Two Member Frame - Fixed/Pin Top Point Load

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two Member Frame - Fixed/Pin Top Point Load formulas

\(\large{ e  \;\;=\;\; \frac{h}{L}  }\)   
\(\large{ \beta \;\;=\;\; \frac{I_h}{I_v}  }\)   
\(\large{ R_A  \;\;=\;\;  \frac{P\;x}{L} \; \left[ 1+ \; \frac{2}{L^2} \; \left( \frac{L^2\;-\;x^2}{ 3\;\beta\;e \;+\; 4} \right) \right] }\)   
\(\large{ R_D  \;\;=\;\;  \frac{P\;\left( L\;-\;x \right)}{L} \; \left[ 1- \; \frac{2\;x}{L^2} \; \left( \frac{L\;+\;x}{ 3\;\beta\;e \;+\; 4} \right) \right]  }\)  
\(\large{ H_A = H_D \;\;=\;\;  \frac{3\;P\;x}{h\;L^2} \; \left( \frac{L^2\;-\;x^2}{ 3\;\beta\;e \;+\; 4} \right) }\)  
\(\large{ M_A \;\;=\;\;  \frac{P\;x}{L^2} \; \left( \frac{L^2\;-\;x^2}{ 3\;\beta\;e \;+\; 4} \right) }\)  
\(\large{ M_B \;\;=\;\;  \frac{2\;P\;x}{L^2} \; \left( \frac{L^2\;-\;x^2}{ 3\;\beta\;e \;+\; 4} \right)   }\)  
\(\large{ M_C \;\;=\;\;  \frac{P\;a \; \left( L\;-\;x \right) }{L} \; \left[ 1- \; \frac{2\;x}{L^2} \; \left( \frac{L\;-\;x}{ 3\;\beta\;e \;+\; 4} \right) \right] }\)  

Where:

 Units English Metric
\(\large{ \Delta }\) = deflection or deformation \(\large{in}\) \(\large{mm}\)
\(\large{ h }\) = height of frame \(\large{in}\) \(\large{mm}\)
\(\large{ x }\) = horizontal distance from reaction to point on beam \(\large{in}\) \(\large{mm}\)
\(\large{ H }\) = horizontal reaction load at bearing point \(\large{lbf}\) \(\large{N}\)
\(\large{ I_h }\) = horizontal member second moment of area (moment of inertia) \(\large{in^4}\) \(\large{mm^4}\)
\(\large{ I_v }\) = vertical member second moment of area (moment of inertia) \(\large{in^4}\) \(\large{mm^4}\)
\(\large{ M }\) = maximum bending moment \(\large{lbf-in}\) \(\large{N-mm}\)
\(\large{ A, B, C, D }\) = point of intrest on frame - -
\(\large{ L }\) = span length under consideration \(\large{in}\) \(\large{mm}\)
\(\large{ P }\) = total concentrated load \(\large{lbf}\) \(\large{N}\)
\(\large{ R }\) = vertical reaction load at bearing point \(\large{lbf}\) \(\large{N}\)

 

diagrams

  • 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.
  • 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.
  • Uniformly distributed load (UDL)  -  A load that is distributed evenly across the entire length of the support area.

 

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Tags: Frame Support