# Beam Fixed at Both Ends - Concentrated Load at Any Point

Written by Jerry Ratzlaff on . Posted in Structural Engineering

## Beam Fixed at Both Ends - Concentrated Load at Any Point formulas

 $$\large{ R_1 = V_1 \; }$$ max. when  $$\large{ \left( a < b \right) = \frac {P\;b^2} {L^3} \; \left( 3\;a + b \right) }$$ $$\large{ R_2 = V_2 \; }$$ max. when  $$\large{ \left( a > b \right) = \frac {P\;a^2} {L^3} \; \left( a + 3\;b \right) }$$ $$\large{ M_1 \; }$$ max. when  $$\large{ \left( a < b \right) = \frac {P\;a\;b^2} {L^2} }$$ $$\large{ M_2 \; }$$ max. when  $$\large{ \left( a > b \right) = \frac {P\;a^2\;b} {L^2} }$$ $$\large{ M_a \; }$$  (at point of load)  $$\large{ = \frac {2\;P\;a^2\;b^2} {L^3} }$$ $$\large{ M_x \; }$$ when  $$\large{ \left( x < a \right) = \frac {P\;a\;b^2} {L^2} }$$ $$\large{ \Delta_{max} \; }$$ when  $$\large{ \left( a > b \right) }$$   at    $$\large{ \left( x = \frac {2\;a\;L}{3\;a \;+\; b} \right) = \frac {2\;P\;a^3\;b^2} {3 \;\lambda\; I \;\left( 3\;a \;+ \;b \right)^2 } }$$ $$\large{ M_a \; }$$  (at point of load)  $$\large{ = \frac {P\;a^3\;b^3} {3\; \lambda\; I \;L^3} }$$ $$\large{ \Delta_x \; }$$ when  $$\large{ \left( x < a \right) = \frac {P\;b^2\;x^2} {6\; \lambda \;I \;L^3} \; \left( 3\;a\;L - 3\;a\;x - b\;x \right) }$$

### 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