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Cantilever Beam - Concentrated Load at Any Point

Cantilever Beam - Concentrated Load at Any Point formulas

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 )  

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

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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.

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