Beam Fixed at Both Ends - Concentrated Load at Any Point

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febe 3B

  

Beam Fixed at Both Ends - Concentrated Load at Any Point formulas

\(\large{ R_1 = V_1  \;  \left(max.\; when \; a < b \right)  \;\;=\;\; \frac {P\;b^2} {L^3} \; \left( 3\;a + b  \right)    }\) 

\(\large{ R_2 = V_2  \;  \left(max.\; when \; a > b \right)  \;\;=\;\; \frac {P\;a^2} {L^3} \; \left( a + 3\;b  \right)    }\) 

\(\large{ M_1  \;  \left(max.\; when \;  a < b \right)  \;\;=\;\; \frac {P\;a\;b^2} {L^2}   }\) 

\(\large{ M_2  \;  \left(max. \;when \; a > b \right)  \;\;=\;\; \frac {P\;a^2\;b} {L^2}   }\)

\(\large{ M_a  \;  \left(at \;point \;of \;load \right)  \;\;=\;\; \frac {2\;P\;a^2\;b^2} {L^3}   }\)

\(\large{ M_x  \; \left( x < a \right) \;\;=\;\; R_1\; x - \frac {P\;a\;b^2} {L^2}   }\)

\(\large{ \Delta_{max}  \;  \left(at \; x =  \frac{2\;a\;L}{3\;a \;+\; b} \;when \;a > b  \right)    \;\;=\;\; \frac {4\;P\;a^3\;b^2} {3 \;\lambda\; I \;\left( 3\;a \;+ \;b  \right)^2 }  }\)

\(\large{ M_a  \; \left(at \;point \;of \;load \right)   \;\;=\;\; \frac {2 \;P\;a^3\;b^3} {6\; \lambda\; I \;L^3}   }\)

\(\large{ \Delta_x  \; \left( x < a \right)  \;\;=\;\; \frac {2 \;P\;b^2\;x^2} {12\; \lambda \;I \;L^3} \; \left( 3\;a\;L - 3\;a\;x - b\;x  \right)    }\)

\(\large{ x  \; \left( point\; of\; contraflexure\;between\;supports \right)   \;\;=\;\; \frac{a\;b^2\;P} {L^2\;R_1}   }\)

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 \(\large{\frac{lbf}{in^2}}\) \(\large{Pa}\)
\(\large{ I }\) = moment of inertia \(\large{in^4}\) \(\large{mm^4}\)
\(\large{ R }\) = reaction load at bearing point \(\large{lbf}\) \(\large{N}\)
\(\large{ L }\) = span length under consideration \(\large{in}\) \(\large{mm}\)
\(\large{ a, b }\) = span length under consideration \(\large{in}\) \(\large{mm}\)
\(\large{ P }\) = 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