Beam Fixed at One End - Concentrated Load at Any Point

on . Posted in Structural Engineering

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

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

\(\large{ R_2 = V_2  \;\;=\;\; \frac {P\;a} {2\;L^3} \; \left(  3\;L^2 - a^2 \right)   }\) 

\(\large{ M_1  \;  \left(at\; point\; of \;load \right)  \;\;=\;\;  R_1 \;a  }\) 

\(\large{ M_2  \;  \left(at\; fixed \;end \right)  \;\;=\;\;  \frac {P\;a\;b} {2\;L^2}  \; \left(  a +L  \right)   }\)

\(\large{ M_x  \;  \left(  x < a   \right)   \;\;=\;\;   R_1\; x    }\)

\(\large{ M_x  \;  \left(  x > a  \right)   \;\;=\;\;  R_1 \;x -  P\; \left( x - a  \right)  }\)

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

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

\(\large{ \Delta_a \;   \left(at\; point\; of\; load \right)  \;\;=\;\;  \frac { P\;a^3 \;b^2} {12\; \lambda\; I \;L^3}\;  \left( 3\;L + b  \right)     }\)

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

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

Symbol English Metric
\(\large{ \Delta }\) = deflection or deformation \(\large{in}\) \(\large{m}\)
\(\large{ x }\) = horizontal distance from reaction to point on beam \(\large{in}\) \(\large{m}\)
\(\large{ a, b }\) = length to point load \(\large{in}\) \(\large{m}\)
\(\large{ M }\) = 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 \(\large{\frac{lbf}{in^2}}\) \(\large{Pa}\)
\(\large{ R }\) = reaction load at bearing point \(\large{lbf}\) \(\large{N}\)
\(\large{ I }\) = second moment of area (moment of inertia) \(\large{in^4}\) \(\large{mm^4}\)
\(\large{ L }\) = span length of the bending member \(\large{in}\) \(\large{m}\)
\(\large{ P }\) = total concentrated load \(\large{lbf}\) \(\large{N}\)

 

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Tags: Beam Support