The Principle of Sloped Armour
The cause for the increased protection of a certain point at a given normal thickness is the increased line-of-sight (LOS) thickness of the armour, which is the thickness along the horizontal plane, along a line describing the oncoming projectile's general direction of travel. For a given thickness of armour plate, a projectile must travel through a greater thickness of armour to penetrate into the vehicle when it is sloped.
The mere fact that the LOS-thickness increases by angling the plate is not however the motive for applying sloped armour in armoured vehicle design. The reason for this is that this increase offers no weight benefit. To maintain a given mass of a vehicle, the area density would have to remain equal and this implies that the LOS-thickness would also have to remain constant while the slope increases, which again implies that the normal thickness decreases. In other words: to avoid increasing the weight of the vehicle, plates have to get proportionally thinner while their slope increases, a process equivalent to shearing the mass.
Sloped armour provides increased protection for armoured fighting vehicles through two primary mechanisms. The most important is based on the fact that to attain a certain protection level a certain volume has to be enclosed by a certain mass of armour and that sloping may reduce the surface to volume ratio and thus allow for either a lesser relative mass for a given volume or more protection for a given weight. If attack were equally likely from all directions, the ideal form would be a sphere; because horizontal attack is in fact to be expected the ideal becomes an oblate spheroid. Angling flat plates or curving cast armour allows designers to approach these ideals. For practical reasons this mechanism is most often applied on the front of the vehicle, where there is sufficient room to slope and much of the armour is concentrated, on the assumption that unidirectional frontal attack is most likely. A simple wedge, such as can be seen in the hull design of the M1 Abrams, is already a good approximation that is often applied.
The second mechanism is that shots hitting sloped armour are more likely to be deflected, ricochet or shatter on impact. Modern weapon and armour technology has significantly reduced this second benefit which initially was the main motive sloped armour was incorporated into vehicle design in the Second World War.
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