# Overhanging Beam - Point Load on Beam End

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 - Point Load on Beam End formulas

$$R_1 \;=\; V_1 \;=\; P\;a\;/\;L$$

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

$$V_2 \;=\; P$$

$$M_{max} \; ( at \;R_2 ) \;=\; P\;a$$

$$M_x \; (between\; supports ) \;=\; P\;a\;x\;/\;L$$

$$M_{x_1} \; (for \;overhang ) \;=\; P\; ( a - x_1 )$$

$$\Delta_x \; (between\; supports ) \;=\; ( -(P\;a\;x) \;/\;6\; \lambda \;I\;L) \; ( L^2 - x^2 )$$

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

$$\Delta_{max} \; ( for\;overhang\; at\; x_1 = a ) \;=\; ( P\;a^2 \;/\;3\; \lambda \;I ) \; ( L + a )$$

$$\Delta_{max} \; ( between\; supports\; at \;x = \frac{L}{\sqrt{3}} ) \;=\; \frac{ -(P\;a\;L^2) }{9\; \sqrt{3} \; \lambda\; I } \;\;=\;\; 0.06415 \; ( P\;a\;L^2 \;/\; \lambda\; I )$$

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