Sloped armour
Encyclopedia
Sloped armour is armour that is neither in a vertical
Vertical direction
In astronomy, geography, geometry and related sciences and contexts, a direction passing by a given point is said to be vertical if it is locally aligned with the gradient of the gravity field, i.e., with the direction of the gravitational force at that point...

nor a horizontal
Horizontal plane
In geometry, physics, astronomy, geography, and related sciences, a plane is said to be horizontal at a given point if it is perpendicular to the gradient of the gravity field at that point— in other words, if apparent gravity makes a plumb bob hang perpendicular to the plane at that point.In...

position. Such "angled" armour is often mounted on tank
Tank
A tank is a tracked, armoured fighting vehicle designed for front-line combat which combines operational mobility, tactical offensive, and defensive capabilities...

s and other armoured fighting vehicle
Armoured fighting vehicle
An armoured fighting vehicle is a combat vehicle, protected by strong armour and armed with weapons. AFVs can be wheeled or tracked....

s (AFVs). Sloping an armour plate makes it harder to penetrate for antitank-weapons, such as armour-piercing shells (kinetic energy penetrator
Kinetic energy penetrator
A kinetic energy penetrator is a type of ammunition which, like a bullet, does not contain explosives and uses kinetic energy to penetrate the target....

s) and rockets, if they take a more or less horizontal path to their target, as is often the case. The better protection is caused by three main effects.

Firstly, a projectile hitting a plate at an angle other than 90° has to move through a greater thickness of armour, compared to hitting the same plate at a right-angle. In the latter case only the plate thickness (the normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

to the surface of the armour) has to be pierced; increasing the armour slope improves, for a given plate thickness, the armour's level of protection at the point of impact by increasing the thickness measured in the horizontal plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

, the angle of attack of the projectile. The protection of an area, instead of just a single point, is indicated by the average horizontal thickness, which is identical to the area density (in this case relative to the horizontal): the relative armour mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

used to protect that area.

If we increase the horizontal thickness by increasing the slope while keeping the plate thickness constant, we need a longer and thus heavier armour plate to protect a certain area. This improvement of protection is simply equivalent to the increase of area density and thus mass, and can offer no weight benefit. Therefore in armoured vehicle design the two other main effects of sloping have been the motive to apply sloped armour.

One of these is a more efficient envelopment of a certain vehicle volume by armour. In general, more rounded forms have a lesser surface relative to their volume. As in an armoured vehicle that surface has to be covered by heavy armour, a more efficient form can lead to a substantial weight reduction or a thicker armour for the same weight. Sloping the armour can lead to a better approximation of an ideal rounded form.

The final effect is that of deflection, deforming and ricochet of a projectile. When it hits a plate under a steep angle, its path might be curved, causing it to move through more armour – or it might bounce off entirely. Also it can be bent, reducing its penetration. However, these effects are strongly dependent on the precise armour materials used and the qualities of the projectile hitting it: sloping might even lead to a better penetration. Shaped charge
Shaped charge
A shaped charge is an explosive charge shaped to focus the effect of the explosive's energy. Various types are used to cut and form metal, to initiate nuclear weapons, to penetrate armor, and in the oil and gas industry...

warheads may fail to penetrate and even detonate when striking armour at a highly oblique angle
Oblique Angle
Oblique angle can refer to:*An angle which is not a multiple of 90°*Another word for "Dutch angle" in cinematography...

.

The sharpest angles are usually seen on the frontal glacis
Glacis
A glacis in military engineering is an artificial slope of earth used in late European fortresses so constructed as to keep any potential assailant under the fire of the defenders until the last possible moment...

plate, both as it is the hull side most likely to be hit and because there is more room to slope in the longitudinal direction of a vehicle.

## 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
Simple shear
In fluid mechanics, simple shear is a special case of deformation where only one component of velocity vectors has a non-zero value:\ V_x=f\ V_y=V_z=0And the gradient of velocity is constant and perpendicular to the velocity itself:...

the mass.

Sloped armour provides increased protection for armoured fighting vehicle
Armoured fighting vehicle
An armoured fighting vehicle is a combat vehicle, protected by strong armour and armed with weapons. AFVs can be wheeled or tracked....

s 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
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

; 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
M1 Abrams
The M1 Abrams is a third-generation main battle tank produced in the United States. It is named after General Creighton Abrams, former Army Chief of Staff and Commander of US military forces in Vietnam from 1968 to 1972. The M1 is a well armed, heavily armored, and highly mobile tank designed for...

