Two Span Continuous Beam - Unequal Spans, Uniformly Distributed Load

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Span Continuous Beam - Unequal Spans, Uniformly Distributed Load formulas

\(\large{ R_1 = V_1   \;\;=\;\; \frac{M_1}{a} + \frac{w\;a}{2}   }\) 

\(\large{ R_2   \;\;=\;\;  w\;a + w\;b - R_1 - R_3    }\) 

\(\large{ R_3 = V_4   \;\;=\;\; \frac{M_1}{b} + \frac{w\;a}{2}   }\) 

\(\large{ V_2   \;\;=\;\;  w\;a - R_1  }\)

\(\large{ V_3   \;\;=\;\;  w\;b - R_3  }\)

\(\large{ M_1  \;\;=\;\; \frac{w\;b^3 \;+\; w\;a^3}{8 \; \left( a\;+ \;b \right) }   }\)

\(\large{ M_1  \;\;=\;\; \frac{w\;b^3 \;+\; w\;a^3}{8 \; \left( a\;+ \;b \right) }   }\)

\(\large{ M_{x_2} \;  \left( x_2 = \frac{R_3}{w} \right)   \;=\; R_3 \;x_2 \; -\;\frac{w\;x_{2}{^2} }{2}    }\)

Symbol English Metric
\(\large{ \Delta }\) = deflection or deformation \(\large{in}\) \(\large{mm}\)
\(\large{ x }\) = horizontal distance from reaction to point on beam \(\large{in}\) \(\large{mm}\)
\(\large{ w }\) = load per unit length \(\large{\frac{lbf}{in}}\) \(\large{\frac{N}{m}}\)
\(\large{ M }\) = maximum bending moment \(\large{lbf-in}\) \(\large{N-mm}\)
\(\large{ V }\) = maximum shear force \(\large{lbf}\) \(\large{N}\)
\(\large{ R }\) = reaction load at bearing point \(\large{lbf}\) \(\large{N}\)
\(\large{ a, b }\) = span length under consideration \(\large{in}\) \(\large{mm}\)

 

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