Boundary conditions of fixed-fixed beam

A beam fixed at both ends (A at $x=0$ and B at $x=L$) cannot move or rotate at either support. For the elastic curve $y(x)$:

So there are four boundary conditions: $y(0)=0$, $y'(0)=0$, $y(L)=0$, $y'(L)=0$. They are enough to evaluate the four constants of integration of $EI\,\dfrac{d^4y}{dx^4}=w(x)$, and they are the compatibility conditions used in the force method. The end moments $M_A$ and $M_B$ and the shears are unknown reactions (the support moments are non-zero).

In the stiffness method the same statement reads $\theta_A=\theta_B=0$ and $v_A=v_B=0$ (no end displacements and no end rotations), so the end moments are the fixed-end moments of the loading, for example $\mp wL^2/12$ for a UDL.