# Overhanging Beam - Point Load on Beam End

on . Posted in Structural Engineering

## Overhanging Beam - Point Load on Beam End formulas

$$\large{ R_1 = V_1 \;\;=\;\; \frac{P\;a}{L} }$$

$$\large{ R_2 = V_1 + V_2 \;\;=\;\; \frac{ P }{L} \; \left( L + a \right) }$$

$$\large{ V_2 \;\;=\;\; P }$$

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

$$\large{ M_x \; \left(between\; supports \right) \;\;=\;\; \frac{P\;a\;x}{L} }$$

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

$$\large{ \Delta_x \; \left(between\; supports \right) \;\;=\;\; \frac{ -\;P\;a\;x }{6\; \lambda \;I\;L} \; \left( L^2 - x^2 \right) }$$

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

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

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

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

## diagrams

• 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. 