# Simple Beam - Central Point 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 - Central Point Load and Variable End Moments formulas

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

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

$$M_3 \; (at\; center ) \;=\; ( P\;L \;/\; 4 ) - ( M_1 + M_2 \;/\; L )$$

$$M_x \; ( x < \frac{L}{2} ) \;=\; [\; ( P \;/\; 2) + ( M_1 - M_2 \;/\; L ) \; x \;] - M_1$$

$$M_x \; ( > \frac{L}{2} ) \;=\; [\;(P\;/\;2) \; ( L - x ) \;] + [\; ( M_1 - M_2 ) \;x \;/\; L \;] - M_1$$

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

$$x \; (first \;point \;of \;contraflexure ) \;=\; 2\;L\; M_1 \;/\; L\;P + 2\;M_1 - 2\;M_2$$

$$x \; (second\; point \;of\; contraflexure ) \;=\; [\; L \; ( L\;P - 2\;M_1 ) \;] \;/\; L\;P - 2\;M_1 + 2\;M_2$$

Symbol English Metric
$$\Delta$$ = deflection or deformation $$in$$ $$mm$$
$$x$$ = horizontal distance from reaction to point on beam $$in$$ $$mm$$
$$M$$ = 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 $$in$$ $$mm$$
$$P$$ = total concentrated load $$lbf$$ $$N$$

Tags: Beam Support