# Beam Fixed at One End - Concentrated Load at Center

Written by Jerry Ratzlaff on . Posted in Structural

### Beam Fixed at One End - Concentrated Load at Center Formula

$$\large{ R_1 = V_1 = \frac {5P} {16} }$$

$$\large{ R_2 = V_2 = \frac {11P} {16} }$$

$$\large{ M_{max} }$$  (at fixed end)  $$\large{ = \frac {3PL} {16} }$$

$$\large{ M_1 }$$  (at point of load)  $$\large{ = \frac {5PL} {32} }$$

$$\large{ M_x \; }$$  when $$\large{ \left( x < \frac {L}{2} \right) = \frac { 5Px} {16} }$$

$$\large{ M_x \; }$$  when $$\large{ \left( x > \frac {L}{2} \right) = P \left( \frac { L} {2} - \frac { 11x} {16} \right) }$$

$$\large{ \Delta_{max} \; }$$  at  $$\large{ \left( x = L \sqrt { \frac {1}{5} } = .4472L \right) = \frac {PL^3} {48 \lambda I \sqrt {5} } = .009317 \frac { PL^3} { \lambda I} }$$

$$\large{ \Delta_x \; }$$  (at point of load)  $$\large{ = \frac { 7PL^3} {768 \lambda I} }$$

$$\large{ \Delta_x \; }$$  when $$\large{ \left( x < \frac {L}{2} \right) = \frac { Px} {96 \lambda I} \left( 3L^2 - 5x^2 \right) }$$

$$\large{ \Delta_x \; }$$  when $$\large{ \left( x > \frac {L}{2} \right) = \frac { P} {96 \lambda I} \left( x - L \right)^2 \left( 11x - 2L \right) }$$

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