, 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
Ricochet
A ricochet is a rebound, bounce or skip off a surface, particularly in the case of a projectile. The possibility of ricochet is one of the reasons for the common firearms safety rule "Never shoot at a flat, hard surface."-Variables:...

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.

### The cosine rule

Even though the increased protection to a point, provided by angling a certain armour plate with a given normal thickness causing an increased line-of-sight (LOS) thickness, is of no consideration in armour vehicle design, it is of great importance when determining the level of protection of a designed vehicle. The LOS-thickness for a vehicle in a horizontal position can be calculated by a simple formula, applying the cosine rule: it is equal to the armour's normal thickness divided by the cosine
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

of the armour's inclination
Inclination
Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...

from perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

ity to the projectile's travel (assumed to be in the horizontal plane) or:

where
• : Line of sight thickness
• : Normal thickness
• : Angle of the sloped armour plate from the vertical

For example, armour sloped sixty degrees back from the vertical presents to a projectile travelling horizontally a line-of-sight thickness twice the armour's normal thickness, as the cosine of 60° is ½. When armour thickness or rolled homogeneous armour
Rolled homogeneous armour
Rolled homogeneous armour is a type of steel which is used to armour vehicles.-Composition:Armoured steel must be hard yet impervious to shock in order to resist high velocity metal projectiles. Steel with these characteristics is produced by processing cast steel billets of appropriate size and...

equivalency (RHAe) values for AFVs are provided without the slope of the armour, the figure provided generally takes into account this effect of the slope, while when the value is in the format of "x units at y degrees", the effects of the slope are not taken into account.

## Deflection

Sloping armour can increase protection by a mechanism such as shattering of a brittle kinetic energy penetrator
Kinetic energy penetrator
A kinetic energy penetrator is a type of ammunition which, like a bullet, does not contain explosives and uses kinetic energy to penetrate the target....

or a deflection of that penetrator away from the surface normal, even though the area density remains constant. These effects are strongest when the projectile has a low absolute weight and is short relative to its width. Armour piercing shells of the Second World War, certainly those of the early years, had these qualities and sloped armour was therefore rather efficient in that period. In the sixties however long-rod penetrators were introduced, projectiles that are both very elongated and very dense in mass. Hitting a sloped thick homogeneous plates such a long-rod penetrator will, after initial penetration into the armour's LOS thickness, bend toward the armour's normal thickness and take a path with a length between the armour's LOS and normal thicknesses. Also the deformed penetrator tends to act as a projectile of a very large diameter and this stretches out the remaining armour, causing it to fail more easily. If these latter effects occur strongly – for modern penetrators this is typically the case for a slope between 55° and 65° – better protection would be provided by vertically mounted armour of the same area density. Another development decreasing the importance of the principle of sloped armour has been the introduction of ceramic armour in the seventies. At any given area density, ceramic armour is also best when mounted more vertically, as maintaining the same area density requires the armour be thinned as it is sloped and the ceramic fractures earlier because of its reduced normal thickness.

Sloped armour can also cause projectiles to ricochet
Ricochet
A ricochet is a rebound, bounce or skip off a surface, particularly in the case of a projectile. The possibility of ricochet is one of the reasons for the common firearms safety rule "Never shoot at a flat, hard surface."-Variables:...

, but this phenomenon is much more complicated and as yet not fully predictable. High rod density, impact velocity, and length-to-diameter ratio are factors that contribute to a high critical ricochet angle (the angle at which ricochet is expected to onset) for a long rod projectile, but different formulae may predict different critical ricochet angles for the same situation.

### Basic physical principles of deflection

The behaviour of a real world projectile, and the armour plate it hits, depends on many effects and mechanisms, involving their material structure and continuum mechanics
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

which are very difficult to predict. Using only a few basic principles will therefore not result in a model that is a good description of the full range of possible outcomes. However, in many conditions most of these factors have only a negligible effect while a few of them dominate the equation. Therefore a very simplified model can be created providing a general idea and understanding of the basic physical principles behind these aspects of sloped armour design.

