There are some who believe the bows on the Mary Rose must have been unfinished staves because of their staggering draw-weights.

Here's a brief video of Joseph Gibbs demonstrating how to shoot a 190lb warbow:

http://www.youtube.com/watch?v=7nPNxTcaZPc

Young Joseph makes it look remarkably easy; but he's immensely strong and has near-perfect technique. He's probably the closest to a true medieval English archer we have today.

Eighteenth-century Manchu archers supposedly drew up to 240lb bows to show off.

I have little doubt there was a fair amount of posturing, posing and general showing off amongst English archers, too.

To be honest, beyond about 150lb draw-weight, there's a significant diminishing of returns from a bow, as the weight of the limbs increases.

I doubt this bow would be more performant than a 150 - 160lb bow.

But still... :D

An English longbow features a stacked cross-section - that is, the limb is (proportionately) deep compared to the width. In order to get the draw-weight the bow limbs need to get thicker and heavier. An amount of the limb's energy must be used to accelerate the limbs themselves, not just the arrow.

As the draw-weight increases the greater mass of the limbs means there is less energy to put into the arrow. It's not so much a sudden change in performance; more that there's a performance 'sweet spot' (if you like) which seems to be around 140 - 150lb; although some mathematical models put it closer to 120lb.

So, yes, there will be some increase in distance / power from the arrow; just not the 27% or so you'd get from upping the draw-weight from 150lb to 190lb.

A flat bow has completely different characteristics and is a much more efficient design for a bow. It's not just the modern materials.

Additionally, there are the extreme compressive loads such a bow imposes on the body, meaning the archer is not capable of getting the best out of the bow.

Put these things together and you get less performance out of the bow than you might expect.

And this is born out in practice. Talking to Joseph, he says he can only just draw the bow to 32" because of the compression on his body; and the performance is not better than his 160lb yew bow.

We shall see what the bow is actually capable of later in the year.

I'm not sure, but I am wondering whether you mean efficiency in terms of just arrow speed, or if you are also considering the possibility of using a heavier arrow? It has been many years but I remember reading the real benefit of the longbow was casting heavier arrows, not necessarily faster arrow speed, and that some of the inefficiencies of the limb construction are compensated for, in terms of energy transfer, by using a heavy arrow.

A wood self bow will only cast an arrow up to a certain speed regardless of the draw weight. You can get a 60lb bow to shoot just as fast as a 120lb bow if they are both shooting arrows matched to the bow and purpose. Wood can only recover so fast. The benefit of a heavy bow is it's ability to cast heavy arrows. So for a 150lb bow vs a 190lb bow you would only see a marginal or even negligible gain is speed if they were shooting the same arrow. But if you switch to a heavier arrow in the 190lber it will suddenly become more efficient because the heavy arrow will absorb more of the bows energy. Glennan is right in what he is saying though wood can only do so much...

Benjamin yes it would have to be light enough to be below the peak efficiency of the bow. If the arrow is right for the heavy bow then it would be too heavy for the light bow and it would cast it slugishly. You would see a definite difference in the arrow speeds then. Now if they were both shooting optimum arrows both designed for the particular bow then the arrow speeds from the bows would be almost identical. Now throw in a 45lb horn and sinew backed flight bow and it will cast arrows faster than both of these bows. The energy's will all be different but we are strictly talking about speeds here.

This has ramiifications for the types of bows used on the battlefield. With campaigns like the Crecy and Agincourt, the crown supplied most of the arrrows. Were these arrows all intended for a similar group of bows - say 120-140 lbs - or were different arrows made for a greater variety of bows? If the former then the upper limit of what an English archer could use is irrelevant since he would have to use a lighter bow for the arrows he was supplied with.

Judging by the tests for The Great Warbow, a 150lb bow can profitably shoot a range of arrow weights: 1.9 ounces up to 3.8 ounces. Historical archers also used different arrow weights. For example, Sir John Smythe wrote that some could reach 400 yards with flight arrows.

I'll try again from a different angle. Considering the arrows that were available to an English archer when on campaign, what benefit would be gained from using a 180-pound bow compared to, say a 140-pound one?

