How much force is produced by control surfaces? Context:

For some context, I'm a game developer and I'm building a flight sim game. My goal is to have realistic -- not arcade -- physics. The game is in Unity. Unity handles the actual application of the forces -- I just calculate the numbers.

The forces applied to the plane are:


$F_T = \text{Thrust}$ applied to both engines directed forward relative to the plane
$F_L = \text{Lift}$ applied to the center of lift directed perpendicular to velocity
$F_{RR} = \text{Rolling Resistance}$ applied to the center of mass and directed opposite to velocity
$F_W = \text{Weight}$ applied down at the center of mass
$F_{AR} = \text{Air Resistance}$ applied to the center of mass and directed opposite to velocity


The center of mass is positioned slightly forward of the rear landing gear. The center of lift is slightly behind and above the center of mass.

I use a constant thrust directed forward relative to the plane, a constant weight force directed down (world), lift calculated with the below equations directed perpendicular to velocity, and drag directed opposite to velocity. I also have a down force provided by the elevators.

My plane is loosely based on an A320 using a wide variety of figures found online. The mass is $72,000 kg$, wing span is about $35m$, wing area is about $122m$, engine thrust is $110,000N$ each, rolling friction coefficient is $0.04$. The lift coefficient is taken from a table graphed against angle of attack. It looks something like the Cessna graph shown here. The drag coefficient is calculated based on the lift coefficient (formula shown here).

Here is a screenshot of the plane and the forces. Pretent it's an A320 -- it's just a placeholder for now. At the time of the screenshot the plane was travelling at $150kn$.


Issue:

When I apply full throttle to the engines, the plane accelerates as normal. However, upon reaching a typical airliner rotation speed of $150kn$, precisely nothing happens. The plane doesn't lift off the ground until reaching almost $300kn$ where $F_L$ finally overcomes $F_W$. The way I see it, there are two possible causes. First is that my lift maths is wrong, and second is the force provided by the elevators is wrong. I calculated lift as shown here in my other question.

This begs the question, how much force do the control surfaces provide? In particular, the elevators. I know the horizontal stabilizer acts essentially as an upside down wing where the elevators act like flaps/spoilers. I tried using the same lift equation but the force was much to high and the plane would spin uncontrollably on the spot. I also tried calculating (estimating) by hand the torque required to overcome the torque lift of the wing due to the center of lift being behind the center of mass, and then a bit extra to make the torques unbalanced causing rotation. However this was not enough force.

Similarly, how do I calculate how much force is provided by the ailerons, flaps and spoilers -- how do I model this mathematically? Does it just modify the coefficients of lift/drag or do I apply a whole new force?
From the OP, it seems like you are treating the aircraft as a point mass. As a point mass, you don't get the rotational degrees of freedom, so control surface can't be accurately modeled. In the case of takeoff, rotation is achieved via elevator up force, resulting in aircraft pitching up, increasing its angle of attack, resulting in increased the overall lift and achieving the initial vertical acceleration. In summary, you can't model it from first principle without the rotational degrees of freedom.

There are a few things you can do. 



You can continue to assume point mass. You can assume some simplified parametric model for takeoff rotation. That is, the trajectory is a parametric relationship of airspeed, time and elevator. It's not a physical model and it won't be accurate to the magnitude of elevator application, but it may suffice. Once in the air, you can do the same thing with banked turn for aileron applications, and airspeed changes with elevator. Essentially, you are working with a performance model (see performance equations of motions), with artificial relationships for control surfaces.

The above may suffice if all you want to do is GTA style flying (as an aside, its takeoff behavior is terrible). It'll look like a plane flying, but won't feel like a plane flying to anyone who's flown a plane.



You can move up to six degrees of freedom. In that case, you'd need more than just lift, drag and thrust; you'll need moment relationships as well. I recommend you to give Etkins, Dynamics of Flight a read. You can skip directly to Chapter 4. You'd be in very good hands once you've read Chapter 4 and 5.