Simple Beam - Load Increasing Uniformly to Center

See Article - Beam Design Formulas

Simple Beam - Load Increasing Uniformly to Center formulas
$R = V_{max} \;=\; \dfrac{ W }{ 2 }$ $V_x \; [ \; x < (L\;/\;2) \;] \;=\; \dfrac{ W }{ 2 \cdot L^2 } \cdot ( L^2 - 4 \cdot x^2 )$ $M_{max} \; (at \;center) \;=\; \dfrac{ W \cdot L }{ 6 }$ $M_x \; [\; x < (L\;/\;2) \;] \;=\; W \cdot x \cdot \left(\dfrac{1}{2} - \dfrac{ 2 \cdot x^2 }{ 3 \cdot L^2 } \right)$ $\Delta_{max} \; (at \;center) \;=\; \dfrac{ W \cdot L^3 }{ 60 \cdot \lambda \cdot I }$ $\Delta_x \; [\; x < (L\;/\;2) \;] \;=\; \dfrac{ W \cdot x }{ 480 \cdot \lambda \cdot I \cdot L^2 } \cdot ( 5 \cdot L^2 - 4 \cdot x^2 )^2$
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	$in$	$mm$
$W$ = total load or $w\;L\;/\;2$	$lbf$	$N$
$w$ = highest load per unit length of UIL	$lbf\;/\;in$	$N\;/\;m$
$L$ = 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.

Simple Beam - Load Increasing Uniformly to Center

Simple Beam - Load Increasing Uniformly to Center formulas

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