A 64 gram arrow from a 180 lb bow will give you approximately 64 m/s
The same arrow from a 150 lb bow will give you approximately 60.4 m/s
In other words, 131 Joules vs. 116.7 Joules and 4.096 N*s vs. 3.8656 N*s

However, if you increase the weight of the arrow, and we know that 20-25 % of the arrows had 8-9 inch fletchings, presumably to stabilize heavy arrows at close range, you get an entire different picture.

A 108 gram arrow from a 180 lb bow will give you at least 55 m/s
The same arrow from a 150 lb bow will give you approximately 52 m/s

That is 146 Joules vs. 163.35 Joules and 5.94 N*s vs. 5.616 N*s

The heavy arrow will have the same distance and/or flat shooting as a lighter arrow from a weaker bow.
The heavy arrow will also have enough kinetic energy to punch through the thinner parts of wrought iron and low carbon steel armour.

Alan WIlliams calculated it like this:
1mm steel with a fracture toughness of 235 kJ/m2 requires 55 Joules.
Wrought iron have a fracture toughness of less than 185 kJ/m2.

W = 185/235 = 0.787

E1 = 55 Joules
t = thickness in millimeters
W = coefficient

E2 = E1*t^1.6*W

E2 = 55*1.5^1.6*0.787
E2 = 82.81 Joules

So here we can see the required energy is way past the threshold for a 1.5 mm plate made of wrought iron, a thickness we see on the sides of many of the helmets we have in museums form this time period. The heavy arrow would have 80.54 Joules left, enough to kill.

If we increase the thickness to 2 mm the result is 131.2 Joules, so these small numbers matter. The difference from 116.7 to 163.35 might seem to be insignificant, but it's not. A heavy arrow at point blank range would still have 32.14 Joules left and will move forward at 24.39 m/s. A lighter arrow, however, would expand all it's kinetic energy. The momentum would be 2.63412 N*s, a significant punch against an arming doublet and possibly mail. A perpendicular strike to the side of the head would possibly kill, and a shot aimed at the face would at least severely wound the man inside the helmet.

Helmets weren't worn against bare skin. An arrow would have to go through a mail coif and significant padding in addition to the side of the helmet.

A cervelliere underneath a great helm might have a coif. A bascinet was not worn with a coif, they had an aventail attached to the vervelles along the lower edge of the helmet. The lining inside the helmet was not especially thick. It's main purpose was to create space between the head and helmet. This was one of the weak spots in armour when english archers shot at men-at-arms from the flanks. It's mentioned in the sources.

"But the French nobility, who had previously advanced in line abreast and had all but come to grip with us, either from fear of the missiles which by their very force pierced the sides and visors of their helmets"
The Gesta Henrici Quinti (c. 1417, Latin), Chapter 13

Arrows shot from the flanks at a tight formation of men-at-arms from a short distance (30-80 meters) will in the majority of the cases strike the top of the body of men, their helmets. Next in line will be the throat and neck, and third will be the shoulders and armpits.

So how does padding create a decent space between skull and helmet without being substantial? An arming doublet isn't any heavier than a winter coat, but helmet padding is a lot thicker and provides significant protection in its own right. Examining the thickness of the steel helmet in isolation will tell us nothing about the potential for an arrow to injure the wearer.

This is the surviving sample we have and it's not substantial relative to padding elsewhere on the body. It doesn't have to be. It's a suspension liner, just like on a modern hard hat.

https://i.pinimg.com/736x/a1/31/af/a131afab0b01ab7e4905417411de4438--medieval-helmets-th-century.jpg
https://d35gqh05wwjv5k.cloudfront.net/media/catalog/product/m/s/msa-fas-trac-liner-hard-hat-suspensions-ff0749-lg.jpg

If the arrow have anywhere from 50 to 80 Joules left after penetrating the side of the helmet, that is the same as a blow from a carpenters hammer.

You are able to swing a hammer at around 15 to 17 m/s if you do your best.
A 16 oz (0.453 kg) will give you 51 Joules with 15 m/s and 65 Joules at 17 m/s
A 20 oz (0.567 kg) will give you 63 Joules with 15 m/s and 82 Joules at 17 m/s

Imagine the hammer-head shaped like a spike with a lozenge-shaped cross-section.
Do you honestly think a person would survive a blow like this to the side of the head, liner or not?

Survival is likely but not guaranteed. Any debilitating effects would be caused by concussion, not penetration.