# Cantilever Beam - Concentrated Load 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.

### Cantilever Beam - Concentrated Load at Any Point formulas

$$R = V \;=\; P$$

$$M_{max} \; (at\; fixed\; end ) \;=\; P\;b$$

$$M_x \; (when \; x > a ) \;=\; P \; ( x - a )$$

$$\Delta_{max} \; (at\; fixed\; end ) \;=\; (P\; b^2\;/\;6\; \lambda\; I) \; ( 3\;L - b )$$

$$\Delta_a \; (at \;point\; of \;load ) \;=\; P\; b^3\;/\;3 \;\lambda\; I$$

$$\Delta_x \; (when \; x < a ) \;=\; (P\; b^2 \;/\;6\; \lambda\; I) \; ( 3\;L - 3\;x - b)$$

$$\Delta_x \; (when \; x > a ) \;=\; [\;P\; ( L - x )^2 \;/\;6 \;\lambda\; I)\;] \; ( 3\;b - L + x )$$

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