Beam Fixed at One End - Concentrated Load at Any Point
Beam Fixed at One End - Concentrated Load at any point formulas |
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\(\large{ R_1 = V_1 \;\;=\;\; \frac {P\;b^2} {2\;L^3} \; \left( a + 2\;L \right) }\) \(\large{ R_2 = V_2 \;\;=\;\; \frac {P\;a} {2\;L^3} \; \left( 3\;L^2 - a^2 \right) }\) \(\large{ M_1 \; \left(at\; point\; of \;load \right) \;\;=\;\; R_1 \;a }\) \(\large{ M_2 \; \left(at\; fixed \;end \right) \;\;=\;\; \frac {P\;a\;b} {2\;L^2} \; \left( a +L \right) }\) \(\large{ M_x \; \left( x < a \right) \;\;=\;\; R_1\; x }\) \(\large{ M_x \; \left( x > a \right) \;\;=\;\; R_1 \;x - P\; \left( x - a \right) }\) \(\large{ \Delta_{max} \; \left( at \;x = L \; \frac{ L^2\; +\; a^2 }{ 3\;L^2 \;-\; a^2 } \; when\; a < 0.414 \;L \right) \;\;=\;\; \frac {P\;a} {3\; \lambda\; I }\; \frac { \left( L^2 \;-\; a^2 \right) ^3 } { \left( 3\;L^2 \;- \;a^2 \right) ^2 } }\) \(\large{ \Delta_{max} \; \left( at \;x = L \;\sqrt{ \frac{ a }{ 2\;L \;+\; a } } \; when\; a > 0.414 \;L \right) \;\;=\;\; \frac {P\;a\;b^2} {6\; \lambda\; I } \; \sqrt{ \frac { a } { 2\;L \;+ \;a } } }\) \(\large{ \Delta_a \; \left(at\; point\; of\; load \right) \;\;=\;\; \frac { P\;a^3 \;b^2} {12\; \lambda\; I \;L^3}\; \left( 3\;L + b \right) }\) \(\large{ \Delta_x \; \left( x < a \right) \;\;=\;\; \frac { P\;b^2\; x} {12 \;\lambda\; I \;L^3}\; \left( 3\;a\;L^2 - 2\;L\;x^2 - a\;x^2 \right) }\) \(\large{ \Delta_x \; \left( x > a \right) \;\;=\;\; \frac { P\;a} {12\; \lambda\; I \;L^3} \; \left( L - x \right)^2 \; \left( 3\;L^2 \;x - a^2 \;x - 2\;a^2 \;L \right) }\) |
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Symbol | English | Metric |
\(\large{ \Delta }\) = deflection or deformation | \(\large{in}\) | \(\large{m}\) |
\(\large{ x }\) = horizontal distance from reaction to point on beam | \(\large{in}\) | \(\large{m}\) |
\(\large{ a, b }\) = length to point load | \(\large{in}\) | \(\large{m}\) |
\(\large{ M }\) = maximum bending moment | \(\large{lbf-in}\) | \(\large{N-mm}\) |
\(\large{ V }\) = maximum shear force | \(\large{lbf}\) | \(\large{N}\) |
\(\large{ \lambda }\) (Greek symbol lambda) = modulus of elasticity | \(\large{\frac{lbf}{in^2}}\) | \(\large{Pa}\) |
\(\large{ R }\) = reaction load at bearing point | \(\large{lbf}\) | \(\large{N}\) |
\(\large{ I }\) = second moment of area (moment of inertia) | \(\large{in^4}\) | \(\large{mm^4}\) |
\(\large{ L }\) = span length of the bending member | \(\large{in}\) | \(\large{m}\) |
\(\large{ P }\) = total concentrated load | \(\large{lbf}\) | \(\large{N}\) |
diagrams
- 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.
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Tags: Beam Support Equations