Simple Beam - Uniform Load Partially Distributed at One End

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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.

 

Simple Beam - Uniform Load Partially Distributed at One End formulas
\( R_1 \;=\; V_1 \;=\; (w \;a\;/\;2\;L)  \; ( 2\;L - a ) \) \( R_2 \;=\; V_2 \;=\; ( w \;a^2 \;/\;2\;L )\) \( V_x  \;  ( x < a )  \;=\; R_1 - w\;x  \) \( M_{max} \; ( at \; x \;=\; R_1\;/\;w )  \;=\; R_{1}{^2} \;/\; 2\;w  \) \( M_x \; (  x < a )  \;=\;   (R_1 \; x) -  (w\;x^2\;/\;2)  \) \( M_x \; (  x > a )  \;=\;   R_2  \; (  L - x )  \) \( \Delta_x  \; (  x < a )  \;=\;   ( w\; x \;/\; 24\; \lambda \;I \;L)  \; [\; [\; a^2  \; ( 2\;L - a )^2\;] - [\; 2\;a\;x^2 \; (  2\;L - a )\;]  + L\;x^3  \;]    \) \( \Delta_x  \; (  x > a )  \;=\; [\; w\; a^2  \; (  L - x ) \;/\; 24\; \lambda \;I \;L \;] \; ( 4\;x\;L - 2\;x^2 - a^2 )    \)
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	\(in\)	\(mm\)
\( w \) = load per unit length	\(lbf\;/\;in\)	\(N\;/\;m\)
\( a \) = width of UDL	\(in\)	\(mm\)
\( L \) = 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)	\(in^4\)	\(mm^4\)