Overhanging Beam - Point Load Between Supports at Any Point

on . Posted in Structural Engineering

ob 6A

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.

 

 

 

 

Overhanging Beam - Point Load Between Supports at Any Point formulas

\( R_1 \;=\; V_1  \; ( max.\; when\; a < b)  \;=\;  P\;b \;/\; L    \) 

\( R_2 \;=\; V_2  \;  (max. \;when\; a > b )  \;=\; P\;a\;/\;L   \) 

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

\( M_x \;  \left( x < a \right)   \;=\; P\;b\;x\;/\;L  \)

\( \Delta_{x_1}  \;=\;  ( P\;a\;b\;x_1 \;/\;6\; \lambda\; I\;L )  \; ( L + a )   \)

\( \Delta_a  \; (at\; point \;of \;load ) \;=\;  P\;a^2\; b^2 \;/\;3\; \lambda\; I\;L   \)

\( \Delta_x  \; (when\;  x < a )  \;=\; ( P\;b\;x \;/\;6\; \lambda\; I\;L)  \; ( L^2 - b^2 - x^2 )   \)

\( \Delta_x  \; ( when\; x > a ) \;=\; [\; P\;a \; ( L - x ) \;/\; 6\; \lambda\; I\;L\;]  \; ( 2\;L\;x - x^2 - a^2 )  \)

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

Symbol English Metric
\( \Delta \) = deflection or deformation \(in\) \(m\)
\( x \) = horizontal distance from reaction to point on beam \(in\) \(m\)
\( M \) = 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) \(in^4\) \(m^4\)
\( L \) = span length of the bending member \(in\) \(m\)
\( P \) = total concentrated load \(lbf\) \(N\)

 

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Tags: Beam Support