Simple Beam - Uniformly Distributed Load and Variable End Moments

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.

 

sb 12D

Simple Beam - Uniformly Distributed Load and Variable End Moments formulas

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

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

\( V_x  \;=\; w \;  [\;( L \;/\; 2 ) - x\;]  + ( M_1 - M_2 \;/\; L ) \) 

\( b \; (inflection\; points) \;=\; \sqrt{ ( L^2 \;/\; 4 ) - ( M_1 + M_2 \;/\; w ) + ( M_1 + M_2 \;/\; w\;L )^2   }  \)

\( M_x  \;=\; [\;( w\;x \;/\; 2 ) \; ( L - x )\;]  + [\;( M_1 - M_2 \;/\; L ) x \;] - M_1  \)

\( M_3 \; ( at\; x =  \frac{ L }{ 2 }  +  \frac{ M_1 - M_2 }{ w\;L } )  \;=\; ( w\;L^2 \;/\; 8 )  - ( M_1 + M_2 \;/\; 2 ) +  [\; (M_1 - M_2)^2 \;/\; 2\;w\;L^2 \;] \)

\( \Delta_x  \;=\; ( \frac{w\;x }{ 48\; \lambda\; I} )  \; [ \;  x^3  -  [\;( 2\;L + \frac { 4\;M_1 } { w\;L }  -   \frac { 4\;M_2 } { w\;L } )  x^2 \;]  +     \frac { 12\;M_1 } { w }  + L^3  + \frac { 8\;M_1 \;L } { w }    -  \frac { 4\;M_2\; L } { w }  \;]  \)

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

 

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Tags: Beam Support