# Simple Beam - Uniformly Distributed Load and Variable End Moments

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 - Uniformly Distributed Load and Variable End Moments formulas

$$R_1 \;=\; V_1 \;=\; ( w\;L \;/\; 2 ) + (M_1 - M_2 \;/\; L)$$

$$R_2 \;=\; V_2 \;=\; ( w\;L \;/\; 2 ) - (M_1 - M_2 \;/\; L)$$

$$V_x \;=\; w \; [\;( L \;/\; 2 ) - x\;] + ( M_1 - M_2 \;/\; L )$$

$$b \; (inflection\; points) \;=\; \sqrt{ ( L^2 \;/\; 4 ) - ( M_1 + M_2 \;/\; w ) + ( M_1 + M_2 \;/\; w\;L )^2 }$$

$$M_x \;=\; [\;( w\;x \;/\; 2 ) \; ( L - x )\;] + [\;( M_1 - M_2 \;/\; L ) x \;] - M_1$$

$$M_3 \; ( at\; x = \frac{ L }{ 2 } + \frac{ M_1 - M_2 }{ w\;L } ) \;=\; ( w\;L^2 \;/\; 8 ) - ( M_1 + M_2 \;/\; 2 ) + [\; (M_1 - M_2)^2 \;/\; 2\;w\;L^2 \;]$$

$$\Delta_x \;=\; ( \frac{w\;x }{ 48\; \lambda\; I} ) \; [ \; x^3 - [\;( 2\;L + \frac { 4\;M_1 } { w\;L } - \frac { 4\;M_2 } { w\;L } ) x^2 \;] + \frac { 12\;M_1 } { w } + L^3 + \frac { 8\;M_1 \;L } { w } - \frac { 4\;M_2\; L } { w } \;]$$

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

Tags: Beam Support