Cantilever Beam - Concentrated Load at Any Point
- See Article - Beam Design Formula
Cantilever Beam - Concentrated Load at Any Point formulas |
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R = V \;=\; P M_{max} \; (at\; fixed\; end ) \;=\; P\cdot b M_x \; (when \; x > a ) \;=\; P \cdot ( x - a ) \Delta_{max} \; (at\; fixed\; end ) \;=\; \dfrac{ P\cdot b^2 }{ 6\cdot \lambda \cdot I } \cdot ( 3\cdot L - b ) \Delta_a \; (at \;point\; of \;load ) \;=\; \dfrac{ P\cdot b^3 }{ 3 \cdot \lambda\cdot I } \Delta_x \; (when \; x < a ) \;=\; \dfrac{ P\cdot b^2 }{ 6\cdot \lambda\cdot I } \cdot ( 3\cdot L - 3\cdot x - b ) \Delta_x \; (when \; x > a ) \;=\; \dfrac{ P\cdot ( L - x )^2 }{ 6 \cdot \lambda\cdot I } \cdot ( 3\cdot b - L + x ) |
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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 |
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.