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Cantilever Beam - Load Increasing Uniformly to One 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.

 

cb 2A

Cantilever Beam - Load Increasing Uniformly to One End formulas

\( R \;=\; V \;=\; W  \) 

\( V_x \;=\; W\; (x^2\;/\;L^2 ) \) 

\( M_{max} \; \left(at\; fixed \;end \right)  \;=\; W\; L\;/\;3  \) 

\( M_x  \;=\;  W\;x^3 \;/\;3\;L^2  \)

\( \Delta_{max} \;  \left(at\; free\; end \right) \;=\; W\; L^3\;/\;15\; \lambda\; I  \)

\( \Delta_x  \;=\; (W\;x^2\;/\;60\; \lambda \;I \;L^2) \; ( x^5 + 5\;L^4 x + 4\;L^5)   \)

C B - Load Increasing Unif to One End - Solve for R

\(\large{R = \frac{w\;L}{2}  }\)

load per unit length, w
span length, L

C B - Load Increasing Unif to One End - Solve for Vx

\(\large{ V_x =  \frac{w\;L}{2}  \; \frac{x^2}{L^2}   }\) 

load per unit length, w
span length, L
distance from reaction, x

C B - Load Increasing Unif to One End - Solve for Mmax

\(\large{ M_{max}  =  \frac{\frac{w\;L}{2}  \; L}{3}  }\) 

load per unit length, w
span length, L

C B - Load Increasing Unif to One End - Solve for Mx

\(\large{ M_x  =   \frac{ \frac{w\;L}{2} \;x^3 }{3\;L^2}   }\)

load per unit length, w
span length, L
distance from reaction, x

C B - Load Increasing Unif to One End - Solve for Δmax

\(\large{ \Delta_{max}  =  \frac{\frac{w\;L}{2} \; L^3}{15\; \lambda\; I}  }\)

load per unit length, w
span length, L
modulus of elasticity, λ
second moment of area, I

C B - Load Increasing Unif to One End - Solve for Δx

\(\large{ \Delta_x  =  \frac{\frac{w\;L}{2} \;x^2}{60\; \lambda \;I \;L^2} \; \left(   x^5 + 5\;L^4 x + 4\;L^5   \right)     }\)

load per unit length, w
span length, L
distance from reaction, x
modulus of elasticity, λ
second moment of area, I

Symbol English Metric
\( R \) = reaction load at bearing point \(lbf\) \(N\)
\( V \) = maximum shear force \(lbf\) \(N\)
\( M \) = maximum bending moment \(lbf-in\) \(N-mm\)
\( \Delta \) = deflection or deformation \(in\) \(mm\)
\( W \) = total load \((w\;L\;/\;2)  \) \(lbf\) \(N\)
\( w \) = highest load per unit length \(lbf\;/\;in\) \(N\;/\;m\)
\( L \) = span length of the bending member \(in\) \(mm\)
\( x \) = horizontal distance from reaction to point on beam \(in\) \(mm\)
\( \lambda  \)   (Greek symbol lambda) = modulus of elasticity \(lbf\;/\;in^2\) \(Pa\)
\( I \) = second moment of area (moment of inertia) \(in^4\) \(mm^4\)

 

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Tags: Beam Support