Simple Beam - Uniform Load Partially Distributed at One End

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 4D

Simple Beam - Uniform Load Partially Distributed at One End formulas

\( R_1 \;=\; V_1 \;=\; (w \;a\;/\;2\;L)  \; ( 2\;L - a ) \)

\( R_2 = V_2 \;=\; ( w \;a^2 \;/\;2\;L )\)

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

\( M_{max} \; ( at \; x = R_1\;/\;w )  \;=\; R_{1}{^2} \;/\; 2\;w  \)

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

\( M_x \; (  x > a )  \;=\;   R_2  \; (  L - x )  \)

\( \Delta_x  \; (  x < a )  \;=\;   ( w\; x \;/\; 24\; \lambda \;I \;L)  \; [\; [\; a^2  \; ( 2\;L - a )^2\;] - [\; 2\;a\;x^2 \; (  2\;L - a )\;]  + L\;x^3  \;]    \)

\( \Delta_x  \; (  x > a )  \;=\; [\; w\; a^2  \; (  L - x ) \;/\; 24\; \lambda \;I \;L \;] \; ( 4\;x\;L - 2\;x^2 - a^2 )    \)

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 \(bf-in\) \(N-mm\)
\( V \) = 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 \(in\) \(mm\)
\( a \) = width of UDL \(in\) \(mm\)

 

Piping Designer Logo 1

Tags: Beam Support