Beam Fixed at One End - Concentrated Load at Any Point
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Beam Fixed at One End - Concentrated Load at Center formulas
| \(\large{ R_1 = V_1 = \frac {P\;b^2} {2\;L^3} \; \left( a + 2\;L \right) }\) | |
| \(\large{ R_2 = V_2 = \frac {P\;a} {2\;L^3} \; \left( 3\;L^2 - a^2 \right) }\) | |
| \(\large{ M_1 }\) (at point of load) \(\large{ = R_1 \;a }\) | |
| \(\large{ M_2 }\) (at fixed end) \(\large{ = \frac {P\;a\;b} {2\;L^2} \; \left( a +L \right) }\) | |
| \(\large{ M_x \; }\) when \(\large{ \left( x < a \right) = R_1\; x }\) | |
| b\(\large{ M_x \; }\) when \(\large{ \left( x > a \right) = R_1 \;x - P\; \left( x - a \right) }\) | |
| \(\large{ \Delta_{max} \; }\) when \(\large{ \left( a < .414\;L \right) \; }\) at \(\large{ L \; \frac { L^2\; +\; a^2 } { 3\;L^2 \;-\; a^2 } = \frac {P\;a} {3\; \lambda\; I }\; \frac { \left( L^2 \;-\; a^2 \right) ^3 } { \left( 3\;L^2 \;- \;a^2 \right) ^2 } }\) | |
| \(\large{ \Delta_{max} \; }\) when \(\large{ \left( a > .414\;L \right) \; }\) at \(\large{ L \;\sqrt{ \frac { a } { 2\;L \;+\; a } } = \frac {P\;a\;b^2} {6\; \lambda\; I } \; \sqrt{ \frac { a } { 2\;L \;+ \;a } } }\) | |
| \(\large{ \Delta_a \; }\) (at point of load) \(\large{ = \frac { P\;a^2 \;b^3} {12\; \lambda\; I \;L^3}\; \left( 3\;L + a \right) }\) | |
| \(\large{ \Delta_x \; }\) when \(\large{ \left( x < a \right) = \frac { P\;b^2\; x} {12 \;\lambda\; I \;L^3}\; \left( 3\;a\;L^2 - 2\;L\;x^2 - a\;x^2 \right) }\) | |
| \(\large{ \Delta_x \; }\) when \(\large{ \left( x > a \right) = \frac { P\;a} {12\; \lambda\; I \;L^3} \; \left( L - x \right)^2 \; \left( 3\;L^2 \;x - a^2 \;x - 2\;a^2 \;L \right) }\) |
Where:
\(\large{ I }\) = moment of inertia
\(\large{ L }\) = span length of the bending member
\(\large{ M }\) = maximum bending moment
\(\large{ P }\) = total concentrated load
\(\large{ R }\) = reaction load at bearing point
\(\large{ V }\) = shear force
\(\large{ w }\) = load per unit length
\(\large{ W }\) = total load from a uniform distribution
\(\large{ x }\) = horizontal distance from reaction to point on beam
\(\large{ \lambda }\) (Greek symbol lambda) = modulus of elasticity
\(\large{ \Delta }\) = deflection or deformation