Beam Fixed at One End - Concentrated Load at Any Point

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diagram Symbols

  • 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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Beam Fixed at One End - Concentrated Load at any point formulas

\( R_1 \;=\; V_1 \;=\; (P\;b^2\;/\;2\;L^3) \; (  a + 2\;L )  \) 

\( R_2 \;=\; V_2  \;=\; (P\;a\;/\;2\;L^3) \; (  3\;L^2 - a^2 )  \) 

\( M_1 \; (at\; point\; of \;load )  \;=\; R_1 \;a  \) 

\( M_2 \; (at\; fixed \;end ) \;=\; (P\;a\;b\;/\;2\;L^2)  \; ( a +L )  \)

\( M_x \; ( x < a ) \;=\; R_1\; x  \)

\( M_x  \; ( x > a ) \;=\; R_1 \;x - [\; P\; ( x - a ) \;] \)

\( \Delta_{max}  \; ( at \;x = L \;  \frac{ L^2 \;+\; a^2 }{ 3\;L^2 \;-\; a^2 } \; when\; a < 0.414 \;L )  \;=\; (P\;a\;/\;3\; \lambda\; I ) \;  \frac{ ( L^2 \;-\; a^2 ) ^3 }{ ( 3\;L^2 \;- \;a^2 ) ^2 }   \)

\( \Delta_{max} \; ( at \;x = L \;\sqrt{ \frac{ a }{ 2\;L \;+\; a } } \; when\; a > 0.414 \;L )  \;=\; (P\;a\;b^2\;/\;6\; \lambda\; I )  \; \sqrt{  a \;/\; 2\;L + a }  \)

\( \Delta_a \; (at\; point\; of\; load ) \;=\; ( P\;a^3 \;b^2\;/\;12\; \lambda\; I \;L^3) \; ( 3\;L + b )   \)

\( \Delta_x  \; ( x < a ) \;=\; ( P\;b^2\; x\;/\;12 \;\lambda\; I \;L^3) \; ( 3\;a\;L^2 - 2\;L\;x^2 - a\;x^2 )  \)

\( \Delta_x  \; ( x > a ) \;=\; ( P\;a\;/\;12\; \lambda\; I \;L^3) \; ( L - x )^2 \; ( 3\;L^2 \;x - a^2 \;x   -  2\;a^2 \;L )   \)

Symbol English Metric
\( \Delta \) = deflection or deformation \(in\) \(m\)
\( x \) = horizontal distance from reaction to point on beam \(in\) \(m\)
\( a, b \) = length to point load \(in\) \(m\)
\( M \) = maximum bending moment \(lbf-in\) \(N-mm\)
\( V \) = maximum shear force \(lbf\) \(N\)
\( \lambda \)   (Greek symbol lambda) = modulus of elasticity \(lbf\;/\;in^2\) \(Pa\)
\( R \) = reaction load at bearing point \(lbf\) \(N\)
\( I \) = second moment of area (moment of inertia) \(in^4\) \(mm^4\)
\( L \) = span length of the bending member \(in\) \(m\)
\( P \) = total concentrated load \(lbf\) \(N\)

 

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Tags: Beam Support