If the projectile travels very fast, and thus is in a state of hypervelocity
Hypervelocity
The term hypervelocity usually refers to a very high velocity, approximately over 3,000 meters per second . In particular, it refers to velocities so high that the strength of materials upon impact is very small compared to inertial stresses. Thus, even metals behave like fluids under hypervelocity...

, the strength of the armour material becomes negligible – as by the impact both projectile and armour will melt and behave like fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

s – and only its area density is an important factor. In this limiting case the projectile after the hit continues to penetrate until it has stopped transferring its momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

to the target matter. In this ideal case only momentum, area cross section, density and LOS-thickness are relevant. The situation of the penetrating metal jet caused by the explosion of the shaped charge
Shaped charge
A shaped charge is an explosive charge shaped to focus the effect of the explosive's energy. Various types are used to cut and form metal, to initiate nuclear weapons, to penetrate armor, and in the oil and gas industry...

of HEAT
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

ammunition, forms a good approximation of this ideal. Therefore, if the angle is not too extreme, and the projectile is very dense and fast, sloping has little effect and no relevant deflection takes place.

On the other extreme, the more light and slow a projectile is, the more relevant sloping becomes. Typical World War II Armoured Piercing shells were bullet-shaped and had a much lower velocity than a shaped charge jet. An impact would not result in a complete melting of projectile and armour. In this condition the strength of the armour material becomes a relevant factor. If the projectile would be very light and slow, the strength of the armour might even cause the hit to result in just an elastic deformation, the projectile being defeated without damage to the target. Sloping will mean the projectile will have to attain a higher velocity to defeat the armour, because on impact on a sloped armour not all kinetic energy is transferred to the target, the ratio depending on the slope angle. The projectile in a process of elastic collision
Elastic collision
An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter...

deflects at an angle of 2 (where denotes the angle between the armour plate surface and the projectile's initial direction), however the change of direction could be virtually divided into a deceleration part, when the projectile is halted when moving in a direction perpendicular to the plate (and will move along the plate after having been deflected at an angle of about ), and a process of elastic acceleration, when the projectile accelerates out of the plate (velocity along the plate is considered as invariant because of negligible friction). Thus the maximum energy accumulated by the plate can be calculated from the deceleration phase of the collision event.

Under the assumption that only elastic deformation takes place and that the target is solid, while disregarding friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

, it is easy to calculate the proportion of energy absorbed by the target if it is hit by projectile, which, if we also disregard more complex deflection effects, after impact bounces off (elastic case) or slides along (idealised inelastic case) the armour plate.

In this very simple model the portion of the energy projected to the target depends on the angle of slope:

where
• : Energy transferred to the target
• : Incident kinetic energy of projectile
• : Angle of the sloped armour plate from the projectile's initial direction

However, in practice the AP-shells were powerful enough that the forces involved reach the plastic deformation limit and the elasticity of the plate could accumulate only a small part of the energy. In that case the armour plate would yield and much of the energy and force be spent by the deformation. As such this means that approximately half the deflection can be assumed (just rather than 2) and the projectile will groove into the plate before it slides along, rather than bounce off. Plasticity surface friction is also very low in comparison to the plastic deformation energy and can be neglected. This implies that the formula above is principally valid also for the plastic deformation case, but because of the gauge grooved into the plate a larger surface angle should be taken into account.

Not only would this imply that the energy transferred to the target would thus be used to damage it; it would also mean that this energy would be higher because the effective angle in the formula is now higher than the angle of the armour slope. The value of the appropriate real ' which should be substituted cannot be derived from this simple principle and can only be determined by a more sophisticated model or simulation.

On the other hand, that very same deformation will also cause, in combination with the armour plate slope, an effect that diminishes armour penetration. Though the deflection is under conditions of plastic deformation smaller, it will nevertheless change the course of the grooving projectile which again will result in an increase of the angle between the new armour surface and the projectile's initial direction. Thus the projectile has to work itself through more armour and, though in absolute terms thereby more energy could be absorbed by the target, it is more easily defeated, the process ideally ending in a complete ricochet.

## Historical application

The principle itself was well known of old and had been in use on warships (one of the earliest recorded uses of sloped armour was on early Confederate ironclads, such as the CSS Virginia
CSS Virginia
CSS Virginia was the first steam-powered ironclad warship of the Confederate States Navy, built during the first year of the American Civil War; she was constructed as a casemate ironclad using the raised and cut down original lower hull and steam engines of the scuttled . Virginia was one of the...

