Overhanging Beam - Uniformly Distributed Load

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.

 

ob 1A

Overhanging Beam - Uniformly Distributed Load formulas

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

\( R_2 \;=\; V_2 + V_3 \;=\; ( w \;/\;2\;L) \; ( L + a )^2  \) 

\(V_2  \;=\;   w\;a  \) 

\( V_3  \;=\; ( w \;/\;2\;L)  \; ( L^2 + a^2 )  \)

\( V_x  \; (between\; supports )  \;=\;     R_1 - w\;x  \)

\(V_{x_1} \; (for \;overhang )  \;=\;    w \; (  a - x_1 )  \)

\( M_x  \; (between\; supports )  \;=\;  (w\;x \;/\;2\;L) \; ( L^2 - a^2 - x\;L )    \)

\( M_{x_1} \; (overhang )  \;=\;  ( w \;/\;2) \; ( a - x_1 )^2    \)

\( M_1  \; [\;at\;  x = \frac{L}{2}  (1 - \frac{a^2}{L^2} )\;] \;=\; (w \;/\;8\; L^2) \; (L + a)^2 \; (L - a)^2  \)

\( M_2 \; (at\; R_2 )  \;=\;  w\;a^2 \;/\;2  \)

\( \Delta_x  \; (between \;supports ) \;=\; \frac { w \;x} { 24 \;\lambda \;I \;L}  \; ( L^4  - 2\;L^2\;x^2  + L\;x^3  - 2\;a^2\;L^2 + 2\;a^2\;x^2 )    \)

\( \Delta_{x_1}  \; (for\; overhang )  \;=\;   \frac { w\; x_1} { 24 \;\lambda\; I }  \;  ( 4\;a^2\;L  - L^3  + 6\;a^2\;x_1 - 4\;a\;x_{1}{^2} +  x_{1}{^3} )    \)

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 \(lbf\;/\;in^2\) \(Pa\)
\( R \) = 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 \(in\) \(mm\)

 

Piping Designer Logo Slide 1

 

 

Tags: Beam Support