Simple Beam - Concentrated Load at Any Point
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Simple Beam - Concentrated Load at Any Point formulas
| \(\large{ R_1 = V_1 }\) max. when \(\large{ \left( a < b \right) = \frac{P\;b}{L} }\) | |
| \(\large{ R_2 = V_2 }\) max. when \(\large{ \left( a > b \right) = \frac{ P\;a}{L} }\) | |
| \(\large{ M_{max} \; }\) (at point of load) \(\large{ = \frac{ P\;a\;b }{ L } }\) | |
| \(\large{ M_x \; }\) when \(\large{ \left( x < b \right) = \frac{ P\;b\;x }{ L } }\) | |
| \(\large{ \Delta_a \; }\) (at point of load) \(\large{ = \frac{ P\;a^2\;b^2 }{ 3\; \lambda \;I \;L } }\) | |
| \(\large{ \Delta_x \; }\) when \(\large{ \left( x < a \right) = \frac{ P\;b\;x }{ 6\; \lambda \;I \;L } \; \left( L^2 - b^2 - x^2 \right) }\) | |
| \(\large{ \Delta_x \; }\) when \(\large{ \left( x > a \right) = \frac{ P\;a\; \left( L \;-\; x \right) } { 6 \; \lambda \;I \;L } \; \left( 2\;L\;x - x^2 - a^2 \right) }\) | |
| \(\large{ \Delta_{max} \; }\) at \(\large{ \left( x = \sqrt{ \frac{ a\; \left( a \;+\; 2\;b \right) }{3} } \right) }\) when \(\large{ \left( a > b \right) = \frac{ P\;a\;b \; \left( a \;+\; 2\;b \right) \; \sqrt{ 3\;a \; \left( a \; 2\;b \right) } } { 27\; \lambda \;I \;L } }\) |
Where:
\(\large{ \Delta }\) = deflection or deformation
\(\large{ a, b }\) = distance to point load
\(\large{ x }\) = horizontal distance from reaction to point on beam
\(\large{ M }\) = maximum bending moment
\(\large{ V }\) = maximum shear force
\(\large{ I }\) = moment of inertia
\(\large{ \lambda }\) (Greek symbol lambda) = modulus of elasticity
\(\large{ R }\) = reaction load at bearing point
\(\large{ L }\) = span length of the bending member
\(\large{ P }\) = total concentrated load