Two Member Frame - Fixed/Free Free End Vertical Point Load

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Two Member Frame - Fixed/Free Free End Vertical Point Load formulas

Support Reaction

\(\large{ R_A  \;\;=\;\; P  }\) 
\(\large{ H_A \;\;=\;\; 0  }\) 

Bending Moment

\(\large{ M_{max}  \left(at \;points\; A\; and \;B\right) \;\;=\;\; P\;L  }\) 

Deflection

\(\large{ \Delta_{Cx}  \;\;=\;\; \frac{P\;L\;h^2}{2\; \lambda \; I}  }\)
\(\large{ \Delta_{Cy}  \;\;=\;\; \frac{P\;L^2}{3\; \lambda \; I} \; \left( L + 3\;h \right)  }\)

Slope

\(\large{ \theta_{C}  \;\;=\;\; \frac{P\;L}{2\; \lambda \; I} \; \left( L + 2\;h \right)  }\)

Where:

 Units English Metric
\(\large{ \Delta }\) = deflection or deformation \(\large{in}\) \(\large{mm}\)
\(\large{ h }\) = height of frame \(\large{in}\) \(\large{mm}\)
\(\large{ H }\) = 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 \(\large{lbf-in}\) \(\large{N-mm}\)
\(\large{ \lambda  }\)   (Greek symbol lambda) = modulus of elasticity \(\large{\frac{lbf}{in^2}}\) \(\large{Pa}\)
\(\large{ A, B, C }\) = point of intrest on frame - -
\(\large{ \theta }\) = slope of member \(\large{rad}\) \(\large{rad}\)
\(\large{ L }\) = 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

  • 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: Frame Support