Simple Beam - Central Point Load and Variable End Moments

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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.

 

sb 13D

Simple Beam - Central Point Load and Variable End Moments formulas

\( R_1 \;=\; V_1  \;=\; ( P \;/\; 2 )  +  ( M_1 - M_2 \;/\; L ) \) 

\( R_2 \;=\; V_2  \;=\; ( P \;/\; 2 )  -  ( M_1 - M_2 \;/\; L )  \) 

\( M_3 \; (at\; center )  \;=\; ( P\;L \;/\; 4 )  -  ( M_1 + M_2 \;/\; L )  \) 

\( M_x \; ( x <  \frac{L}{2}  )  \;=\; [\; ( P \;/\; 2) + ( M_1 - M_2 \;/\; L ) \; x \;] - M_1  \)

\( M_x \; ( > \frac{L}{2} )  \;=\; [\;(P\;/\;2) \; ( L - x ) \;] + [\; ( M_1 - M_2 ) \;x \;/\; L \;] - M_1   \)

\( \Delta_x  ( x <  \frac{L}{2} )  \;=\; \frac{ P\;x }{ 48\; \lambda\; I }  \; [ \; 3\;L^2  -  4\;x^2 -  \frac{  8\; ( L - x ) }{ P\;L }  \; [ \;M_1 ( 2\;L - x ) + M_2\; ( L + x ) \; ]  \;]  \)

\( x \; (first \;point \;of \;contraflexure ) \;=\;   2\;L\; M_1 \;/\; L\;P + 2\;M_1 - 2\;M_2   \)

\( x \; (second\; point \;of\; contraflexure ) \;=\; [\; L \; ( L\;P - 2\;M_1 ) \;] \;/\; L\;P - 2\;M_1 + 2\;M_2   \)

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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Tags: Beam Support