Simple Beam - Concentrated Load at Any Point

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• Tags: Beam Support

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.

 

Simple Beam - Concentrated Load at Any Point formulas
\( R_1 \;=\; V_1 \; ( max.\; when \;\; a < b ) \;=\; P\;b\;/\;L      \) \( R_2 \;=\; V_2 \; ( max.\; when \;\; a > b ) \;=\; P\;a\;/\;L     \) \( M_{max} \;  (at \;point \;of \;load )  \;=\; P\;a\;b \;/\; L   \) \( M_x \; (  x < a )  \;=\; P\;b\;x \;/\; L   \) \( \Delta_a  \; (at \;point \;of \;load )  \;=\; P\;a^2\;b^2 \;/\; 3\; \lambda \;I  \;L   \) \(  \Delta_x  \; (  x < a )  \;=\; ( P\;b\;x \;/\; 6\; \lambda \;I  \;L ) \; (  L^2 - b^2 - x^2 )    \) \( \Delta_{max} \; (at \; x = \sqrt{  \frac{ a\; ( a \;+\; 2\;b )  }{3}  } \; when \;  a > b )  \;=\;   P\;a\;b \; (  a + 2\;b) \; \sqrt{ 3\;a \; ( a + 2\;b ) }   \;/\; 27\; \lambda \;I  \;L  \)
Symbol	English	Metric
\( R \) = reaction load at bearing point	\(lbf\)	\(N\)
\( V \) = maximum shear force	\(lbf\)	\(N\)
\( M \) = maximum bending moment	\(lbf - in\)	\(N - mm\)
\( \Delta \) = deflection or deformation	\(in\)	\(mm\)
\( a, b \) = distance to point load	\(in\)	\(mm\)
\( P \) = total concentrated load	\(lbf\)	\(N\)
\( L \) = span length of the bending member	\(in\)	\(mm\)
\( x \) = horizontal distance from reaction to point on beam	\(in\)	\(mm\)
\( \lambda  \)   (Greek symbol lambda) = modulus of elasticity	\(lbf\;/\;in^2\)	\(Pa\)
\( I \) = second moment of area (moment of inertia)	\(in^4\)	\(mm^4\)