Statically Indeterminate Beams

Consider a beam with a fixed support at $A$ and roller support at $B$, with a uniformly distributed load.

From the FBD, there are four unknown equations, but we only have three equations for static equilibrium ($\sum F_x=0,\sum F_y=0,\sum M_A=0$).

The beam is statically indeterminate. Also, the beam deflection equation:

$$EIy={\int }_0^{x}dx{\int }_0^{x}M(x)dx+C_{1}x+C_2$$

has two unknowns but provides three additional equations from the boundary conditions:

• At $x=0$, we have $\theta =0$ and $y=0$
• At $x=L$, $y=0$