# Cantilever Beam - Concentrated Load at Any Point

on . Posted in Structural Engineering

## Cantilever Beam - Concentrated Load at Any Point formulas

$$\large{ R = V \;\;=\;\; P }$$

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

$$\large{ M_x \; \left(when \; x > a \right) \;\;=\;\; P \; \left( x - a \right) }$$

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

$$\large{ \Delta_a \; \left(at \;point\; of \;load \right) \;\;=\;\; \frac {P\; b^3} {3 \;\lambda\; I} }$$

$$\large{ \Delta_x \; \left(when \; x < a \right) \;\;=\;\; \frac {P\; b^2 } {6\; \lambda\; I} \; \left( 3\;L - 3\;x - b \right) }$$

$$\large{ \Delta_x \; \left(when \; x > a \right) \;\;=\;\; \frac {P\; \left( L\; - \;x \right)^2 } {6 \;\lambda\; I} \; \left( 3\;b - L + x \right) }$$

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.