Simple Beam - Concentrated Load at Any Point

See Article - Beam Design Formulas

Simple Beam - Concentrated Load at Any Point formulas
$R_1 \;=\; V_1 \; ( max.\; when \;\; a < b ) \;=\; \dfrac{ P \cdot b }{ L }$ $R_2 \;=\; V_2 \; ( max.\; when \;\; a > b ) \;=\; \dfrac{ P \cdot a }{ L }$ $M_{max} \; (at \;point \;of \;load ) \;=\; \dfrac{ P \cdot a \cdot b }{ L }$ $M_x \; ( x < a ) \;=\; \dfrac{ P \cdot b \cdot x }{ L }$ $\Delta_a \; (at \;point \;of \;load ) \;=\; \dfrac{ P \cdot a^2 \cdot b^2 }{ 3 \cdot \lambda \cdot I \cdot L }$ $\Delta_x \; ( x < a ) \;=\; \dfrac{ P \cdot b \cdot x }{ 6\cdot \lambda \cdot I \cdot L } \cdot ( L^2 - b^2 - x^2 )$ $\Delta_{max} \; (at \; x = \sqrt{ \frac{ a\; ( a \;+\; 2\;b ) }{3} } \; when \; a > b ) \;=\; \dfrac{ P\cdot a\cdot b \cdot ( a + 2\cdot b) \cdot \sqrt{ 3\cdot a \cdot ( a + 2\cdot b ) } }{ 27\cdot \lambda \cdot I \cdot L }$
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)	$in^4$	$mm^4$

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.