# Beam Fixed at One End - Concentrated Load at Any Point

Written by Jerry Ratzlaff on . Posted in Structural

## 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