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Simple Beam - Central Point Load and Variable End Moments

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Simple Beam - Central Point Load and Variable End Moments formulas

\(\large{ R_1 = V_1  \;\;=\;\;  \frac { P }  { 2 }  +  \frac { M_1 \;- \;M_2 }  { L }   }\) 

\(\large{ R_2 = V_2  \;\;=\;\;  \frac { P }  { 2 }  -  \frac { M_1 \;-\; M_2 }  { L }    }\) 

\(\large{ M_3  \;  \left(at\; center \right)  \;\;=\;\;  \frac { P\;L }  { 4 }  -  \frac { M_1 \;+\; M_2 }  { L }    }\) 

\(\large{ M_x   \left(  x <  \frac{L} {2}    \right)   \;\;=\;\;   \left(     \frac { P }  { 2 }  +  \frac { M_1\; -\; M_2 }  { L }  \right)  x - M_1    }\)

\(\large{ M_x   \left(  >  \frac{L} {2}    \right)   \;\;=\;\;  \frac {P}{2} \; \left( L - x  \right)  +   \frac { \left(  M_1 \;-\; M_2  \right) \;x }  { L } \; -\; M_1    }\)

\(\large{ \Delta_x   \left( x <  \frac{L} {2}    \right)    \;\;=\;\;    \frac { P\;x } { 48\; \lambda\; I }   \;      \left[        3\;L^2  -  4\;x^2 -    \frac {  8\;  \left( L\; -\; x  \right) }  { P\;L }  \;    \left[ M_1 \left( 2\;L - x \right)   +  M_2\; \left( L + x \right)  \right]       \right]  }\)

\(\large{ x \; \left(first \;point \;of \;contraflexure \right)  \;\;=\;\;  \frac { 2\;L\; M_1 }  { L\;P \;+ \;2\;M_1\; - \;2\;M_2 }    }\)

\(\large{ x \; \left(second\; point \;of\; contraflexure \right) \;\;=\;\;  \frac {  L \; \left( L\;P\; - \;2\;M_1     \right)       }  { L\;P \;- \;2\;M_1 \;+\; 2\;M_2 }    }\)
Symbol English Metric
\(\large{ \Delta }\) = deflection or deformation \(\large{in}\) \(\large{mm}\)
\(\large{ x }\) = horizontal distance from reaction to point on beam \(\large{in}\) \(\large{mm}\)
\(\large{ M }\) = 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 \(\large{\frac{lbf}{in^2}}\) \(\large{Pa}\)
\(\large{ R }\) = reaction load at bearing point \(\large{lbf}\) \(\large{N}\)
\(\large{ I }\) = second moment of area (moment of inertia) \(\large{in^4}\) \(\large{mm^4}\)
\(\large{ L }\) = 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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