Simple Beam - Two Unequal Point Loads Unequally Spaced

on 24 April 2018. Posted in Structural Engineering

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 - Two Unequal Point Loads Unequally Spaced Formulas
\( R_1 \;=\; V_1 \;=\; [\; P_1 \; ( L \;- \;a ) \;] + (P_2\; b \;/\; L) \) \( R_2 \;=\; V_2 \;=\; P_1 \;a +[\; P_2 \; ( L - b ) \;] \;/\; L \) \( V_x \; [\; a < x < ( L - b ) \;] \;=\; R_1 - P_1 \) \( M_1 \; (max.\; when\; R_1 < P_1 ) \;=\; R_1\; a \) \( M_2 \; (max.\; when\; R_2 < P_2 ) \;=\; R_2\; b \) \( M_x \; (max.\; when\; x < a ) \;=\; R_1\; x \) \( M_x \; [\; max.\; when\; a < x < \;( L - b ) \;] \;=\; R_1\; x - [\;P_1\; ( x - a ) \;] \)
Symbol	English	Metric
\( \Delta \) = deflection or deformation	\(in\)	\(mm\)
\( x \) = horizontal distance from reaction to point on beam	\(in\)	\(mm\)
\( a, b \) = length to point load	\(in\)	\(mm\)
\( M \) = maximum bending moment	\(lbf-in\)	\(N-mm\)
\( V \) = maximum shear force	\(lbf\)	\(N\)
\( \lambda \) (Greek symbol lambda) = modulus of elasticity	\(lbf\;/\;in^2\)	\(Pa\)
\( I \) = second moment of area (moment of inertia)	\(in^4\)	\(mm^4\)
\( R \) = reaction load at bearing point	\(lbf\)	\(N\)
\( L \) = span length of the bending member	\(in\)	\(mm\)
\( P \) = total concentrated load	\(lbf\)	\(N\)

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