Overhanging Beam - Point Load Between Supports at Any Point

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ob 6A

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Overhanging Beam - Point Load Between Supports at Any Point formulas

\(\large{ R_1 = V_1  \;  \left( max.\; when\; a < b \right)     \;\;=\;\;  \frac{P\;b}{L}    }\) 

\(\large{ R_2 = V_2  \;  \left(max. \;when\; a > b \right)    \;\;=\;\;  \frac{P\;a}{L}    }\) 

\(\large{ M_{max}  \;   \left(at \;point \;of \;load \right)  \;\;=\;\; \frac{P\;a\;b}{L}    }\) 

\(\large{ M_x \;  \left( x < a \right)   \;\;=\;\; \frac{P\;b\;x}{L}    }\)

\(\large{ \Delta_{x_1}   \;\;=\;\;   \frac{ P\;a\;b\;x_1 }{6\; \lambda\; I\;L }  \; \left( L + a  \right)   }\)

\(\large{ \Delta_a  \;   \left(at\; point \;of \;load \right) \;\;=\;\;  \frac{ P\;a^2\; b^2 }{3\; \lambda\; I\;L}    }\)

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

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

\(\large{ \Delta_{max}    \;  \left(at\; x =   \sqrt{  \frac{ a \; \left(a \;+\; 2\;b \right)  }{3}  }  \; when\; a > b \right) \;\;=\;\;   \frac{    P\;a\;b \; \left( a \;+\; 2\;b \right) \;  \sqrt{ 3\;a \; \left( a \;+\; 2\;b \right) }  }   {27\; \lambda \;I\;L }          }\)

Symbol English Metric
\(\large{ \Delta }\) = deflection or deformation \(\large{in}\) \(\large{m}\)
\(\large{ x }\) = horizontal distance from reaction to point on beam \(\large{in}\) \(\large{m}\)
\(\large{ M }\) = 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) \(\large{in^4}\) \(\large{m^4}\)
\(\large{ L }\) = span length of the bending member \(\large{in}\) \(\large{m}\)
\(\large{ P }\) = 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.

 

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Tags: Beam Support Equations