Beam Bending

Equations from classical (Euler-Bernoulli) beam theory.

Symbols used:

$E$	is Young's Modulus for the material (Pa);
$I$	is the second moment of area (m⁴);
$M$	is the bending moment (Nm);
$L$	is the length of the beam (m);
$P$	is the force applied (N);
$Q$	is the shear force in the beam (N);
$w$	is the deflection of the beam (m);
$y$	is the distance from the central axis of the beam (m);

Simply supported beam

Maximum bending moment:

M_{L / 2} = \frac{P L}{4}

Shear force:

Q = \frac{P}{2}

Maximum deflection:

w_{L / 2} = \frac{P L^{3}}{48 E I}

Maximum bending moment:

M_{0} = P L

Shear force:

Q = P

Maximum deflection:

w_{L} = \frac{P L^{3}}{3 E I}

Bending stress:

σ (y) = \frac{M y}{I}

Shear stress:

τ_{a v g} = \frac{P}{A}