) and partially implemented on the first French tank, the Schneider CA1
Schneider CA1
The Schneider CA1 was the first French tank. It was inspired by the need to overcome the stalemate of the trench warfare of the Great War.-Caterpillar development:...

in the First World War, but the first tanks to be completely fitted with sloped armour were the French SOMUA S35 and other contemporary French tanks like the Renault R35, which had fully cast hulls and turrets. It was also used to a greater effect on the famous Soviet T-34
T-34
The T-34 was a Soviet medium tank produced from 1940 to 1958. Although its armour and armament were surpassed by later tanks of the era, it has been often credited as the most effective, efficient and influential design of World War II...

battle tank by the Soviet tank design team of the Kharkov Locomotive Factory
Malyshev Factory

, led by Mikhail Koshkin
Mikhail Koshkin
Mikhail Ilyich Koshkin was a Soviet tank designer, chief designer of the famous T-34 medium tank. The T-34 was the most effective and most produced tank of World War II. He started out in life as a candy maker, but then studied engineering...

. It was a technological response to the more effective anti-tank guns being put into service at this time.
The T-34 had profound impact on German WWII tank design. Pre- or early war designs like the Panzer IV
Panzer IV
The Panzerkampfwagen IV , commonly known as the Panzer IV, was a medium tank developed in Nazi Germany in the late 1930s and used extensively during the Second World War. Its ordnance inventory designation was Sd.Kfz...

and Tiger differ clearly from post 1941 vehicles like for example the Panther
Panther tank
Panther is the common name of a medium tank fielded by Nazi Germany in World War II that served from mid-1943 to the end of the European war in 1945. It was intended as a counter to the T-34, and to replace the Panzer III and Panzer IV; while never replacing the latter, it served alongside it as...

, the King Tiger, the Jagdpanzer
Jagdpanzer
Jagdpanzer , German for "hunting tank", is a name given to German self-propelled anti-tank guns.It typically refers to anti-tank variants of existing tank chassis with a well-armoured casemate fixed superstructure, mounting an anti-tank gun with limited traverse in the front, and usually classed by...

and the Hetzer
Hetzer
The Jagdpanzer 38 , later known as Hetzer , was a German light tank destroyer of the Second World War based on a modified Czechoslovakian Panzer 38 chassis. The project was inspired by the Romanian "Mareşal" tank destroyer.The name Hetzer was at the time not commonly used for this vehicle...

, which all had sloped armour. This is especially evident because German tank armour was generally not cast but consisted of welded plates.
Sloped armour became very much the fashion after World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...

, its most pure expression being perhaps the British Chieftain
Chieftain tank
The FV 4201 Chieftain was the main battle tank of the United Kingdom during the 1960s and 1970s. It was one of the most advanced tanks of its era, and at the time of its introduction in 1966 had the most powerful main gun and heaviest armour of any tank in the world...

.
However, the latest main battle tanks use perforated and composite armour
Composite armour
Composite armour is a type of vehicle armour consisting of layers of different material such as metals, plastics, ceramics or air. Most composite armour are lighter than their all-metal equivalent, but instead occupy a larger volume for the same resistance to penetration...

, which attempts to deform and abrade a penetrator rather than deflecting it, as deflecting a long rod penetrator is so difficult. These tanks have a more blocky appearance. Examples include the Leopard 2
Leopard 2
The Leopard 2 is a main battle tank developed by Krauss-Maffei in the early 1970s for the West German Army. The tank first entered service in 1979 and succeeded the earlier Leopard 1 as the main battle tank of the German Army. Various versions have served in the armed forces of Germany and twelve...

and M1 Abrams
M1 Abrams
The M1 Abrams is a third-generation main battle tank produced in the United States. It is named after General Creighton Abrams, former Army Chief of Staff and Commander of US military forces in Vietnam from 1968 to 1972. The M1 is a well armed, heavily armored, and highly mobile tank designed for...

. An exception is the Israeli Merkava
Merkava
The Merkava is a main battle tank used by the Israel Defense Forces. The tank began development in 1974 and was first introduced in 1978. Four main versions of the tank have been deployed. It was first used extensively in the 1982 Lebanon War...

.