Why does maneuvering speed vary with weight? Aircraft certified by the FAA under 14 CFR Part 23 have many V speeds, one of them which manufacturers are required to determine and publish in the POH is maneuvering speed, or Va.


  In aviation, the maneuvering speed (Va) of an aircraft is an airspeed limitation selected by the designer of the aircraft. At speeds close to, and faster than, the maneuvering speed, full deflection of any flight control surface should not be attempted because of the risk of damage to the aircraft structure.


This speed varies with the weight of the aircraft. Why does this speed vary with weight? Additionally since manufacturers often only provide that value at max gross weight, how does one figure out what that Va airspeed is when not at max gross weight?
At the normal stall speeds given by $V_s$, the load on the aircraft is 1g, and the lift is equal to weight. i.e $L = W$.

In case of maneuvering, the load factor is greater than one, and we have $L = nW$, with $n$ being the load factor.

We have,  $L = nW = \frac{1}{2}  C_L  \rho  V^2 S$.

This gives maneuvering speed , $V_a$ = $V_s \sqrt{n}$ 

It can also be written as $V_s$ = $\sqrt{\frac{2 n W}{\rho C_{Lmax} S}}$

At maximum weight, this gives, 
$V_A$ = $\sqrt{\frac{2 n W_{max}}{\rho C_Lmax S}}$

For other weights, we have,
$V_a$ = $\sqrt{\frac{2 n W_{a}}{\rho C_Lmax S}}$

For the same load factor, we then have,
$V_{a} = V_{A} \sqrt{\frac{W_{a}}{W_{max}}}$,
where
$V_{a}$ is the maneuvering speed at weight $W_{a}$ and $V_{A}$ is the maneuvering speed at maximum weight $W_{max}$

As the maneuvering speed depends on the load factor and the stall speed, it depends on the weight of the aircraft (which decides the stall speed). It is basically a structural limit.

Another thing to note is that the aircraft can sustain structural failure even below maneuvering speed when multiple large control inputs are given.