# Simple Beam - Uniform Load Partially Distributed at Any Point

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.

### Simple Beam - Uniform Load Partially Distributed at Any Point formulas

$$R_1 \;=\; V_1 \; ( max. \;when\; a < c ) \;=\; (w \;b\;/\;2\;L) \; ( 2\;c + b )$$

$$R_2 \;=\; V_2 \; ( max. \;when\; a > c ) \;=\; ( w \;b \;/\;2\;L) \; ( 2\;a + b )$$

$$V_x \; [ \; a < x < ( a + b ) \;] \;=\; R_1 - [\;w \; ( x - a )\;]$$

$$M_{max} \; [\; at \; x = a + (R_1\;/\;w) \;] \;=\; R_1 \; [\; a + ( R_1 \;/\; 2\;w ) \;]$$

$$M_x \; ( x < a ) \;=\; R_1 \;x$$

$$M_x \; [ \; a < x < \; ( a + b ) \;] \;=\; (R_1 \;x) - [\; (w\;/\;2) \; ( x - a )^2 \;]$$

$$M_x \; [ \; x > ( a + b ) \;] \;=\; R_2 \; ( L - x )$$

Symbol English Metric
$$x$$ = 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 $$lbf$$ $$N$$
$$R$$ = reaction load at bearing point $$lbf$$ $$N$$
$$L$$ = span length of the bending member $$in$$ $$mm$$
$$a, b, c$$ = width and seperation of UDL $$in$$ $$mm$$

Tags: Beam Support