Simple Beam - Central Point Load and Variable End Moments
Simple Beam - Central Point Load and Variable End Moments formulas |
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\(\large{ R_1 = V_1 \;\;=\;\; \frac { P } { 2 } + \frac { M_1 \;- \;M_2 } { L } }\) \(\large{ R_2 = V_2 \;\;=\;\; \frac { P } { 2 } - \frac { M_1 \;-\; M_2 } { L } }\) \(\large{ M_3 \; \left(at\; center \right) \;\;=\;\; \frac { P\;L } { 4 } - \frac { M_1 \;+\; M_2 } { L } }\) \(\large{ M_x \left( x < \frac{L} {2} \right) \;\;=\;\; \left( \frac { P } { 2 } + \frac { M_1\; -\; M_2 } { L } \right) x - M_1 }\) \(\large{ M_x \left( > \frac{L} {2} \right) \;\;=\;\; \frac {P}{2} \; \left( L - x \right) + \frac { \left( M_1 \;-\; M_2 \right) \;x } { L } \; -\; M_1 }\) \(\large{ \Delta_x \left( x < \frac{L} {2} \right) \;\;=\;\; \frac { P\;x } { 48\; \lambda\; I } \; \left[ 3\;L^2 - 4\;x^2 - \frac { 8\; \left( L\; -\; x \right) } { P\;L } \; \left[ M_1 \left( 2\;L - x \right) + M_2\; \left( L + x \right) \right] \right] }\) \(\large{ x \; \left(first \;point \;of \;contraflexure \right) \;\;=\;\; \frac { 2\;L\; M_1 } { L\;P \;+ \;2\;M_1\; - \;2\;M_2 } }\) \(\large{ x \; \left(second\; point \;of\; contraflexure \right) \;\;=\;\; \frac { L \; \left( L\;P\; - \;2\;M_1 \right) } { L\;P \;- \;2\;M_1 \;+\; 2\;M_2 } }\) |
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Symbol | English | Metric |
\(\large{ \Delta }\) = deflection or deformation | \(\large{in}\) | \(\large{mm}\) |
\(\large{ x }\) = horizontal distance from reaction to point on beam | \(\large{in}\) | \(\large{mm}\) |
\(\large{ M }\) = 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 | \(\large{in}\) | \(\large{mm}\) |
\(\large{ P }\) = 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