2 \;=\; w\; a^2 \;/\;2\;L$$

$$R_2 \;=\; V_1 + V_2 \;=\; (w\; a \;/\;2\;L) \; ( 2\;L + a)$$

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

$$V_{x _1} \;=\; w \; ( a - x_1 )$$

$$M_{max} \; ( at\; R_2 ) \;=\; w \;a^2 \;/\;2$$

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

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

$$\Delta_x \; (between\; supports ) \;=\; ( - \;w \;a^2\; x \;/\;12\; \lambda\; I \;L) \; ( L^2 - x^2 )$$

$$\Delta_{max} \; (between\; supports \;at\; x = \frac{L}{\sqrt{3}} ) \;=\; \frac{ - \;w\; a^2 \;L^2 }{18 \; \sqrt{3} \; \lambda\; I } \;=\; 0.03208 \; ( w \;a^2 \; L^2 \;/\; \lambda\; I)$$

$$\Delta_{max} \; (for \;overhang \;at\; x_1 = a ) \;=\; ( w\; x^3 \;/\;24\; \lambda\; I ) \; ( 4\;L + 3\;a )$$

$$\Delta_{x1} \; (for \;overhang ) \;=\; ( w\; x_1 \;/\;24\; \lambda\; I ) \; ( 4\;a^2 \;L + 6\;a^2\; x_1 - 4\;a \;x_{1}{^2} + x_{1}{^3} )$$

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