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

Written by Jerry Ratzlaff on . Posted in Structural

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

$$\large{ R_1 = V_1 \; }$$ max. when  $$\large{ \left( a < b \right) = \frac {Pb^2} {L^3} \left( 3a + b \right) }$$

$$\large{ R_2 = V_2 \; }$$ max. when  $$\large{ \left( a > b \right) = \frac {Pa^2} {L^3} \left( a + 3b \right) }$$

$$\large{ M_1 \; }$$ max. when  $$\large{ \left( a < b \right) = \frac {Pab^2} {L^2} }$$

$$\large{ M_2 \; }$$ max. when  $$\large{ \left( a > b \right) = \frac {Pa^2b} {L^2} }$$

$$\large{ M_a \; }$$  (at point of load)  $$\large{ = \frac {2Pa^2b^2} {L^3} }$$

$$\large{ M_x \; }$$ when  $$\large{ \left( x < a \right) = \frac {Pab^2} {L^2} }$$

$$\large{ \Delta_{max} \; }$$ when  $$\large{ \left( a > b \right) }$$   at    $$\large{ \left( x = \frac {2aL}{3a + b} \right) = \frac {2Pa^3b^2} {3 \lambda I \left( 3a + b \right)^2 } }$$

$$\large{ M_a \; }$$  (at point of load)  $$\large{ = \frac {Pa^3b^3} {3 \lambda I L^3} }$$

$$\large{ \Delta_x \; }$$ when  $$\large{ \left( x < a \right) = \frac {Pb^2x^2} {6 \lambda I L^3} \left( 3aL - 3ax - bx \right) }$$

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