Simple Beam - Uniform Load Partially Distributed at Each End

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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 6D

Simple Beam - Uniform Load Partially Distributed at Each End formulas

\( R_1 = V_1 \;=\; [\;w_1 \;a \; (  2\;L - a ) \;]  + (w_2 \;c^2 \;/\; 2\;L) \)

\( R_2 = V_2 \;=\; [\;w_2 \;c  \; (  2\;L - c ) \;]  + ( w_1\; a^2 \;/\; 2\;L) \)

\( V_x  \; (  x < a )  \;=\; R_1 - w_1 \;x  \)

\( V_x  \; [ \; a < x < ( a + b) \;]  \;=\; R_1 - w_1 \;a   \)

\( V_x   \;  [ \; x >  (  a + b ) \;] \;=\; R_2 - [\; w_2 \;( 1 - x ) \;]   \)

\( M_{max} \; [\; at \; x = (R_1\;/\;w_1) \; when \; R_1 < w_1 \;a \;]  \;=\; R_{1}{^2} \;/\; 2\;w_1   \)

\( M_{max} \; [\; at \; x = L - (R_2\;/\;w_2) \; when \;  R_2 < w_2 \;c \;]  \;=\; R_{2}{^2} \;/\; 2\;w_2  \)

\( M_x  \; ( w < a )  \;=\; (R_1 \;x)  - ( w_1 \;x^2\;/\; 2 )  \)

\( M_x  \; [\;  a < x < (  a + b ) \;] \;=\; (R_1\; x) - [\;( w_1 \;a\;/\; 2 )  \; (  2\;x - a )\;]   \)

\( M_x  \; [\; x > ( a + b ) \;]  \;=\; [\; R_2 \; (  L - x ) \;] - [\; w_2 \; ( L - x )^2   \;/\; 2 \;] \)

Symbol English Metric
\( x \) = 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 \(lbf\) \(N\)
\( R \) = reaction load at bearing point \(lbf\) \(N\)
\( L \) = span length of the bending member \(in\) \(mm\)
\( a, b, c \) = width and seperation of UDL \(in\) \(mm\)

 

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Tags: Beam Support