Beam Fixed at Both Ends - 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.

 

febe 3B

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

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

\( R_2 = V_2 \; (max.\; when \; a > b ) \;=\; (P\;a^2\;/\;L^3) \; ( a + 3\;b )  \) 

\( M_1 \; (max.\; when \;  a < b ) \;=\; P\;a\;b^2\;/\;L^2 \) 

\( M_2 \; (max. \;when \; a > b ) \;=\; P\;a^2\;b\;/\;L^2 \)

\( M_a \; (at \;point \;of \;load ) \;=\;  2\;P\;a^2\;b^2\;/\;L^3 \)

\( M_x  \; ( x < a ) \;=\;  (R_1 \; x) - (P\;a\;b^2\;/\;L^2) \)

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

\( M_a \; (at \;point \;of \;load ) \;=\; 2 \;P\;a^3\;b^3\;/\;6\; \lambda\; I \;L^3 \)

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

\( x  \; ( point\; of\; contraflexure\;between\;supports ) \;=\; a\;b^2\;P\;/\;L^2\;R_1  \)

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

 

Piping Designer Logo Slide 1

Tags: Beam Support