Beam Fixed at One End - Uniformly Distributed Load

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.

Beam Fixed at One End - Uniformly Distributed Load formulas
\( R_1 \;=\; V_1 \;=\; 3\; w \;L\;/\;8 \) \( R_2 \;=\; V_2 max \;=\; 5\; w\; L\;/\;8 \) \( V_x \;=\; R_1 - w\;x \) \( M_{max} \;=\; w\; L^2\;/\;8 \) \( M_1 \; \left(at\; x = \frac {3\;L}{8} \right) \;=\; 9\;w\;L^2 \;/\;128 \) \( M_x \;=\; (R_1\; x) - (w\; x^2\;/\;2) \) \( \Delta_{max} \;=\; [\; at \; x = \frac{L}{16}\; ( 1 + \sqrt{33} \;) \;or\; x = 0.4215\;L \;] \;=\; w\; L^4\;/\;185\; \lambda\; I \) \( \Delta_x \;=\; (w \;x\;/\;48\; \lambda\; I) \; ( L^3 - 3\;L\;x^2 + 2\;x^3 ) \)
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	\(lbf\;/\;in\)	\(N\;/\;m\)
\( L \) = 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)	\(in^4\)	\(mm^4\)

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