Simple Beam - Two Unequal Point Loads Unequally Spaced

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Simple Beam - Two Unequal Point Loads Unequally Spaced Formulas

\(\large{ R_1 = V_1  \;\;=\;\;  \frac {P_1 \; \left(  L \;- \;a  \right)  \;+\; P_2\; b     }  { L }   }\) 

\(\large{ R_2 = V_2  \;\;=\;\;  \frac {P_1 \;a \;+\; P_2 \; \left(  L\; - \;b  \right)   }  { L }    }\) 

\(\large{ V_x  \; \left[ a < x  <  \left(  L - b  \right) \right] \;\;=\;\;  R_1 - P_1  }\) 

\(\large{ M_1 \; \left(max.\; when\;  R_1 < P_1  \right)  \;\;=\;\;  R_1\; a     }\)

\(\large{ M_2 \; \left(max.\; when\;  R_2 < P_2  \right)  \;\;=\;\; R_2\; b     }\)

\(\large{ M_x \; \left(max.\; when\;  x < a  \right)  \;\;=\;\;  R_1\; x     }\)

\(\large{ M_x \;  \left[ max.\; when\;  a < x <  \;\left(  L - b  \right) \right] \;\;=\;\;  R_1\; x  - P_1\; \left(  x - a  \right)   }\)

Symbol English Metric
\(\large{ \Delta }\) = deflection or deformation  \(\large{in}\) \(\large{mm}\)
\(\large{ x }\) = horizontal distance from reaction to point on beam \(\large{in}\) \(\large{mm}\)
\(\large{ a, b }\) = length to point load \(\large{in}\) \(\large{mm}\)
\(\large{ M }\) = maximum bending moment \(\large{lbf-in}\) \(\large{N-mm}\)
\(\large{ V }\) = maximum shear force \(\large{lbf}\) \(\large{N}\)
\(\large{ \lambda  }\)   (Greek symbol lambda) = modulus of elasticity \(\large{\frac{lbf}{in^2}}\) \(\large{Pa}\)
\(\large{ I }\) = second moment of area (moment of inertia) \(\large{in^4}\) \(\large{mm^4}\)
\(\large{ R }\) = reaction load at bearing point \(\large{lbf}\) \(\large{N}\)
\(\large{ L }\) = span length of the bending member \(\large{in}\) \(\large{mm}\)
\(\large{ P }\) = total concentrated load \(\large{lbf}\) \(\large{N}\)

 

diagrams

  • 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.

 

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Tags: Beam Support Equations